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Operational Markov condition for quantum processes
Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, Kavan Modi
TL;DR
The paper asks how to define quantum Markovianity so that it captures memory effects that classical and existing quantum criteria can miss. It formulates an operational causal-break condition and proves its equivalence with the classical condition in the appropriate setting. Examples show that CP-divisibility and monotonic trace-distance decrease can coexist with operationally non-Markovian dynamics.
Problem
Existing quantum Markovianity definitions extend necessary classical conditions, producing criteria that do not coincide with the classical condition and can miss detectable memory effects.
Method
The paper defines Markovianity by requiring the post-causal-break system state to depend only on the fresh input state, and proves the condition using process-tensor expansions over linearly independent controls.
Results
The condition is necessary and sufficient, while examples show that CP-divisibility and monotonic trace-distance decrease can label operationally non-Markovian processes as Markovian.
Takeaways & Limitations
Operational causal breaks reveal memory stored in the environment or multi-time correlations even when common reduced-dynamics witnesses do not.
Abstract
from arXiv · showhide
We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process, or the experimental falsifiability of a Markovian hypothesis.
Appendix A: Proof of quantum Markov condition (main Theorem)
The proof shows that equality of conditional states after a causal break for a complete linearly independent control set extends to arbitrary prior controls, establishing Markovianity.
- Appendix A: Proof of quantum Markov condition (main Theorem): The converse proof expands arbitrary pre-break control sequences in a basis of operations and measurement elements.It also assumes no further operations between time steps k and l, while noting straightforward generalization when later operations occur.
- Appendix A: Proof of quantum Markov condition (main Theorem): Joint probabilities weight the conditional states associated with the measurement outcome and all preceding basis operations.The resulting expression rewrites the process tensor in terms of conditional states and their joint probabilities.
- Appendix A: Proof of quantum Markov condition (main Theorem): If the conditional state is identical for every element of a finite linearly independent basis, it can be taken outside the relevant sum.The proof then uses normalization of the conditional state and divides by the corresponding probability.
- Appendix A: Proof of quantum Markov condition (main Theorem): Therefore, the causal-break condition holds for any possible input before the break, so the process is Markovian.This completes the converse implication from basis controls to arbitrary controls.
Appendix B: Examples
The examples show that operational witnesses can efficiently detect non-Markovianity, including a process that is CP-divisible without being Markovian.
- Appendix B: Examples: The paper presents examples where several non-Markovianity witnesses fail to detect non-Markovian behavior.These witnesses nevertheless provide efficient criteria for determining non-Markovianity in many cases.
- Appendix B: Examples: A qubit’s reduced dynamics can be pure dephasing and CP-divisible when uninterrupted, yet an intermediate X operation reverses the evolution and reveals memory.After the reversal period, the state returns to its initial state up to another X operation, followed by renewed dephasing.
1. Divisibility
The divisibility example demonstrates that snapshot-based GKSL tests can label a process Markovian even when its history-dependent reversal makes it non-Markovian.
- 1. Divisibility: The qubit dynamics are fully CP-divisible with a time-independent GKSL generator and positive rates, yet two X operations reverse the exponential decay.The reversal occurs over a duration determined by the system’s history, indicating memory in multi-time correlations.
- 1. Divisibility: A snapshot definition that accepts Λ = e^L classifies this process as Markovian despite its operational non-Markovian behavior.Thus, the snapshot method has a demonstrated limitation for detecting memory.
2. Trace distance
A partial-swap process can monotonically decrease trace distance while retaining memory that becomes visible after a causal break and fresh preparation.
- 2. Trace distance: The process uses partial-swap evolution of a qubit system and qubit environment, with reduced dynamics described by a depolarising channel.The setup compares different initial system states and later applies a measurement followed by pure-state preparation.
- 2. Trace distance: For ω(t3 − t1) ≤ π/2, trace-distance distinguishability decreases monotonically, so the trace-distance measure labels the process Markovian.This conclusion concerns the interval in which the distinguishability function is monotonic.
- 2. Trace distance: After the causal break resets the system to a fresh pure state, the environment still depends on the initial system state and measurement outcome.The post-break system states initially have zero trace distance, but later evolution depends on that environment state.
- 2. Trace distance: Consequently, the process is operationally non-Markovian because the later state depends on the initial choice and measurement outcome through the environment.Operational Markovianity would require dependence only on the fresh preparation, the interaction, and an environment state independent of earlier system history.
3. Non-Markovianity without correlations
A quantum process can be non-Markovian even when the system and environment never develop correlations. In the SWAP example, the later system state is independent of the intermediate operation because information remains accessible through the environment.
- The SWAP process is non-Markovian despite the joint system-environment state remaining a product at all times.The example therefore shows that the absence of system-environment correlations does not ensure Markovian dynamics.
- After the first SWAP, the system contains ρE while the environment contains the system’s initial state ρS.This exchanges the system and environment states before the intermediary operation.
- Whatever operation, including a causal break, is applied at the intermediate step, the second SWAP returns the system to its initial state ρS.The state at the next step is independent of the intermediary preparation.