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Equivalence Of Qubit-environment Entanglement And Discord Generation Via Pure Dephasing Interactions And The Consequences Thereof

We find that when a qubit initialized in a pure state experiences pure dephasing due to interaction with an environment, separable qubit-environment states generated during the evolution also have zero quantum discord with respect to the environment. What follows is that the set of separable states which can be reached during the evolution has zero volume and hence, such effects as sudden death of qubit-environment entanglement are very unlikely. In case of the discord with respect to the qubit, a vast majority of separable states qubit-environment is discordant, but in specific situations zero-discord states are possible. This is conceptually important since there is a connection between the discordance with respect to a given subsystem and the possibility of describing the evolution of this subsystem using completely positive maps. Finally, we use the formalism to find an exemplary evolution of an entangled state of two qubits that is completely positive, occurs solely due to interaction of only one of the qubits with its environment (so one could guess that it corresponds to a local operation, since it is local in a physical sense), but which nevertheless causes the enhancement of entanglement between the qubits. While this simply means that the considered evolution is completely positive, but does not belong to LOCC, it shows how much caution has to be exercised when identifying evolution channels that belong to that class.

There is little or no ambiguity in the study of quantum correlations for pure states, as long as the potentially correlated parties are well defined and completely distinguishable. Such correlations can be fully described by entanglement and none of the pure separable states (states with no entanglement) exhibit any type of behaviors which can be associated with quantum correlations. The two main characteristics of pure entangled states are that (a) it is not possible to prepare an entangled state via local operations and classical communication (criterion of preparation) and (b) it is not possible to fully determine the state of either of the entangled subsystems by local measurements on this subsystem and classical communication alone without disturbing it (criterion of measurement). It is this second characteristic which results in the property of entangled states that appropriately chosen measurements on one subsystem determine the state of the other subsystem, which underlie many applications of entangled states such as quantum algorithms Deutsch and Jozsa (1992); Ekert and Jozsa (1996) or quantum teleportation Bennett et al. (1993); Popescu (1994). In case of mixed states, the situation becomes more complicated. Mixed state entanglement Werner (1989); Plenio and Virmani (2007); Horodecki et al. (2009) is defined using the criterion of preparation of the previous paragraph, meaning that a state of two subsystems is entangled, if and only if it cannot be written as a statistical mixture of product states of the two subsystems - equivalently, if it cannot be prepared by local operations and classical communication (LOCC). All entangled states satisfy the criterion of measurement as well, but there exist separable (not entangled) states which satisfy the criterion of measurement for one or both subsystems (while obviously not satisfying the criterion of preparation). A measure of quantum correlations which is based on the criterion of measurement is called the quantum discord Ollivier and Zurek (2002); Henderson and Vedral (2001); Modi (2014). The set of discordant states is larger than the set of entangled states and in fact, it includes the set of entangled states, so although there do not exist entangled states with zero discord, there do exist separable discordant states Modi (2014). It is important to note that there is an inherent asymmetry in the definition of the quantum discord with respect to the potentially correlated subsystems, since the criterion of measurement can be fulfilled for one of the subsystems while it is not fulfilled for the other. Entanglement and the quantum discord differ significantly when it comes to the qualitative and quantitative features of their evolution. This is partly because the set of zero-discord states has zero volume Ferraro et al. (2010), while the volume of separable states is finite. The characteristic property for entanglement evolutions, the possibility for it to undergo sudden death Rajagopal and Rendell (2001); Życzkowski et al. (2001); Yu and Eberly (2006) (the complete disappearance of entanglement at a certain finite time, while the continuous decoherence of the entangled subsystems is not complete), which is sometimes followed by sudden birth (the reemergence of entanglement after a state is separable for a finite amount of time), is a direct result of the geometry of separable states. Since they have finite volume, there exist separable states which are completely surrounded by other separable states and may not be approached by means of a continuous evolution otherwise than from another separable state. Any evolution which approaches such a state has to display sudden death of entanglement. Since the volume of zero-discord states is not finite, any zero-discord state can be reached directly from a discordant state and sudden-death-type behavior in the discord evolution is much less likely. On the other hand, entanglement is symmetric with respect to both entangled subsystems, while the discord does not have to be symmetric with respect to the systems under study Modi (2014) (it is fairly common that the measurement of one of the correlated subsystems yields information about its state with less damage to the state itself than the other - some of the geometric measures are artificially symmetrized Dakić et al. (2010), or that a state is discordant only with respect to one of the subsystems). Another characteristic property of discord evolutions is the occurrence of points of indifferentiability (for which the time-dependence of the discord function is continuous, but not smooth). This quality of the discord should not be dismissed as an artifact of the mathematical properties of the geometric measures used to quantify the discord, since it has been observed in quantum discord curves calculated using the original discord definition Roszak and Cywiński (2015); Mazzola et al. (2010) and in case of states which do have parity symmetry Roszak et al. (2013); Mazurek et al. (2014a). In the following, we show that for a class of quantum systems composed of a qubit (Q) initialized in a pure state, and an arbitrarily large environment (E), for which the evolution of the qubit alone is of pure dephasing type (the occupations of the qubit are not disturbed due to the interaction with the environment), strong quantum correlations described by entanglement and weaker quantum correlations described by the quantum discord with respect to the environment are operationally the same. This means that the class of separable qubit-environment (Q-E) states that can be reached during such joint evolution, is the same as the class of reachable zero-discord states from the point of view of E. Hence, all separable Q-E states obtained during the evolution are automatically one-sidedly zero-discord states, so they posses the specifically non-discordant quality of being a zero-volume set of states. This suppresses the possibility for such Q-E evolutions to display characteristic behaviors for entanglement, such as its sudden death. We also study the property of discordance with respect to the qubit subsystem, which turns out to have qualitatively different properties than the discord discussed in the previous paragraph. In fact most of the separable qubit-environment evolutions are discordant in this sense, and only in very specific situations are zero-discord points possible during the evolution. This means that in terms of weak quantum correlations, two types of evolutions are possible. After achieving such an understanding of Q-E discord generation, in the last part of the paper we use these results to shed light on issues related to role of system-environment quantum discord in dynamics of open systems. We generalize our results on Q-E evolution to the case of a class of entangled two-qubit states subjected to pure dephasing due to E𝐸Eitalic_E. Then we construct an example of a system in which only one qubit interacts with E𝐸Eitalic_E, so that the resulting decoherence is local, and use the evolution due to the interaction with E𝐸Eitalic_E to find an example of a state ρ^SEminsuperscriptsubscript^𝜌SEmin\hat\rho_\mathrmSE^\mathrmminover^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT roman_SE end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT with zero discord between the qubits and E𝐸Eitalic_E (zero with respect to the two-qubit subsystem), for which entanglement between the qubits is minimal, but subsequent evolution leads to an increase of interqubit entanglement. It is known that the lack of system-environment discord with respect to the system implies that the system’s evolution starting from such a state may be described using completely positive (CP) maps Rodríguez-Rosario et al. (2008), so the qubits’ evolution starting from ρ^SEminsuperscriptsubscript^𝜌SEmin\hat\rho_\mathrmSE^\mathrmminover^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT roman_SE end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT state of the whole system is CP, but, despite the fact that only one of the qubits is interacting with its local environment, it is does not belong to the LOCC class. The paper is organized as follows. In Sec. II we introduce the notion of the quantum discord further, including the original definition of the discord, and focusing on the differences and similarities between separable states and zero-discord states. In Sec. II.2 we state the criteria for zero-discord states following Ref. Huang et al. (2011), which we will later use to obtain the main results of this paper. The class of systems under study is described in Sec. III, and the separability criterion specific for this system is the topic of Sec. III.1. The equivalence of the class of separable states and zero-discord states with respect to the environment for the class of systems under study is shown in Sec. IV. The properties of the quantum discord with respect to the qubit are discussed in Sec. V, while the an extension of the results to entangled two-qubit states is the topic of Sec. VI. In Sec. VII we discuss the implications of our results for the understanding of open quantum systems dynamics.

The quantification of the quantum discord is in general complicated, even in comparison with the stronger measure of quantum correlations, entanglement. In case of entangled mixed states, some means of quantification of the amount of mixed state entanglement have been available for two decades. This includes entanglement witnesses, a multitude of two-qubit measures Wootters (1998); Vedral and Plenio (1998); Horodecki (2001); Plenio and Virmani (2007), which allow for the calculation of two-qubit or qubit-qutrit entanglement directly from the density matrix. Contrarily, the first geometric measures of the quantum discord (measures based on the calculation of the smallest distance between a given mixed quantum state and the set of zero-discord states; the distance measures used vary) Dakić et al. (2010); Luo and Fu (2010); Nakano et al. (2013); Paula et al. (2013); Spehner and Orszag (2013, 2014) and methods of estimating their upper and lower bounds are about five years old Dakić et al. (2010); Miranowicz et al. (2012); Tufarelli et al. (2013). Note, that only the methods for the calculation of the limits on the quantum discord are direct ones, allowing for calculation from the density matrix of the studied system - the calculation of precise values of discord still requires minimization over the set of all zero-discord states.

II.1 Separable states vs. zero-discord states

The class of separable states can be generally represented mathematically as the set of states, which can be written in the form,

ρABsep=∑αpαρAα⊗ρBα.superscriptsubscript𝜌ABsepsubscript𝛼tensor-productsubscript𝑝𝛼superscriptsubscript𝜌𝐴𝛼superscriptsubscript𝜌𝐵𝛼\rho_\mathrmAB^\mathrmsep=\sum_\alphap_\alpha\rho_A^\alpha% \otimes\rho_B^\alpha.italic_ρ start_POSTSUBSCRIPT roman_AB end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_sep end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT . (1)

Here, the density matrices on the left side of the tensor product correspond to subsystem A𝐴Aitalic_A and the ones on the right side correspond to subsystem B𝐵Bitalic_B. The only constraint is on the parameters of the decomposition pαsubscript𝑝𝛼p_\alphaitalic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, which have to be probabilities, 0<pα<10subscript𝑝𝛼10<0 <italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT <1 and ∑αpα=1subscript𝛼subscript𝑝𝛼1\sum_\alphap_\alpha=1∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = 1. Hence there are no constraints on the states ρAαsuperscriptsubscript𝜌𝐴𝛼\rho_A^\alphaitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT and ρBαsuperscriptsubscript𝜌𝐵𝛼\rho_B^\alphaitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT, which do not have to be pure (although there does exist an equivalent definition using projectors) or form an orthonormal basis for subsystem A𝐴Aitalic_A or B𝐵Bitalic_B. The lack of the orthonormality requirement is in fact the reason, why checking for entanglement between two subsystems is in general complicated. The class of zero-discord states can be represented in an analogous way Modi (2014). The only difference is that there is an additional constraint on states of one or both subsystems. If the system state has zero discord with respect to subsystem A𝐴Aitalic_A (B𝐵Bitalic_B), then there must exist a decomposition of the joint state of systems A𝐴Aitalic_A and B𝐵Bitalic_B such, that the density matrices ρA(B)αsuperscriptsubscript𝜌𝐴𝐵𝛼\rho_A(B)^\alphaitalic_ρ start_POSTSUBSCRIPT italic_A ( italic_B ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT in eq. (1) can be written as projectors,

ρA(B)α=|aα⟩⟨aα|,superscriptsubscript𝜌𝐴𝐵𝛼ketsubscript𝑎𝛼brasubscript𝑎𝛼\rho_A(B)^\alpha=|a_\alpha\rangle\langle a_\alpha|,italic_ρ start_POSTSUBSCRIPT italic_A ( italic_B ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT = | italic_a start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⟩ ⟨ italic_a start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | , (2)

where ketsubscript𝑎𝛼\a_\alpha\rangle\ forms an ortonormal set in the subspace of subsystem A𝐴Aitalic_A (B𝐵Bitalic_B). If both the zero-discord criteria for subsystems A𝐴Aitalic_A and B𝐵Bitalic_B are fulfilled then the state is completely zero-discordant (there is no discord with respect to either subsystem). Note, that the set of zero-discord states is obviously a subset of separable states regardless of whether it is discordant with respect to one or both subsystems.

II.2 The criteria for zero-discord states

Contrarily to the case of entanglement, for which even the determination, if a mixed-state density matrix is entangled or not, is complicated for bipartite entanglement of larger systems (for which at least one is not a qubit or qutrit), the determination, if the quantum discord is present in a system is fairly straightforward even in case of two arbitrarily large systems Dakić et al. (2010); Huang et al. (2011). In the following we used the criterion of Ref. Huang et al. (2011), which is more suitable for the class of systems under study (both criteria allow to check, if a state is discordant with respect to one of the potentially correlated systems at a time). The criterion introduced in the paper states that a bipartite state (where the parties are of arbitrary dimension N𝑁Nitalic_N and M𝑀Mitalic_M) has zero quantum discord with respect to the system M𝑀Mitalic_M, if and only if all blocks of its density matrix, after the bipartite (NM)×(NM)𝑁𝑀𝑁𝑀(NM)\times(NM)( italic_N italic_M ) × ( italic_N italic_M ) density matrix is partitioned into N2superscript𝑁2N^2italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT matrices of dimension M×M𝑀𝑀M\times Mitalic_M × italic_M (the particulars of the partition are described below), are normal matrices and commute with each other. Here, the partition is performed starting from a bipartite density matrix

σ^=∑kq∑nmPkqnm|k⟩⟨q|⊗|n⟩⟨m|,^𝜎subscript𝑘𝑞subscript𝑛𝑚tensor-productsuperscriptsubscript𝑃𝑘𝑞𝑛𝑚ket𝑘bra𝑞ket𝑛bra𝑚\hat\sigma=\sum_kq\sum_nmP_kq^nm|k\rangle\langle q|\otimes|n\rangle% \langle m|,over^ start_ARG italic_σ end_ARG = ∑ start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n italic_m end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_m end_POSTSUPERSCRIPT | italic_k ⟩ ⟨ italic_q | ⊗ | italic_n ⟩ ⟨ italic_m | , (3)

where the indices (and states labeled by them) k,q𝑘𝑞k,qitalic_k , italic_q correspond to one of the subsystems (say, the one of dimension N𝑁Nitalic_N), while the indices (and states) n,m𝑛𝑚n,mitalic_n , italic_m correspond to the other subsystem (of dimension M𝑀Mitalic_M). Note, that the parameters Pkqnmsuperscriptsubscript𝑃𝑘𝑞𝑛𝑚P_kq^nmitalic_P start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_m end_POSTSUPERSCRIPT must fulfill a number of conditions for the matrix σ𝜎\sigmaitalic_σ to be a density matrix, but we will not concern ourself with those here, since in the following a state obtained via a unitary evolution from an initial product state of two density matrices will be considered, which can obviously always be described by a density matrix). The partition of this density matrix into N2superscript𝑁2N^2italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blocks of dimension M×M𝑀𝑀M\times Mitalic_M × italic_M is done as follows

σ^kq=⟨k|σ^|q⟩,subscript^𝜎𝑘𝑞quantum-operator-product𝑘^𝜎𝑞\hat\sigma_kq=\langle k|\hat\sigma|q\rangle,over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT = ⟨ italic_k | over^ start_ARG italic_σ end_ARG | italic_q ⟩ , (4)

for all k,q𝑘𝑞k,qitalic_k , italic_q. The criterion of normality means that for all k,q𝑘𝑞k,qitalic_k , italic_q

[σ^kq,σ^kq†]=0,subscript^𝜎𝑘𝑞subscriptsuperscript^𝜎†𝑘𝑞0[\hat\sigma_kq,\hat\sigma^\dagger_kq]=0,[ over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT , over^ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT ] = 0 , (5)

while the commutation criterion means that for all k,q𝑘𝑞k,qitalic_k , italic_q and k′,q′superscript𝑘′superscript𝑞′k^\prime,q^\primeitalic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT

[σ^kq,σ^k′q′]=0.subscript^𝜎𝑘𝑞subscript^𝜎superscript𝑘′superscript𝑞′0[\hat\sigma_kq,\hat\sigma_k^\primeq^\prime]=0.[ over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_k italic_q end_POSTSUBSCRIPT , over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ] = 0 . (6)

Both criteria are fulfilled, if and only if the state has zero discord with respect to subsystem M𝑀Mitalic_M (meaning that the state of subsystem of size M𝑀Mitalic_M can be fully determined by local measurements performed on this subsystem and classical communication alone without disturbing it).

III The class of systems under study

We study a class of systems consisting of a qubit and an environment which, when only the qubit is a system of interest, always lead to pure dephasing evolutions of the qubit (the occupations of the qubit remain unchanged). In general, the Hamiltonian of such a system can be written as

H^^𝐻\displaystyle\hatHover^ start_ARG italic_H end_ARG =\displaystyle== H^Q+H^E+|0⟩⟨0|⊗V0^+|1⟩⟨1|⊗V1^.subscript^𝐻Qsubscript^𝐻Etensor-productket0bra0^subscript𝑉0tensor-productket1bra1^subscript𝑉1\displaystyle\hatH_\mathrmQ+\hatH_\mathrmE+|0\rangle\langle 0|% \otimes\hatV_0+|1\rangle\langle 1|\otimes\hatV_1\,\,.over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT roman_Q end_POSTSUBSCRIPT + over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT roman_E end_POSTSUBSCRIPT + | 0 ⟩ ⟨ 0 | ⊗ over^ start_ARG italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG + | 1 ⟩ ⟨ 1 | ⊗ over^ start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG . (7)

The first term of the Hamiltonian describes the qubit and is given by H^Q=∑i=0,1εi|i⟩⟨i|subscript^𝐻Qsubscript𝑖01subscript𝜀𝑖ket𝑖bra𝑖\hatH_\mathrmQ=\sum_i=0,1\varepsilon_i|i\rangle\langle i|over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT roman_Q end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 0 , 1 end_POSTSUBSCRIPT italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ ⟨ italic_i |, the second describes the environment and is arbitrary, while the remaining terms describe the qubit-environment interaction with the qubit states written on the left side of each term and the environmental operators V0^^subscript𝑉0\hatV_0over^ start_ARG italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG and V1^^subscript𝑉1\hatV_1over^ start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG are also arbitrary. The full qubit-environment evolution operator U^(t)=exp(-iH^t)^𝑈𝑡𝑖^𝐻𝑡\hatU(t)\!=\!\exp(-i\hatHt)over^ start_ARG italic_U end_ARG ( italic_t ) = roman_exp ( - italic_i over^ start_ARG italic_H end_ARG italic_t ) resulting from the Hamiltonian (7) can be written as

U^(t)=|0⟩⟨0|⊗w^0(t)+|1⟩⟨1|⊗w^1(t),^𝑈𝑡tensor-productket0bra0subscript^𝑤0𝑡tensor-productket1bra1subscript^𝑤1𝑡\hatU(t)=|0\rangle\langle 0|\otimes\hatw_0(t)+|1\rangle\langle 1|\otimes% \hatw_1(t)\,\,,over^ start_ARG italic_U end_ARG ( italic_t ) = | 0 ⟩ ⟨ 0 | ⊗ over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) + | 1 ⟩ ⟨ 1 | ⊗ over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , (8)

where we have defined the operators

w^i(t)=exp(-iH^it),subscript^𝑤𝑖𝑡𝑖subscript^𝐻𝑖𝑡\hatw_i(t)=\exp(-i\hatH_it)\,\,,over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = roman_exp ( - italic_i over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t ) , (9)

with i=0,1𝑖01i=0,1italic_i = 0 , 1, and H^i=H^E+V^isubscript^𝐻𝑖subscript^𝐻Esubscript^𝑉𝑖\hatH_i\!=\!\hatH_\mathrmE+\hatV_iover^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT roman_E end_POSTSUBSCRIPT + over^ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. We study the joint state of a qubit and an environment which are initially in a product state σ^(0)=ρ^Q(0)⊗R^(0)^𝜎0tensor-productsubscript^𝜌Q0^𝑅0\hat\sigma(0)=\hat\rho_\mathrmQ(0)\otimes\hatR(0)over^ start_ARG italic_σ end_ARG ( 0 ) = over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT roman_Q end_POSTSUBSCRIPT ( 0 ) ⊗ over^ start_ARG italic_R end_ARG ( 0 ) and evolve according to the operator (8). The qubit is initially in a pure state |ψ⟩=α|0⟩+β|1⟩ket𝜓𝛼ket0𝛽ket1|\psi\rangle=\alpha|0\rangle+\beta|1\rangle| italic_ψ ⟩ = italic_α | 0 ⟩ + italic_β | 1 ⟩, with |α|2+|β|2=1superscript𝛼2superscript𝛽21|\alpha|^2+|\beta|^2=1| italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 and α,β≠0𝛼𝛽0\alpha,\beta

eq 0italic_α , italic_β ≠ 0 (a superposition is needed for dephasing to occur as well as entanglement and discord generation), so the density matrix ρ^Q(0)=|ψ⟩⟨ψ|subscript^𝜌Q0ket𝜓bra𝜓\hat\rho_\mathrmQ(0)=|\psi\rangle\langle\psi|over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT roman_Q end_POSTSUBSCRIPT ( 0 ) = | italic_ψ ⟩ ⟨ italic_ψ |. We impose no restrictions on the initial density matrix of the environment and write it in terms of its eigenstates, R^(0)=∑ncn|n⟩⟨n|^𝑅0subscript𝑛subscript𝑐𝑛ket𝑛bra𝑛\hatR(0)=\sum_nc_n|n\rangle\langle n|over^ start_ARG italic_R end_ARG ( 0 ) = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n ⟩ ⟨ italic_n |. The time-evolved qubit-environment density matrix takes the form

σ^(t)=(|α|2∑ncn|n0(t)⟩⟨n0(t)|αβ*∑ncn|n0(t)⟩⟨n!(t)|α*β∑ncn|n1(t)⟩⟨n0(t)||β|2∑ncn|n1(t)⟩⟨n1(t)|),^𝜎𝑡absentsuperscript𝛼2subscript𝑛subscript𝑐𝑛ketsubscript𝑛0𝑡brasubscript𝑛0𝑡𝛼superscript𝛽subscript𝑛subscript𝑐𝑛ketsubscript𝑛0𝑡brasubscript𝑛𝑡superscript𝛼𝛽subscript𝑛subscript𝑐𝑛ketsubscript𝑛1𝑡brasubscript𝑛0𝑡superscript𝛽2subscript𝑛subscript𝑐𝑛ketsubscript𝑛1𝑡brasubscript𝑛1𝑡\beginarray[]l\hat\sigma(t)=\\ \left(\beginarray[]cc|\alpha|^2\sum_nc_n|n_0(t)\rangle\langle n_0% (t)|&\alpha\beta^*\sum_nc_n|n_0(t)\rangle\langle n_!(t)|\\ \alpha^*\beta\sum_nc_n|n_1(t)\rangle\langle n_0(t)|&|\beta|^2\sum_% nc_n|n_1(t)\rangle\langle n_1(t)|\endarray\right),\endarraystart_ARRAY start_ROW start_CELL over^ start_ARG italic_σ end_ARG ( italic_t ) = end_CELL end_ROW start_ROW start_CELL ( start_ARRAY start_ROW start_CELL | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) | end_CELL start_CELL italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUBSCRIPT ! end_POSTSUBSCRIPT ( italic_t ) | end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) | end_CELL start_CELL | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) | end_CELL end_ROW end_ARRAY ) , end_CELL end_ROW end_ARRAY (10)

where the matrix is written in the basis of the eigenstates of the free qubit Hamiltonian, and |ni(t)⟩=w^i(t)|n⟩ketsubscript𝑛𝑖𝑡subscript^𝑤𝑖𝑡ket𝑛|n_i(t)\rangle=\hatw_i(t)|n\rangle| italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ⟩ = over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) | italic_n ⟩ with w^i(t)subscript^𝑤𝑖𝑡\hatw_i(t)over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) given by Eq. (12). Note, that the density matrix written as in eq. (10) is already decomposed into four blocks (with respect to the states of the qubit). Each element of the matrix (10) written as it is in block form, is the type of block that allows to check for the presence of the discord with respect to the environment following the criteria of the previous section.

III.1 The zero-entanglement criterion for pure dephasing

The problem of separability for the class of systems described in the previous section has been solved in Ref. Roszak and Cywiński (2015). A joint state of the qubit and its environment (10) which is generated by the evolution operator given by eq. (8) (which itself comes from the Hamiltonian (7)) is separable at time t𝑡titalic_t, if and only if

[R^(0),w^(t)]=0,^𝑅0^𝑤𝑡0[\hatR(0),\hatw(t)]\!=\!0\,\,,[ over^ start_ARG italic_R end_ARG ( 0 ) , over^ start_ARG italic_w end_ARG ( italic_t ) ] = 0 , (11)

where

w^(t)=exp(iH^0t)exp(-iH^1t)=w^0(t)w^1(t).^𝑤𝑡𝑖subscript^𝐻0𝑡𝑖subscript^𝐻1𝑡subscript^𝑤0𝑡subscript^𝑤1𝑡\hatw(t)=\exp(i\hatH_0t)\exp(-i\hatH_1t)=\hatw_0(t)\hatw_1(t% )\,\,.over^ start_ARG italic_w end_ARG ( italic_t ) = roman_exp ( italic_i over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t ) roman_exp ( - italic_i over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t ) = over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) . (12)

This criterion can be equivalently stated as

w^0†(t)R^(0)w^0(t)=w^1†(t)R^(0)w^1(t).subscriptsuperscript^𝑤†0𝑡^𝑅0subscript^𝑤0𝑡subscriptsuperscript^𝑤†1𝑡^𝑅0subscript^𝑤1𝑡\hatw^\dagger_0(t)\hatR(0)\hatw_0(t)=\hatw^\dagger_1(t)\hat% R(0)\hatw_1(t).over^ start_ARG italic_w end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) = over^ start_ARG italic_w end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) . (13)

In this form, the criterion is particularly easy to compare with the results of applying the zero-discord criteria to the system under study.

IV Equivalence of separability and zero-discord with respect to the environment

Using the criterion introduced in Sec. II.2 to check, if the system-environment density matrix given by eq. (10) is discordant with respect to the environment at time t𝑡titalic_t is uncomplicated. We have the density matrix of the qubit and environment written in such a way that it is already divided into the aforementioned blocks, meaning that each of the four matrices in the subspace of the environment σ^ij(t)=⟨i|σ^(t)|j⟩subscript^𝜎𝑖𝑗𝑡quantum-operator-product𝑖^𝜎𝑡𝑗\hat\sigma_ij(t)=\langle i|\hat\sigma(t)|j\rangleover^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = ⟨ italic_i | over^ start_ARG italic_σ end_ARG ( italic_t ) | italic_j ⟩ is a separate block (|i⟩,|j⟩=|0⟩,|1⟩formulae-sequenceket𝑖ket𝑗ket0ket1|i\rangle,|j\rangle=|0\rangle,|1\rangle| italic_i ⟩ , | italic_j ⟩ = | 0 ⟩ , | 1 ⟩ are qubit states). It is straightforward to show that σ^00(t)subscript^𝜎00𝑡\hat\sigma_00(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) and σ^11(t)subscript^𝜎11𝑡\hat\sigma_11(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) are always normal, since σ^00(t)=σ^00†(t)subscript^𝜎00𝑡superscriptsubscript^𝜎00†𝑡\hat\sigma_00(t)=\hat\sigma_00^\dagger(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) = over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) and σ^11(t)=σ^11†(t)subscript^𝜎11𝑡superscriptsubscript^𝜎11†𝑡\hat\sigma_11(t)=\hat\sigma_11^\dagger(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) = over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ), so [σ^ii(t),σ^ii†(t)]=0subscript^𝜎𝑖𝑖𝑡superscriptsubscript^𝜎𝑖𝑖†𝑡0[\hat\sigma_ii(t),\hat\sigma_ii^\dagger(t)]=0[ over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ] = 0. For the blocks corresponding to the diagonal elements of the qubit density matrix, σ^01(t)=σ^10†(t)subscript^𝜎01𝑡superscriptsubscript^𝜎10†𝑡\hat\sigma_01(t)=\hat\sigma_10^\dagger(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) = over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ), the part of the normality criterion for zero-discord states, is not so easy to check and in fact these matrices are not always normal. We will return to this in the next paragraph, since this normality criterion is equivalent to one of the commutation criteria.

Before we continue, let us denote

R^ij(t)=w^i(t)R^(0)w^j†(t).subscript^𝑅𝑖𝑗𝑡subscript^𝑤𝑖𝑡^𝑅0superscriptsubscript^𝑤𝑗†𝑡\hatR_ij(t)=\hatw_i(t)\hatR(0)\hatw_j^\dagger(t).over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) . (14)

The criterion that all σ^ij(t)subscript^𝜎𝑖𝑗𝑡\hat\sigma_ij(t)over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) must commute, obviously reduces to the criterion that all R^ij(t)subscript^𝑅𝑖𝑗𝑡\hatR_ij(t)over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) must commute. For the density matrix (10), this leads to the following commutation conditions

[R^00(t),R^11(t)]subscript^𝑅00𝑡subscript^𝑅11𝑡\displaystyle\left[\hatR_00(t),\hatR_11(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== 0,0\displaystyle 0,0 , (15a)

[R^00(t),R^01(t)]subscript^𝑅00𝑡subscript^𝑅01𝑡\displaystyle\left[\hatR_00(t),\hatR_01(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== R^01(t)(R^11(t)-R^00(t))=0,subscript^𝑅01𝑡subscript^𝑅11𝑡subscript^𝑅00𝑡0\displaystyle\hatR_01(t)\left(\hatR_11(t)-\hatR_00(t)\right)=0,over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) ( over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) ) = 0 , (15b)

[R^00(t),R^10(t)]subscript^𝑅00𝑡subscript^𝑅10𝑡\displaystyle\left[\hatR_00(t),\hatR_10(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== (R^00(t)-R^11(t))R^10(t)=0,subscript^𝑅00𝑡subscript^𝑅11𝑡subscript^𝑅10𝑡0\displaystyle\left(\hatR_00(t)-\hatR_11(t)\right)\hatR_10(t)=0,( over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) ) over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ) = 0 , (15c)

[R^11(t),R^01(t)]subscript^𝑅11𝑡subscript^𝑅01𝑡\displaystyle\left[\hatR_11(t),\hatR_01(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== (R^11(t)-R^00(t))R^01(t)=0,subscript^𝑅11𝑡subscript^𝑅00𝑡subscript^𝑅01𝑡0\displaystyle\left(\hatR_11(t)-\hatR_00(t)\right)\hatR_01(t)=0,( over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) ) over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) = 0 , (15d)

[R^11(t),R^10(t)]subscript^𝑅11𝑡subscript^𝑅10𝑡\displaystyle\left[\hatR_11(t),\hatR_10(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== R^10(t)(R^00(t)-R^11(t))=0,subscript^𝑅10𝑡subscript^𝑅00𝑡subscript^𝑅11𝑡0\displaystyle\hatR_10(t)\left(\hatR_00(t)-\hatR_11(t)\right)=0,over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ) ( over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) ) = 0 , (15e)

[R^01(t),R^10(t)]subscript^𝑅01𝑡subscript^𝑅10𝑡\displaystyle\left[\hatR_01(t),\hatR_10(t)\right][ over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) , over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ) ] =\displaystyle== R^00(t)2-R^11(t)2=0,subscript^𝑅00superscript𝑡2subscript^𝑅11superscript𝑡20\displaystyle\hatR_00(t)^2-\hatR_11(t)^2=0,over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 , (15f)

since

R^00(t)R^01(t)subscript^𝑅00𝑡subscript^𝑅01𝑡\displaystyle\hatR_00(t)\hatR_01(t)over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) =\displaystyle== w^0(t)R^(0)w^0†(t)w^0(t)R^(0)w^1†(t)subscript^𝑤0𝑡^𝑅0superscriptsubscript^𝑤0†𝑡subscript^𝑤0𝑡^𝑅0superscriptsubscript^𝑤1†𝑡\displaystyle\hatw_0(t)\hatR(0)\hatw_0^\dagger(t)\hatw_0(t)% \hatR(0)\hatw_1^\dagger(t)over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) (16)

=\displaystyle== w^0(t)R^(0)w^1†(t)w^1(t)R^(0)w^1†(t)subscript^𝑤0𝑡^𝑅0superscriptsubscript^𝑤1†𝑡subscript^𝑤1𝑡^𝑅0superscriptsubscript^𝑤1†𝑡\displaystyle\hatw_0(t)\hatR(0)\hatw_1^\dagger(t)\hatw_1(t)% \hatR(0)\hatw_1^\dagger(t)over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t )

=\displaystyle== R^01(t)R^11(t),subscript^𝑅01𝑡subscript^𝑅11𝑡\displaystyle\hatR_01(t)\hatR_11(t),over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) ,

etc.. Note, that the criteria of normality for matrices σ^01subscript^𝜎01\hat\sigma_01over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT and σ^10subscript^𝜎10\hat\sigma_10over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT are always equivalent to commutation criterion (15f), since R^01†(t)=R^10(t)superscriptsubscript^𝑅01†𝑡subscript^𝑅10𝑡\hatR_01^\dagger(t)=\hatR_10(t)over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) = over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_t ). Furthermore, all of the conditions (IV) are always satisfied when the state (10) is separable, since then R^00(t)-R^11(t)=0subscript^𝑅00𝑡subscript^𝑅11𝑡0\hatR_00(t)-\hatR_11(t)=0over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ( italic_t ) - over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_t ) = 0 as shown in eq. (13). This means, quite surprisingly, that the class of separable (zero-entanglement) qubit-environment states which can be obtained during a pure-dephasing evolution is equivalent to the class of states with zero discord with respect to the environment for this type of evolution, since entangled states are always also discordant Modi (2014). Consequently, for this type of evolution there is little or no difference between entanglement and the environmental quantum discord. The discord may still display qualities resulting from points of indifferentiability, but entanglement evolution is unlikely to display its most characteristic feature, namely sudden death of entanglement, because for the class of systems under study not only the set of zero-discord states has zero volume, but also the set of separable states has zero volume (even “one-sided” zero-discord states possess the zero-volume quality).

V Separability and zero-discord with respect to the qubit

The relationship between separability and the lack of discord with respect to the qubit subspace is much more ambiguous. Since local unitary operations cannot change the amount of discord in a system Hassan and Joag (2013) and specifically no local operations on the environment can change, whether a state is discordant with respect to the qubit or not, as is evident from the definition of “one-sided” zero-discord states in Sec. II.1, nor do they change the purity of the reduced density matrix of the qubit, let us work with the qubit-environment density matrix transformed by an unitary operation on the environment (as we did in Roszak and Cywiński (2015)), namely

σ~(t)=w^0†(t)σ^(t)w^0(t).~𝜎𝑡superscriptsubscript^𝑤0†𝑡^𝜎𝑡subscript^𝑤0𝑡\tilde\sigma(t)=\hatw_0^\dagger(t)\hat\sigma(t)\hatw_0(t).over~ start_ARG italic_σ end_ARG ( italic_t ) = over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_σ end_ARG ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) . (17)

Since separability at time t𝑡titalic_t indicates that there exists a (time-dependent) basis ketsuperscript𝑛′𝑡\ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ in which both the initial density matrix of the environment R^(0)^𝑅0\hatR(0)over^ start_ARG italic_R end_ARG ( 0 ) and the operator w^0†(t)w^1(t)superscriptsubscript^𝑤0†𝑡subscript^𝑤1𝑡\hatw_0^\dagger(t)\hatw_1(t)over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) are diagonal, they can be written as R^(0)=∑ncn′(t)|n′(t)⟩⟨n′(t)|^𝑅0subscript𝑛superscriptsubscript𝑐𝑛′𝑡ketsuperscript𝑛′𝑡brasuperscript𝑛′𝑡\hatR(0)=\sum_nc_n^\prime(t)|n^\prime(t)\rangle\langle n^\prime(t)|over^ start_ARG italic_R end_ARG ( 0 ) = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | and w^0†(t)w^1(t)=∑nexp(-iϕn(t))|n′(t)⟩⟨n′(t)|superscriptsubscript^𝑤0†𝑡subscript^𝑤1𝑡subscript𝑛𝑖subscriptitalic-ϕ𝑛𝑡ketsuperscript𝑛′𝑡brasuperscript𝑛′𝑡\hatw_0^\dagger(t)\hatw_1(t)=\sum_n\exp(-i\phi_n(t))|n^\prime(% t)\rangle\langle n^\prime(t)|over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_exp ( - italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ) | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) |. Consequently the density matrix (17) can be written as

σ~(t)=∑ncn′(t)(|α|2αβ*eiϕn(t)α*βe-iϕn(t)|β|2)⊗|n′(t)⟩⟨n′(t)|.~𝜎𝑡subscript𝑛tensor-productsuperscriptsubscript𝑐𝑛′𝑡superscript𝛼2𝛼superscript𝛽superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝛼𝛽superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝛽2ketsuperscript𝑛′𝑡brasuperscript𝑛′𝑡\tilde\sigma(t)=\sum_nc_n^\prime(t)\left(\beginarray[]cc|\alpha|^% 2&\alpha\beta^*e^i\phi_n(t)\\ \alpha^*\beta e^-i\phi_n(t)&|\beta|^2\endarray\right)\otimes|n^% \prime(t)\rangle\langle n^\prime(t)|.over~ start_ARG italic_σ end_ARG ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ( start_ARRAY start_ROW start_CELL | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_CELL start_CELL | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY ) ⊗ | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | . (18)

If the separable system density matrix corresponding to time t𝑡titalic_t, eq. (18), is partitioned into N2superscript𝑁2N^2italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (where N𝑁Nitalic_N is the dimension of the environment) 2×2222\times 22 × 2 matrices in terms of the eigenbasis ketsuperscript𝑛′𝑡\ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ , then the diagonal matrices (the ones corresponding to |k⟩=|q⟩=|n′(t)⟩ket𝑘ket𝑞ketsuperscript𝑛′𝑡|k\rangle=|q\rangle=|n^\prime(t)\rangle| italic_k ⟩ = | italic_q ⟩ = | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ in eq. (4)) are of the form

⟨n′(t)|σ~(t)|n′(t)⟩=cn′(t)(|α|2αβ*e-iϕn(t)α*βeiϕn(t)|β|2),quantum-operator-productsuperscript𝑛′𝑡~𝜎𝑡superscript𝑛′𝑡superscriptsubscript𝑐𝑛′𝑡superscript𝛼2𝛼superscript𝛽superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝛼𝛽superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝛽2\langle n^\prime(t)|\tilde\sigma(t)|n^\prime(t)\rangle=c_n^\prime(t)% \left(\beginarray[]cc|\alpha|^2&\alpha\beta^*e^-i\phi_n(t)\\ \alpha^*\beta e^i\phi_n(t)&|\beta|^2\endarray\right),⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | over~ start_ARG italic_σ end_ARG ( italic_t ) | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ = italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ( start_ARRAY start_ROW start_CELL | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT end_CELL start_CELL | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY ) , (19)

while the off-diagonal matrices are equal to zero. Hence, all of the matrices of this partition fulfill the normality requirement for the discord, while the commutation requirements are reduced to

[⟨n′(t)|σ~(t)|n′(t)⟩,⟨m′(t)|σ~(t)|m′(t)⟩]=cn′(t)cm′(t)(2i|α|2|β|2sin[ϕn(t)-ϕm(t)]-αβ*(|α|2-|β|2)(eiϕn(t)-eiϕm(t))αβ*(|α|2-|β|2)(e-iϕn(t)-e-iϕm(t))2i|α|2|β|2sin[ϕn(t)-ϕm(t)])=0quantum-operator-productsuperscript𝑛′𝑡~𝜎𝑡superscript𝑛′𝑡quantum-operator-productsuperscript𝑚′𝑡~𝜎𝑡superscript𝑚′𝑡absentsuperscriptsubscript𝑐𝑛′𝑡superscriptsubscript𝑐𝑚′𝑡2𝑖superscript𝛼2superscript𝛽2subscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ𝑚𝑡𝛼superscript𝛽superscript𝛼2superscript𝛽2superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝑒𝑖subscriptitalic-ϕ𝑚𝑡𝛼superscript𝛽superscript𝛼2superscript𝛽2superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡superscript𝑒𝑖subscriptitalic-ϕ𝑚𝑡2𝑖superscript𝛼2superscript𝛽2subscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ𝑚𝑡0\beginarray[]l\left[\langle n^\prime(t)|\tilde\sigma(t)|n^\prime(t)% \rangle,\langle m^\prime(t)|\tilde\sigma(t)|m^\prime(t)\rangle\right]\\ =c_n^\prime(t)c_m^\prime(t)\left(\beginarray[]cc2i|\alpha|^2|% \beta|^2\sin\left[\phi_n(t)-\phi_m(t)\right]&-\alpha\beta^*(|\alpha|^% 2-|\beta|^2)\left(e^i\phi_n(t)-e^i\phi_m(t)\right)\\ \alpha\beta^*(|\alpha|^2-|\beta|^2)\left(e^-i\phi_n(t)-e^-i\phi_m% (t)\right)&2i|\alpha|^2|\beta|^2\sin\left[\phi_n(t)-\phi_m(t)\right]% \endarray\right)=0\endarraystart_ARRAY start_ROW start_CELL [ ⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | over~ start_ARG italic_σ end_ARG ( italic_t ) | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ , ⟨ italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | over~ start_ARG italic_σ end_ARG ( italic_t ) | italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ ] end_CELL end_ROW start_ROW start_CELL = italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ( start_ARRAY start_ROW start_CELL 2 italic_i | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_sin [ italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ] end_CELL start_CELL - italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) end_CELL start_CELL 2 italic_i | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_sin [ italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ] end_CELL end_ROW end_ARRAY ) = 0 end_CELL end_ROW end_ARRAY (20)

for all n𝑛nitalic_n and m𝑚mitalic_m. Firstly, let us identify the trivial solutions that lead to no pure dephasing of the qubit (when no correlations of any type between a qubit and its environment are formed and the qubit-environment state remains a product). These include an initial state of the qubit which is not a superposition, i.e. α=0𝛼0\alpha=0italic_α = 0 or β=0𝛽0\beta=0italic_β = 0, and the situation when for all n𝑛nitalic_n and m𝑚mitalic_m for which cn(t)′≠0subscript𝑐𝑛superscript𝑡′0c_n(t)^\prime

eq 0italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ 0 and cm′(t)≠0superscriptsubscript𝑐𝑚′𝑡0c_m^\prime(t)

eq 0italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ≠ 0, ϕn(t)=ϕm(t)subscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ𝑚𝑡\phi_n(t)=\phi_m(t)italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) mod 2π2𝜋2\pi2 italic_π. Furthermore, the condition stemming from the off-diagonal elements of the matrices (20) that needs to be taken into account when |α|≠|β|𝛼𝛽|\alpha|

eq|\beta|| italic_α | ≠ | italic_β |, implies that for all n𝑛nitalic_n and m𝑚mitalic_m for which cn′(t)≠0superscriptsubscript𝑐𝑛′𝑡0c_n^\prime(t)

eq 0italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ≠ 0 and cm′(t)≠0superscriptsubscript𝑐𝑚′𝑡0c_m^\prime(t)

eq 0italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ≠ 0 we must have exp(iϕn(t))=exp(iϕm(t))𝑖subscriptitalic-ϕ𝑛𝑡𝑖subscriptitalic-ϕ𝑚𝑡\exp(i\phi_n(t))=\exp(i\phi_m(t))roman_exp ( italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ) = roman_exp ( italic_i italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ); if this condition is fulfilled, it is easy to see that the qubit does not undergo pure dephasing (and only a phase shift between the elements of its superposition). Hence pure dephasing is always accompanied by discord generation with respect to the qubit, as long as the initial state of the qubit is not an equal superposition state. The only case when the qubit can experience pure dephasing due to an interaction with the environment which is not accompanied by discord with respect to the qubit state is, if it is initially in an equal superposition state, |α|=|β|=1/2𝛼𝛽12|\alpha|=|\beta|=1/\sqrt2| italic_α | = | italic_β | = 1 / square-root start_ARG 2 end_ARG. Then the set of commutation conditions for zero discord are reduced to sin[ϕn(t)-ϕm(t)]=0subscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ𝑚𝑡0\sin\left[\phi_n(t)-\phi_m(t)\right]=0roman_sin [ italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ] = 0 for all n𝑛nitalic_n and m𝑚mitalic_m. To see that such a situation is possible let us study the simplest example, with the dimension of the environment N=2𝑁2N=2italic_N = 2. Imagine that at a certain time t𝑡titalic_t, the exponential factors are exp(iϕ0(t))=1𝑖subscriptitalic-ϕ0𝑡1\exp(i\phi_0(t))=1roman_exp ( italic_i italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ) = 1 and exp(iϕ1(t))=-1𝑖subscriptitalic-ϕ1𝑡1\exp(i\phi_1(t))=-1roman_exp ( italic_i italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) ) = - 1, respectively. The level of coherence of the qubit (the amplitude of the off-diagonal element of its reduced density matrix) is governed by the function |c0′(t)-c1′(t)|superscriptsubscript𝑐0′𝑡superscriptsubscript𝑐1′𝑡|c_0^\prime(t)-c_1^\prime(t)|| italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | and the qubit is fully coherent for |c0′(t)-c1′(t)|=1superscriptsubscript𝑐0′𝑡superscriptsubscript𝑐1′𝑡1|c_0^\prime(t)-c_1^\prime(t)|=1| italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | = 1 and in a completely mixed state for |c0′(t)-c1′(t)|=0superscriptsubscript𝑐0′𝑡superscriptsubscript𝑐1′𝑡0|c_0^\prime(t)-c_1^\prime(t)|=0| italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | = 0. Obviously, regardless of the values of c0′(t)superscriptsubscript𝑐0′𝑡c_0^\prime(t)italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) and c1′(t)superscriptsubscript𝑐1′𝑡c_1^\prime(t)italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ), there is no discord of any kind between the qubit and the environment, but only for c0′(t)=0superscriptsubscript𝑐0′𝑡0c_0^\prime(t)=0italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) = 0 or c1′(t)=0superscriptsubscript𝑐1′𝑡0c_1^\prime(t)=0italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) = 0 is the qubit fully coherent, while the coherence of the qubit depends only on the mixedness of the initial density matrix of the environment. In the case of a single qubit environment it is not possible for a pure dephasing evolution to have zero qubit discord for all times, because this would require the initial environmental state to be pure, and for such a state the dephasing of the qubit is equivalent to creation of qubit-environment entanglement Żurek (2003).

In general, in order to have zero discord with respect to the qubit initialized in an equal superposition state which undergoes non-entangling evolution the following conditions have to be met at time t𝑡titalic_t. For all n𝑛nitalic_n and m𝑚mitalic_m that correspond to nonzero coefficients cn′(t)superscriptsubscript𝑐𝑛′𝑡c_n^\prime(t)italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) and cm′(t)superscriptsubscript𝑐𝑚′𝑡c_m^\prime(t)italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ), the phase differences fulfill ϕn(t)-ϕm(t)=pπsubscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ𝑚𝑡𝑝𝜋\phi_n(t)-\phi_m(t)=p\piitalic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_p italic_π, where p𝑝pitalic_p is an integer. For an environment with N>2𝑁2N>2italic_N >2 this implies a rigid condition on all phases corresponding to non-zero coefficients cn′(t)superscriptsubscript𝑐𝑛′𝑡c_n^\prime(t)italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ), which must all have the form ϕn(t)=ϕ0(t)+qπsubscriptitalic-ϕ𝑛𝑡subscriptitalic-ϕ0𝑡𝑞𝜋\phi_n(t)=\phi_0(t)+q\piitalic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) + italic_q italic_π, again with integer q𝑞qitalic_q. The situation is peculiar, since decoherence is almost always accompanied by a buildup of the quantum discord with respect to the qubit. The one (very notable) exception allows for discord-less dephased states, when the qubit state is of high symmetry and this symmetry is mirrored by the state of the environment. In this case any level of decoherence is possible, so it is not only a minor aberration, but a different type of decoherence process in terms of correlations with an environemnt.

VI Enhancement of two-qubit entanglement under local decoherence

It has been recently shown Rodríguez-Rosario et al. (2008) that lack of the quantum discord with respect to one of the subsystems in an initial state implies complete positivity of the reduced dynamics of this subsystem. Hence, in case of evolution of the type discussed here, in the case of zero Q-E entanglement being generated, and thus zero discord with respect to the environment being generated, the evolution of the environment can be described by CP maps not only from the studied initial product state, but also setting the initial time to any time t𝑡titalic_t. As we have shown in the previous section this is not usually the case for the evolution of the qubit, that can certainly be described by CP maps from the initial product state, but nothing can be said for most evolved states. Note that the full understanding of conditions for initial system-environment correlations that make the subsequent evolution of the reduced state of the system completely positive is a subject of ongoing research Vacchini and Amato (2016); Dominy et al. (2016); Dominy and Lidar (2016). The formalism presented in Ref. Roszak and Cywiński (2015) in general cannot be used to study two-qubit states undergoing pure dephasing due to an interaction with an environment, but it turns out that it is viable, if the initial qubit state is a superposition of only two states that are product states in the pointer states bases of the qubits (i.e. the bases singled out by the pure dephasing couplings to the respective environments). Such a state is “operationally” a two-level system, since during pure dephasing it will never leave the subspace of these two states, and the two-qubit entanglement is then simply proportional to the modulus of the single nonzero coherence present in the reduced density matrix of the qubits Yu and Eberly (2007); Mazurek et al. (2014b); Szańkowski et al. (2015); Bragar and Cywiński (2015). This allows for the study of the generation of entanglement between any initial two-qubit Bell state and its environment, while the state undergoes pure dephasing and its entanglement is diminished. Furthermore, if the qubits are initialized in a pure state, each qubit interacts with a separate environment, the initial state of these environments is a product of density operators of the two environments, and there is no interaction between the qubits and between the environments, then the evolution of each qubit is local, so the qubits evolve under local operations and classical communication (LOCC). In the case of a product initial state of the two-qubit state and the environments, It is known that such evolution cannot lead to enhancement of entanglement between the initial qubit state and any qubit state at time t𝑡titalic_t - this is follows from definition of entanglement as a quantity that cannot be increased by LOCC Plenio and Virmani (2007); Horodecki et al. (2009). However, it has to be stressed that while the above-listed conditions for LOCC dynamics are too strong (they are definitely sufficient, but not all of them are necessary) breaking of any of them could make the evolution become nonlocal. Full understanding of the necessary conditions is hampered by the fact that LOCC transformations are notoriously difficult to characterize (see Plenio and Virmani (2007); Aolita et al. (2015) and references therein). It is easier to consider a larger set (containing LOCC within it) of so-called separable maps that are CP and for which all the Kraus operators defining them can be written as products of operators acting on relevant local subsystems. This however comes at a price: there exist separable maps that can increase entanglement of states Chitambar and Duan (2009) that are neither “generically” separable nor pure Gheorghiu and Griffiths (2008), e.g. certain states that were obtained from pure entangled states by subjecting them to decoherence. We have reminded the reader about those known results in order to stress the fact that the mathematical conditions for an evolution not to increase entanglement are highly nontrivial, and consequently simple intuitions about what kind of evolution is “local” (and thus cannot increase entanglement) often prove wrong. It is known that local operations can enhance entanglement with respect to an initial state having correlations between the entangled system and the environment (for an example see D’Arrigo et al. (2014)), if these correlations make the subsequent dynamics not completely positive, and thus not belonging to LOCC, as LOCC is a subset of separable CP operations. As shown recently, nonzero discord between the qubits and their environment can be Rodríguez-Rosario et al. (2008) (but does not have to be, see Dominy et al. (2016)) a correlation that leads to subsequent non-CP dynamics. Let us see then if the two-qubit generalization of the previously obtained results about system-environment discord generated during decoherence, can shed some light on the behavior of two-qubit entanglement dynamics.

To this end let us study the initial two-qubit Bell state |ψ⟩=1/2(|00⟩+|11⟩)ket𝜓12ket00ket11|\psi\rangle=1/\sqrt2(|00\rangle+|11\rangle)| italic_ψ ⟩ = 1 / square-root start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ). The choice of the Bell state is arbitrary. We assume (for simplicity) that only one of the qubits interacts with an environment and that this environment is a qubit itself in some initial state R(0)=c0|0⟩⟨0|+c1|1⟩⟨1|𝑅0subscript𝑐0ket0quantum-operator-product0subscript𝑐11bra1R(0)=c_0|0\rangle\langle 0|+c_1|1\rangle\langle 1|italic_R ( 0 ) = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | 0 ⟩ ⟨ 0 | + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 1 ⟩ ⟨ 1 |. The product initial state of the whole state system is therefore σ^(0)=|ψ⟩⟨ψ|⊗R(0)^𝜎0tensor-productket𝜓bra𝜓𝑅0\hat\sigma(0)=|\psi\rangle\langle\psi|\otimes R(0)over^ start_ARG italic_σ end_ARG ( 0 ) = | italic_ψ ⟩ ⟨ italic_ψ | ⊗ italic_R ( 0 ). The Hamiltonian of this system is

H^=εA|1⟩AA⟨1|+εB|1⟩BB⟨1|+|1⟩AA⟨1|⊗V^A+H^E,^𝐻subscript𝜀𝐴subscriptket1𝐴𝐴subscriptquantum-operator-product1subscript𝜀𝐵1𝐵𝐵bra1tensor-productsubscriptket1𝐴𝐴bra1subscript^𝑉𝐴subscript^𝐻𝐸\hatH=\varepsilon_A|1\rangle_AA\langle 1|+\varepsilon_B|1\rangle_BB% \langle 1|+|1\rangle_AA\langle 1|\otimes\hatV_A+\hatH_E,over^ start_ARG italic_H end_ARG = italic_ε start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | 1 ⟩ start_POSTSUBSCRIPT italic_A italic_A end_POSTSUBSCRIPT ⟨ 1 | + italic_ε start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | 1 ⟩ start_POSTSUBSCRIPT italic_B italic_B end_POSTSUBSCRIPT ⟨ 1 | + | 1 ⟩ start_POSTSUBSCRIPT italic_A italic_A end_POSTSUBSCRIPT ⟨ 1 | ⊗ over^ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT , (21)

where the indices A𝐴Aitalic_A and B𝐵Bitalic_B differentiate between the qubits, V^Asubscript^𝑉𝐴\hatV_Aover^ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is an operator acting in the subspace of the environment, while H^Esubscript^𝐻𝐸\hatH_Eover^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT is the free Hamiltonian of the environment. The interaction of qubit A𝐴Aitalic_A with the environment has been asymmetrized for convenience, since the aim of this section is to show an exemplary evolution of a certain type and not to quantify all possible evolutions of this type. Although this Hamiltonian is of larger dimensionally in terms of the qubits than the Hamiltonian of eq. (7), the resulting evolution is equivalent to the evolution discussed in Sec. III, if the assumptions introduced in the previous paragraph are taken into account. Although the evolution operator is different and is equal to

U^(t)^𝑈𝑡\displaystyle\hatU(t)over^ start_ARG italic_U end_ARG ( italic_t ) =\displaystyle== |00⟩⟨00|⊗𝕀+|01⟩⟨01|⊗𝕀tensor-productket00bra00𝕀tensor-productket01bra01𝕀\displaystyle|00\rangle\langle 00|\otimes\mathbbI+|01\rangle\langle 01|% \otimes\mathbbI| 00 ⟩ ⟨ 00 | ⊗ blackboard_I + | 01 ⟩ ⟨ 01 | ⊗ blackboard_I

+|10⟩⟨10|⊗w^(t)+|11⟩⟨11|⊗w^(t),tensor-productket10bra10^𝑤𝑡tensor-productket11bra11^𝑤𝑡\displaystyle+|10\rangle\langle 10|\otimes\hatw(t)+|11\rangle\langle 11|% \otimes\hatw(t),+ | 10 ⟩ ⟨ 10 | ⊗ over^ start_ARG italic_w end_ARG ( italic_t ) + | 11 ⟩ ⟨ 11 | ⊗ over^ start_ARG italic_w end_ARG ( italic_t ) ,

where w^(t)=exp(-i(H^E+V^)t)^𝑤𝑡𝑖subscript^𝐻𝐸^𝑉𝑡\hatw(t)=\exp\left(-i(\hatH_E+\hatV)t\right)over^ start_ARG italic_w end_ARG ( italic_t ) = roman_exp ( - italic_i ( over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT + over^ start_ARG italic_V end_ARG ) italic_t ), the density matrix of the whole system evolves according to

σ^(t)=12(R^(0)00R^(0)w^(t)00000000w^†(t)R^(0)00w^†(t)R^(0)w^(t)),^𝜎𝑡12^𝑅000^𝑅0^𝑤𝑡00000000superscript^𝑤†𝑡^𝑅000superscript^𝑤†𝑡^𝑅0^𝑤𝑡\hat\sigma(t)=\frac12\left(\beginarray[]cccc\hatR(0)&0&0&\hatR(0% )\hatw(t)\\ 0&0&0&0\\ 0&0&0&0\\ \hatw^\dagger(t)\hatR(0)&0&0&\hatw^\dagger(t)\hatR(0)\hatw(t)% \endarray\right),over^ start_ARG italic_σ end_ARG ( italic_t ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( start_ARRAY start_ROW start_CELL over^ start_ARG italic_R end_ARG ( 0 ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG ( italic_t ) end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL over^ start_ARG italic_w end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL over^ start_ARG italic_w end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) over^ start_ARG italic_R end_ARG ( 0 ) over^ start_ARG italic_w end_ARG ( italic_t ) end_CELL end_ROW end_ARRAY ) , (23)

which is the same as in case of a single qubit interacting with the type of environment under study. Let us now additionally assume that the evolution is non-entangling (as always in this paper), so it is possible to write R^(0)^𝑅0\hatR(0)over^ start_ARG italic_R end_ARG ( 0 ) and w^(t)^𝑤𝑡\hatw(t)over^ start_ARG italic_w end_ARG ( italic_t ) in the common eigenbasis ketsuperscript𝑛′𝑡

^\prime(t)\rangle\ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ at every time t𝑡titalic_t. Hence, the state σ^(t)^𝜎𝑡\hat\sigma(t)over^ start_ARG italic_σ end_ARG ( italic_t ) can be written in the form

σ^(t)=∑n=01cn′(t)|ψn(t)⟩⟨ψn(t)|⊗|n′(t)⟩⟨n′(t)|,^𝜎𝑡superscriptsubscript𝑛01tensor-productsuperscriptsubscript𝑐𝑛′𝑡ketsubscript𝜓𝑛𝑡brasubscript𝜓𝑛𝑡ketsuperscript𝑛′𝑡brasuperscript𝑛′𝑡\hat\sigma(t)=\sum_n=0^1c_n^\prime(t)|\psi_n(t)\rangle\langle\psi_% n(t)|\otimes|n^\prime(t)\rangle\langle n^\prime(t)|,over^ start_ARG italic_σ end_ARG ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | ⊗ | italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ ⟨ italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) | , (24)

where |ψn(t)⟩=1/2(|00⟩+eiϕn(t)|11⟩)ketsubscript𝜓𝑛𝑡12ket00superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡ket11|\psi_n(t)\rangle=1/\sqrt2\left(|00\rangle+e^i\phi_n(t)|11\rangle\right)| italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ⟩ = 1 / square-root start_ARG 2 end_ARG ( | 00 ⟩ + italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT | 11 ⟩ ) and the factors eiϕn(t)superscript𝑒𝑖subscriptitalic-ϕ𝑛𝑡e^i\phi_n(t)italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT are eigenvalues of w^(t)^𝑤𝑡\hatw(t)over^ start_ARG italic_w end_ARG ( italic_t ) corresponding to the eigenstates |n′(t)⟩ketsuperscript𝑛′𝑡|n^\prime(t)\rangle| italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ⟩ appropriately. It is straightforward to quantify inter-qubit entanglement in this state (after tracing-out the degrees of freedom of the environment) and the entanglement measure concurrence Wootters (1998) of such a state is equal to CQ=|c0′(t)eiϕ0(t)+c1′(t)eiϕ1(t)|subscript𝐶𝑄superscriptsubscript𝑐0′𝑡superscript𝑒𝑖subscriptitalic-ϕ0𝑡superscriptsubscript𝑐1′𝑡superscript𝑒𝑖subscriptitalic-ϕ1𝑡C_Q=|c_0^\prime(t)e^i\phi_0(t)+c_1^\prime(t)e^i\phi_1(t)|italic_C start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = | italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) end_POSTSUPERSCRIPT |.

In the simplest case, the basis n′(t)⟩ketsuperscript𝑛′𝑡

^\prime(t)\rangle\ is time independent and is the same as the eigenbasis of the initial state of the environment, 0⟩,ket0ket1\0\rangle, 1 ⟩ , so c0′(t)=c0superscriptsubscript𝑐0′𝑡subscript𝑐0c_0^\prime(t)=c_0italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and c1′(t)=c1superscriptsubscript𝑐1′𝑡subscript𝑐1c_1^\prime(t)=c_1italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) = italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, while the time-dependence of the phase factors reduces to ϕn(t)=φntsubscriptitalic-ϕ𝑛𝑡subscript𝜑𝑛𝑡\phi_n(t)=\varphi_ntitalic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_t. Here, zero discord with respect to the qubit system is obtained only in two situations; firstly, at times tpsubscript𝑡𝑝t_pitalic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT when |ψ0(tp)⟩=|ψ1(tp)⟩ketsubscript𝜓0subscript𝑡𝑝ketsubscript𝜓1subscript𝑡𝑝|\psi_0(t_p)\rangle=|\psi_1(t_p)\rangle| italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ⟩ = | italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ⟩, so eiφ0tp=eiφ1tpsuperscript𝑒𝑖subscript𝜑0subscript𝑡𝑝superscript𝑒𝑖subscript𝜑1subscript𝑡𝑝e^i\varphi_0t_p=e^i\varphi_1t_pitalic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT and the qubit is in a pure, maximally entangled state, and secondly, at times tqsubscript𝑡𝑞t_qitalic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT when ⟨ψ0(tq)|ψ1(tq)⟩=0inner-productsubscript𝜓0subscript𝑡𝑞subscript𝜓1subscript𝑡𝑞0\langle\psi_0(t_q)|\psi_1(t_q)\rangle=0⟨ italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) | italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ⟩ = 0, so eiφ0tq=-eiφ1tqsuperscript𝑒𝑖subscript𝜑0subscript𝑡𝑞superscript𝑒𝑖subscript𝜑1subscript𝑡𝑞e^i\varphi_0t_q=-e^i\varphi_1t_qitalic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = - italic_e start_POSTSUPERSCRIPT italic_i italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT and the qubit decoherence is maximal while inter-qubit entanglement is minimal. Times tpsubscript𝑡𝑝t_pitalic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and tqsubscript𝑡𝑞t_qitalic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT appear interchangeably, since tp=2πp/|φ1-φ0|subscript𝑡𝑝2𝜋𝑝subscript𝜑1subscript𝜑0t_p=2\pi p/|\varphi_1-\varphi_0|italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 2 italic_π italic_p / | italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | and tq=2π(q+1)/|φ1-φ0|subscript𝑡𝑞2𝜋𝑞1subscript𝜑1subscript𝜑0t_q=2\pi(q+1)/|\varphi_1-\varphi_0|italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = 2 italic_π ( italic_q + 1 ) / | italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT |, with p,q=0,1,2…formulae-sequence𝑝𝑞012…p,q=0,1,2...italic_p , italic_q = 0 , 1 , 2 … and the evolution of inter-qubit entanglement is periodically repeated every 2π/|φ1-φ0|2𝜋subscript𝜑1subscript𝜑02\pi/|\varphi_1-\varphi_0|2 italic_π / | italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_φ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT |. The evolution of such entanglement from a certain time tpsubscript𝑡𝑝t_pitalic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT to time tp+1subscript𝑡𝑝1t_p+1italic_t start_POSTSUBSCRIPT italic_p + 1 end_POSTSUBSCRIPT (capturing a full cycle of entanglement evolution) is shown in Fig. (1) for three different initial states of the environment (the time tqsubscript𝑡𝑞t_qitalic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT which appears midcycle is marked on the figure). If we now choose a certain time τ=tq𝜏subscript𝑡𝑞\tau=t_qitalic_τ = italic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT as a new initial time, the evolution of the new initial state σ^(τ)^𝜎𝜏\hat\sigma(\tau)over^ start_ARG italic_σ end_ARG ( italic_τ ), which has minimal entanglement (zero entanglement for c0=c1=1/2subscript𝑐0subscript𝑐112c_0=c_1=1/2italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 / 2) to any later state t+τ𝑡𝜏t+\tauitalic_t + italic_τ can be descibed using CP maps, since the state is zero-discordant with respect to the two qubits Rodríguez-Rosario et al. (2008). Note that all later states have greater or equal inter-qubit entanglement than the new initial state, so the evolution discussed in this section, which could easily be mistakenly believed to be local, as it occurs due to interaction with an environment of only one of the qubits, is also entangling, and thus it does not belong to LOCC. Apparently, the initial nonlocality (entanglement) of the qubits’ state makes the total system state at time tqsubscript𝑡𝑞t_qitalic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT incompatible with subsequently local dynamics, although, as we have shown, this incompatibility does not follow from nonzero discord. Finally, let us note that we have so far failed at finding a separable representation of the CP evolution of the qubits from time tqsubscript𝑡𝑞t_qitalic_t start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT onwards, but have not proven that such a representation is impossible. It remains then an open question if the example evolution described above is CP and separable, but not LOCC, or if it is simply CP but nonseparable.

VII Conclusion

We have studied pure dephasing evolution of a qubit, intialized in a pure state, interacting with an environment and found that, if no qubit-environment entanglement is generated at a given time, then automatically no qubit-environment quantum discord with respect to the environment is generated. Hence, the set of separable states which can be obtained due to an evolution described by the class of Hamiltonians studied is zero-volume, and behaviors such as sudden death of qubit-environment entanglement are unlikely. Furthermore, the evolution of the environment alone between two arbitrary times may be described using completely positive maps, as follows from connection between zero discord with respect to one subsystem and complete positivity of subsequent evolution of the reduced state of this system Rodríguez-Rosario et al. (2008). We have also looked at the qubit-environment quantum discord with respect to the qubit and it turns out that the situation is very different here. For times at which the qubit and the environment are separable, this type of discord is usually still present in the system. It is only possible for such discord to be zero, if the qubit is initially in an equal superposition state. discord server Then the discord can vanish at certain times when the relative phases of the environmental states evolving due to the interaction with the qubit specifically align - something that can be expected to happen only for rather small environments. Furthermore, an evolution for which this type of quantum correlations would never appear is impossible if we ignore the trivial cases of evolutions not leading to any decoherence. Lastly, we were able to compare an exemplary evolution of two-qubit entanglement under the influence of a local interaction with an environment, with qubit-environment discord generation. We have shown that such an evolution which displays zero-discordant points in time with respect to the two qubit subsystem is possible (but not common). This means that the evolution starting from one of the zero-discord times can be described using completely positive maps. Since at such points the qubits are either fully entangled (as they return at these times to their initial state) or have minimal entanglement possible, the evolution starting from times corresponding to the former case are trivial, but those corresponding to the latter situation, lead to enhancement of inter-qubit entanglement due to a local interaction whilst the evolution can be described by completely positive maps. This interesting observation illustrates how hard it is to identify a LOCC evolution when one abandons the common assumption of complete lack of correlations between the initial states of the entangled system and the environment.

This work was funded from the Reseach Project number DEC-2012/07/B/ST3/03616 financed by the National Science Centre (NCN).

Website: https://coderwall.com/p/nuvmqa/12-best-nft-discord-server-communities-in-2022

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