[Paper Review] Zero Discord for Markovian Bipartite Systems

M. Arsenijević, J. Jeknić-Dugić+1 more|arXiv (Cornell University)|Apr 12, 2012

Quantum Information and Cryptography1 references4 citations

TL;DR

This paper constructs a Markovian bipartite quantum system model that maintains zero two-way quantum discord indefinitely, demonstrating 'Markovian classicality'—a state of persistent classical correlations. By structuring the composite system as center-of-mass (S) and relative (S') degrees of freedom, with S decoupled from its environment and S' interacting locally, the model ensures the total density matrix remains a time-independent tensor product, satisfying both C- and P-criteria for classicality and proving that zero discord is dynamically stable under Markovian evolution for specific system structures.

ABSTRACT

Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro et al, Phys. Rev. A 81, 052328 (2010)] may seem to imply that the Markovian bipartite systems are practically deprived of zero discord states. Nevertheless, complementary to the result of Ferraro et al, we construct a model of a Markovian bipartite system providing zero discord for arbitrary long time interval, that we term 'Markovian classicality'. Our model represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. Interestingly enough, we are not able to offer any alternative to the model. Physical relevance of the model is twofold. First, the model is in intimate relation to the topics of quantum information locality, quantum discord saturation and quantum decorrelation. Second, the model is of the general physical interest. It pertains to a specific structure (decomposition into parts/subsystems) of a composite system, not to a special physical kind of composite systems. Being a characteristic of a structure, by definition, the model of Markovian classicality is not a model of sudden death of discord. We emphasize wide-range implications of our results.

Motivation & Objective

• To investigate whether Markovian dynamics can sustain zero quantum discord indefinitely, countering the prevailing view that discord vanishes only asymptotically.
• To construct a physically realizable model where zero two-way discord is a constant of motion, not just transient or asymptotic.
• To clarify the role of system structure (e.g., center-of-mass vs. relative degrees of freedom) in enabling long-term classical correlations.
• To establish that zero discord is not incompatible with Markovian evolution when the system's decomposition respects specific dynamical decoupling.
• To link the model to foundational quantum information concepts such as quantum locality, decorrelation, and discord saturation.

Proposed method

• The model decomposes a composite system C = S + S' into two non-interacting subsystems: S (center-of-mass) and S' (relative degrees of freedom), with S isolated from environment.
• The system evolves under a master equation where only S' interacts with a Markovian bath, while S remains decoupled and unitary.
• The initial state is a tensor product ρ(S) ⊗ ρ(S'), which remains invariant under time evolution due to the dynamical decoupling of S.
• The model satisfies both the C-criterion (classicality via diagonal density matrix in a fixed basis) and P-criterion (classicality via local operations and classical communication) for classicality.
• The analysis uses the two-way discord measure D^←(S|S') = 0 = D^←(S'|S), ensuring the state is a classical-classical (CC) state at all times.
• The construction relies on canonical transformations that separate the system into CM and relative coordinates, reflecting physical systems like Brownian particles.

Experimental results

Research questions

• RQ1 Can Markovian dynamics sustain zero quantum discord for an arbitrary long time interval?
• RQ2 What structural conditions on a bipartite quantum system allow for persistent zero discord under Markovian evolution?
• RQ3 Is there a physically meaningful model where zero discord is not just asymptotic but a constant of motion?
• RQ4 How does the decomposition of a composite system (e.g., CM + relative motion) affect the stability of classical correlations?
• RQ5 Can such a model be linked to known phenomena in quantum information, such as quantum decorrelation or discord saturation?

Key findings

• The model realizes Markovian classicality: zero two-way discord is preserved for all times, providing a sufficient condition for persistent classical correlations in open quantum systems.
• The system's structure—specifically, the separation into center-of-mass and relative degrees of freedom—is essential for maintaining zero discord under Markovian evolution.
• The tensor-product initial state ρ(S) ⊗ ρ(S') remains invariant under time evolution due to the dynamical decoupling of S from its environment.
• The model satisfies both the C- and P-criteria for classicality, confirming that the state is genuinely classical in the quantum information sense.
• The result shows that zero discord is not incompatible with Markovian dynamics when the system's decomposition respects dynamical isolation of one subsystem.
• The model is not a model of sudden death of discord but rather a formal proof of existence of long-term classical behavior in open quantum systems.