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arXiv:1907.12574 · Entropy

Limits to Perception by Quantum Monitoring with Finite Efficiency

Luis Pedro García-Pintos, Adolfo del Campo

Published Nov 17, 2021• 3 citations
arXivHTMLPDF

Abstract

We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and the actual state of the system. These bounds are expressed solely in terms of the purity and von Neumann entropy of the state assigned by the agent, and are shown to characterize how an agent's perception of the system is altered by access to additional information. We apply our results to Gaussian states and to the dynamics of a system embedded in an environment illustrated on a quantum Ising chain.

DOI
10.3390/e23111527

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We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and the actual state of the system. These bounds are expressed solely in terms of the purity and von Neumann entropy of the state assigned by the agent, and are shown to characterize how an agent's perception of the system is altered by access to additional information. We apply our results to Gaussian states and to the dynamics of a system embedded in an environment illustrated on a quantum Ising chain.
We study the descriptions that different agents monitoring a quantum system provide of it, by comparing the state that an agent assigns to a system given partial knowledge of measurement outcomes and the actual state of the system. We do this by obtaining a) bounds on the trace distance, and b) the relative entropy, between the respective states. The results have simple expressions solely in terms of the purity and von Neumann entropy of the state assigned by the agent. These results can be interpreted as limits on the awareness that agents can have of the state of a system given incomplete knowledge. By considering the case of an agent with partial access to information of the outcomes of the monitoring process, we study how a transition from ignorance to awareness of the state of a system affects its description. In the setting of a system interacting with an environment, our results provide estimates on how ones description of a system is refined as information encoded in the environment is incorporated into the picture.
In quantum information theory, communication capacities are mostly given in terms of entropic formulas. Continuity of such entropic quantities are significant, as they ensure uniformity of measures against perturbations of quantum states. Traditionally, continuity bounds have been provided in terms of the trace distance, which is a bonafide metric on the set of quantum states. In the present contribution we derive continuity bounds for various information measures based on the difference in purity of the concerned quantum states. In a finite-dimensional system, we establish continuity bounds for von Neumann entropy which depend only on purity distance and dimension of the system. We then obtain uniform continuity bounds for conditional von Neumann entropy in terms of purity distance which is free of the dimension of the conditioning subsystem. Furthermore, we derive the uniform continuity bounds for other entropic quantities like relative entropy distance, quantum mutual information and quantum conditional mutual information. As an application, we investigate the variation in squashed entanglement with respect to purity. We also obtain a bound to the quantum conditional mutual information of a quantum state which is arbitrarily close to a quantum Markov chain.
We derive a universal inequality that provides a lower bound on the ensemble-averaged von Neumann entropy change in a quantum system subject to continuous measurement and dissipation. Our result clarifies how entropy production is fundamentally constrained by three distinct contributions: (i) the non-Hermitian structure of the dissipation operator, (ii) the standard variance associated with measurement-induced fluctuations, and (iii) a generalized quantum variance reflecting the noncommutativity between the measurement observable and the quantum state. This third term vanishes when the state and observable commute, and thus represents a purely quantum contribution arising from coherence disturbance and measurement backaction. The derived inequality generalizes classical information-thermodynamic relations, such as the Sagawa--Ueda inequality, to the quantum regime, providing a new perspective on the trade-offs between information acquisition, control, and entropy production in continuously monitored open quantum systems.
Abstract Information is often considered classical in a definite state rather than in a superposition of states. It seems rather strange to consider information in superpositions. Some people would, on the basis of this argument, conclude that quantum information can never exist and we can only have access to classical information. It turns out, however, that quantum information can be quantified in the same way as classical information using Shannon's prescription. There is a unique measure (up to a constant additive or multiplicative term) of quantum information such that S (the von Neumann entropy) is purely a function of the probabilities of outcomes of measurements made on a quantum system (that is, a function of a density operator); S is a continuous function of probability; S is additive. This chapter discusses the fidelity of pure quantum states, Helstrom's discrimination, quantum data compression, entropy of observation, conditional entropy and mutual information, relative entropy, and statistical interpretation of relative entropy.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.