Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes

In order to engineer an open quantum system and its evolution, it is essential to identify and control the memory effects. These are formally attributed to the non-Markovianity of dynamics that manifests itself by the evolution being indivisible in time, a property which can be witnessed by a non-monotonic behavior of contractive functions or correlation measures. We show that by monitoring directly the entanglement behavior of a system in a tripartite setting it is possible to witness all invertible non-Markovian dynamics, as well as all (also non-invertible) qubit evolutions. This is achieved by using negativity, a computable measure of entanglement, which in the usual bipartite setting is not a universal non-Markovianity witness. We emphasize further the importance of multipartite states by showing that non-Markovianity cannot be faithfully witnessed by any contractive function of single qubits. We support our statements by an explicit example of eternally non-Markovian qubit dynamics, for which negativity can witness non-Markovianity at arbitrary time scales. arXiv:1903.08663v2 [quant-ph]

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“…Once ρ 1 and ρ 2 are determined, by setting the initial tripartite state introduced in Eqs. (16)(17) as…”

Section: Appendix F: Eternally Non-markovian Qubit Dynamicsmentioning

confidence: 99%

Entanglement negativity as a universal non-Markovianity witness

2020

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“…This interpretation is based on the following. The proof is given in appendix D. We expect that the network min-entropy defined in equation (45) will play a role in the study non-Markovian quantum evolutions, along the lines of the entropic characterization of Markovianity provided in [89,90]. Intuitively, the idea is that one can evaluate how the correlations build up from one step to the next and use this information to infer properties of the internal memory used by the network.…”

Section: The Conditional Min-entropy Of a Quantum Causal Networkmentioning

confidence: 99%

Optimal quantum networks and one-shot entropies

Chiribella

Ebler

2016

New J. Phys.

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We develop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interative tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of applications, including the inversion, charge conjugation, and controlization of an unknown unitary dynamics. In the non-causal setting, we show a proof-of-principle application to the maximization of the winning probability in a non-causal quantum game. IntroductionAdvances in quantum communication [1][2][3] and in the integration of quantum hardware [4-8] are pushing towards the realization of networked quantum information systems, such as quantum communication networks [9-13] and distributed quantum computing [14][15][16]. Networks of interacting quantum devices are attracting interest also at the theoretical level, providing a framework for quantum games [17] and protocols [18][19][20], insights on the foundations of quantum mechanics [18, 21-23], a starting point for a general theory of Bayesian inference [24-31] and for the development of models of higher-order quantum computation [32][33][34].The network scenario motivates a new set of optimization problems, where the goal is not to optimize individual devices, but rather to optimize how different devices interact with one another. In many situations, the devices operate in a well-defined causal order-this is the case, for example, in the circuit model of quantum computing, where computations are implemented by sequences of gates [35,36]. Recently, researchers have started to investigate more general situations, where the causal order can be in a quantum superposition [20,33,34,[37][38][39] or can be indefinite in other more exotic ways, in principle compatible with quantum mechanics [34,[40][41][42][43][44][45]. In these new situations, optimizing quantum networks is important, for at least three reasons: First, in order to establish an advantage, one has to first find the optimal performances achievable in a definite causal order. Second, finding the maximum advantage requires an optimization over all non-causal networks. This is an essential step for assessing the power of the new, non-causal models of information processing. Third, identifying the ultimate performances achieved in the absence of pre-defined causal structure is expected to shed light on the interplay between quantum mechanics and spacetime.In this paper we develop a semidefinite programming (SDP) approach to the optimization of quantum networks. We start by analyzing scenarios with definite causal order, choosing an operational measure of performance, quantified by how much the network scores in a giv...

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“…This question was answered already in 1953 by Blackwell [7] for the classical case, and in 1980 by Alberti and Uhlmann [8] for the qubit case. More recently, it was solved for pure states in [9], characterized in [10], [11], [12], [13], and finally, in [14] it was fully solved (for finite dimensions) with semidefinite programming. In [14], [15] (see also references therein) it was also shown that this pre-order can be characterized completely in terms of a family of distinguishability measures that are given in terms of the conditional min-entropy [16], [17].…”

Section: Introductionmentioning

confidence: 99%

Comparison of Quantum Channels by Superchannels

Gour

2019

IEEE Trans. Inform. Theory

123

156

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We extend the definition of the conditional minentropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a preorder that extends the definition of "quantum majorization" from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels, and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.

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Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes

Cited by 85 publications

References 48 publications

Entanglement negativity as a universal non-Markovianity witness

Entanglement negativity as a universal non-Markovianity witness

Optimal quantum networks and one-shot entropies

Comparison of Quantum Channels by Superchannels

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