Markovianizing Cost of Tripartite Quantum States

Abstract

We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of $n$ copies of a tripartite state ${\rho }^{ABC}$, and is transformed to a quantum Markov chain conditioned by $B^n$ with a small error, using a random unitary operation on $A^n$. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.

19 pages, 6 figures. Definition 6 is added; the statements of Theorem 1, Theorem 7, Lemma 11, Lemma 12, and Remark in the end of Section II are modified; and the proofs in Appendix B-C and B-D are modified. The main results are unchanged

References in corpus (5)

• On the quantum, classical and total amount of correlations in a quantum state
• Quantum information can be negative
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• The mother of all protocols: Restructuring quantum information's family tree
• The structure of preserved information in quantum processes

Cited by in corpus (13)

• Quantum Resource Theories
• Journeys from Quantum Optics to Quantum Technology
• Deconstruction and conditional erasure of quantum correlations
• Disentanglement Cost of Quantum States
• Conditional Decoupling of Quantum Information
• A Coding Theorem for Bipartite Unitaries in Distributed Quantum Computation
• Complexity of causal order structure in distributed quantum information processing and its trade-off with entanglement
• Symmetrizing Cost of Quantum States
• One-Shot Randomized and Nonrandomized Partial Decoupling
• Quantum State Merging for Arbitrarily Small-Dimensional Systems
• Distributed Encoding and Decoding of Quantum Information over Networks
• The Cost of Randomness for Converting a Tripartite Quantum State to be Approximately Recoverable
• Classical and quantum parts of conditional mutual information for open quantum systems