We present a conservative reformulation of the complementarity analysis containedin the underlying manuscript materials. The claim is restricted to the part that isdirectly supported by the available derivations: an exact reduced-state balance for localpredictability, local visibility, and entanglement in an exactly solvable mini-reservoir modelfor non-Markovian two-qubit dynamics. For each local qubit, the reduced budget is definedas Bk(t) = P2k (t) + V 2k (t) + C2(t) and is completed by a remainder Mk(t) = 1 − Bk(t).Within the Hilbert–Schmidt bookkeeping adopted from the source material, this remainderis written as the nonlocal contribution beyond concurrence, Mk(t) = 2Cnlhs(t)− C2(t). Theresult is model bound and should not be read as a universal conservation law for arbitrarycombinations of coherence, discord, and entanglement measures. Instead, it isolates a precisecomplementarity identity inside a controlled Markov-embedded open-system setting.
We present a conservative reformulation of the complementarity analysis containedin the underlying manuscript materials. The claim is restricted to the part that isdirectly supported by the available derivations: an exact reduced-state balance for localpredictability, local visibility, and entanglement in an exactly solvable mini-reservoir modelfor non-Markovian two-qubit dynamics. For each local qubit, the reduced budget is definedas Bk(t) = P2k (t) + V 2k (t) + C2(t) and is completed by a remainder Mk(t) = 1 − Bk(t).Within the Hilbert–Schmidt bookkeeping adopted from the source material, this remainderis written as the nonlocal contribution beyond concurrence, Mk(t) = 2Cnlhs(t)− C2(t). Theresult is model bound and should not be read as a universal conservation law for arbitrarycombinations of coherence, discord, and entanglement measures. Instead, it isolates a precisecomplementarity identity inside a controlled Markov-embedded open-system setting.
Non-Markovian optical channels can redistribute interferometric visibility,which-path information, and entanglement in ways that are not captured byconcurrence dynamics alone. Motivated by cavity-QED, waveguide-feedback,and related photonic reservoir architectures, we study an exactly solvabletwo-qubit mini-reservoir model and interpret its reduced dynamics as an effectivelocal photonic feedback channel. For the reduced two-qubit state, predictabilityPk, visibility Vk, and concurrence C obey the mixed-state relationP2k (t)+V 2k (t)+C2(t) = Bk(t) ≤ 1, where Bk(t) is the locally accessible complementaritybudget. Within the Hilbert–Schmidt formulation of completecomplementarity, we show that the corresponding deficit is closed exactly byMk(t) ≡ 1 − Bk(t) = 2Cnlhs(t) − C2(t). The remainder therefore identifiesthe part of the nonlocal optical coherence sector that is not already encodedin concurrence. Exact trajectories for Bell, nonmaximally entangled parity,superposition-of-Bell, and Werner initial states reveal memory-inducedcollapse-and-revival structure in the accessible budget, in the visibility sectorwhen local coherence is present, and in the remainder itself. Comparisonswith one-sided discord and with the purity deficit show that Mk(t)is a distinct reduced-state diagnostic rather than a relabeling of entanglement,discord, or mixedness. Because all quantities are reconstructed fromthe same reduced two-qubit state, the framework is naturally compatible with interferometric and tomographic readout in memory-bearing quantumopticalplatforms.
Abstract We develop a complementarity-based description of open two-qubit dynamics built aroundthe budget Bk(t) = P2k(t)+V 2k(t)+C2(t) and the associated deficit Dk(t) = 1−Bk(t). The mainoutcome is not a new universal witness of non-Markovianity, but a more useful split betweenstructure observables and flow observables. Using explicit prototype amplitude-damping anddephasing channels, together with asymmetric couplings and mixed initial states, we showthat deficit-based distances such as ∆D = maxt|Dk(t) − SL(t)| mainly encode state- andchannel-dependent geometry, whereas positive-variation functionals such as NC and N−Drespond more directly to memory-driven recovery. In addition, for an exactly solvable Bell-statedephasing sector we prove that the revival windows of C and −DA coincide and obtain closedexpressions for BA and DA. The paper reframes complementarity dynamics as a two-levelproblem: geometry of resource partition on one side and temporal backflow on the other. Theframework is tied to a same-dataset BLP-like benchmark and to two analytical propositionsthat anchor the relation between concurrence revival and deficit flow in solvable sectors and ongeneral revival intervals
We study complementarity dynamics in open two-qubit systems through the local budgetBA = P2A+ V 2A + C2 and its deficit DA = 1 − BA. The purpose is not to define a newnon-Markovianity measure, but to distinguish two diagnostic roles. Structure observablesdescribe how predictability, visibility, concurrence, and deficit are arranged along a statespacetrajectory, whereas flow observables record temporal recovery. Amplitude-damping anddephasing prototypes, finite parameter scans, BLP-like comparisons, and two analyticallysolvable Bell-sector results show that ΔD is mainly structural, while NC and N−D respond torevival windows. The framework therefore clarifies what complementarity budgets certify inopen two-qubit dynamics.
We study the decoherence properties of an entangled bipartite qubit system, represented by two two-level atoms that are individually coupled to non-Markovian reservoirs. This coupling ensures that the dynamical equations of the atoms can be treated independently. The non-Markovian reservoirs are described by a model which leads to an exact non-Markovian master equation of the Nakajima-Zwanzig form [J. Salo, S. M. Barnett, and S. Stenholm, Optics Commun. 259, 772 (2006)]. We consider the evolution of the entanglement of a two-atom state that is initially completely entangled, quantified by its concurrence. Collapses and revivals in the concurrence, induced by the memory effects of the reservoir, govern the dynamics of the entangled quantum system. These collapses and revivals in the concurrence are a strong manifestation of the non-Markovian reservoir.
We study the entanglement dynamics of a two-qubit system, each of which is embedded into its local reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components and particularly it also satisfies an identity firstly given by Yonac, Yu and Eberly (J. Phys. B 40, S45 (2007)) for a double J-C model without decoherence. While the explicit distribution of the entanglement is dependent of the detail of the model, even the approximation used, the identity remains unchanged. Our unified treatment includes the previous results in the literature as special cases. The result reveals the profound nature of the entanglement and should have significant implications for quantum information processing. below) has been satisfied at any time. This may be not surprising since the double J-C model has no decoherence and any initial information can be preserved in the time evolution. However, it would be surprising if the identity is still valid in the presence of the decoherence, in which a non-equilibrium relaxation process is involved. In this paper, we show that it is indeed true for such a system consisted of two qubits locally interacting with two reser- voirs. We find that the distribution of the entanglement among the bipartite subsystems is dependent of the ex- plicit property of the environment and its coupling with the qubit. The rich dynamical behaviors obtained pre- viously in the literature can be regarded as the special cases of our present result or Markovian approximation. Particularly, we find that the entanglement can stably distribute among all the bipartite subsystems if the qubit and its environment can form a bound state and the non- Markovian effect is important. Irrespective of how dis- tributes the entanglement, it is found that the identity about the entanglement in the whole system can be sat- isfied at any time, which reveals the profound physics of the entanglement dynamics. We consider two qubits interacting with two uncorre- lated vacuum reservoirs. Due to the dynamical indepen- dence between the two local subsystems, we can firstly solve the single subsystem, then apply the result obtained to the double-qubit case. The Hamiltonian of each local subsystem is (16)
We consider an entangled but non-interacting qubit pair a 1 and b 1 that is independently coupled to a set of local qubit systems, a I and b J , of 0 bit value, respectively. We derive rules for the transfer of entanglement from the pair a 1 – b 1 to an arbitrary pair a I – b J , for the case of qubit-number conserving local interactions. It is shown that the transfer rule depends strongly on the initial entangled state. If the initial entanglement is in the form of the Bell state corresponding to anti-correlated qubits, the sum of the square of the nonlocal pairwise concurrences is conserved. If the initial state is the Bell state with correlated qubits, this sum can be reduced, even to zero in some cases, to reveal a complete and abrupt loss of all nonlocal pairwise entanglement. We also identify that for the nonlocal bipartitions A – b J involving all qubits at one location, with one qubit b J at the other location, the concurrences satisfy a simple addition rule for both cases of the Bell states that the sum of the square of the nonlocal concurrences is conserved.