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Limits to Perception in the Quantum World

Luis Pedro García-Pintos, Adolfo del Campo

Published Jul 29, 2019• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
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.
A resolution of the quantum measurement problem(s) using the consistent histories interpretation yields in a rather natural way a restriction on what an observer can know about a quantum system, one that is also consistent with some results in quantum information theory. This analysis provides a quantum mechanical understanding of some recent work that shows that certain kinds of quantum behavior are exhibited by a fully classical model if by hypothesis an observer's knowledge of its state is appropriately limited.
Abstract The traditional formalism of quantum measurement (hereafter ‘TQM’) describes processes where some properties of quantum states are extracted and stored as classical information. While TQM is a natural and appropriate description of how humans interact with quantum systems, it is silent on the question of how a more general, quantum , agent would do so. How do we describe the observation of a system by an agent with the ability to store not only classical information but quantum states in its memory? In this paper, we extend the idea of measurement to a more general class of sensors for quantum agents which interact with a system in such a way that the agent’s memory stores information (classical or quantum) about the system under study. For appropriate sensory interactions, the quantum agent may ‘learn’ more about the system than would be possible under any set of classical measurements—but as we show, this comes at the cost of additional measurement disturbance. We experimentally demonstrate such a system and characterize the tradeoffs by considering the channel capacity required to erase the effect of a measurement.
Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.