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Instrument-Specific Finite Quantum Markov Order

Claim/Theorem

Taranto, Pollock, Milz, Tomamichel, and Modi show that quantum finite Markov order is inherently instrument-specific: there do not exist non-Markovian quantum processes with finite Markov order relative to all possible probing instruments. Taranto, Milz, Pollock, and Modi further analyze the structure of such finite-memory processes and show that vanishing quantum conditional mutual information is only a special case. Therefore, any bounded-Markov-order reduction of Conjecture 1 must specify the admissible intervention class rather than treating finite memory as an instrument-independent property.

Dependencies

  • [[process-tensor-framework.md]]

Conflicts/Gaps

  • Conjecture 1 states memory decay through a uniform diamond-norm bound over prior control histories, which is stronger and differently phrased than instrument-specific finite Markov order.
  • This node clarifies what can and cannot be meant by “bounded memory” in quantum process language, but it does not yet produce a Markovian dilation usable inside a random-circuit moment calculation.
  • The relevant probing class for local random circuits still has to be chosen and justified.

Sources

  • 10.1103/PhysRevLett.122.140401 (Theorem 4, p. 4)
  • 10.1103/PhysRevA.99.042108 (abstract; Sec. V and App. C on finite Markov order vs. quantum CMI)