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General measure for macroscopic quantum states beyond "dead and alive"

Pavel Sekatski, Benjamin Yadin, Marc-Olivier Renou, Wolfgang Dür, Nicolas Gisin, Froewis, Florian

Published Jan 1, 2018• 4 citations
OpenAlex

Abstract

We consider the characterization of quantum superposition states beyond the pattern "dead and alive". We propose a measure that is applicable to superpositions of multiple macroscopically distinct states, superpositions with different weights as well as mixed states. The measure is based on the mutual information to characterize the distinguishability between multiple superposition states. This allows us to overcome limitations of previous proposals, and to bridge the gap between general measures for macroscopic quantumness and measures for Schr"odinger-cat type superpositions. We discuss a number of relevant examples, provide an alternative definition using basis-dependent quantum discord and reveal connections to other proposals in the literature. Finally, we also show the connection between the size of quantum states as quantified by our measure and their vulnerability to noise.

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We consider the characterization of quantum superposition states beyond the pattern "dead and alive". We propose a measure that is applicable to superpositions of multiple macroscopically distinct states, superpositions with different weights as well as mixed states. The measure is based on the mutual information to characterize the distinguishability between multiple superposition states. This allows us to overcome limitations of previous proposals, and to bridge the gap between general measures for macroscopic quantumness and measures for Schr"odinger-cat type superpositions. We discuss a number of relevant examples, provide an alternative definition using basis-dependent quantum discord and reveal connections to other proposals in the literature. Finally, we also show the connection between the size of quantum states as quantified by our measure and their vulnerability to noise.
Abstract In the Schrödinger cat gedanken experiment a “cat” is in a quantum superposition of two macroscopically distinct states. There is the apparent interpretation that the “cat” is not in one state or the other, “alive” or “dead”. Here this interpretation is proved objectively. I propose the following definition of macroscopic reality: first, that the “cat” is either dead or alive, the measurement revealing which; second, that measurements on other “cats” some distance away cannot induce the macroscopic change, “dead” to “alive” and vice versa, to the “cat”. The predictions of quantum mechanics are shown to be incompatible with this premise.
We propose a criterion which defines whether a superposition of two photonic components is macroscopic. It is based on the ability to discriminate these components with a particular class of ``classical'' detectors, namely, a photon number measurement with a resolution coarse-grained by noise. We show how our criterion can be extended to a measure of the size of macroscopic superpositions by quantifying the amount of noise that can be tolerated and taking the distinctness of two Fock states differing by N photons as a reference. After applying our measure to several well-known examples, we demonstrate that the superpositions which meet our criterion are very sensitive to phase fluctuations. This suggests that quantifying the macroscopicity of a superposition state through the distinguishability of its components with ``classical'' detectors not only is a natural measure but also explains why it is difficult to observe superpositions at the macroscopic scale.
Based on phase-space structures of quantum states, we propose a novel measure to quantify macroscopic quantum superpositions. Our measure simultaneously quantifies two different kinds of essential information for a given quantum state in a harmonious manner: the degree of quantum coherence and the effective size of the physical system that involves the superposition. It enjoys remarkably good analytical and algebraic properties. It turns out to be the most general and inclusive measure ever proposed that it can be applied to any types of multipartite states and mixed states represented in phase space.
In the Schrödinger-cat gedanken experiment a cat is in a quantum superposition of two macroscopically distinct states, alive and dead.The paradoxical interpretation of quantum mechanics is that the cat is not in one state or the other, alive or dead, immediately prior to its measurement. Because of this apparent defiance of macroscopic reality, quantum superpositions of states macroscopically distinct have generated much interest. Here we address the issue of proving a contradiction with macroscopic reality objectively, through the testable predictions of quantum mechanics. We consider the premises of macroscopic reality (that the ``cat'' is either ``dead'' or ``alive'', the measurement indicating which) and macroscopic locality (that simultaneous measurements some distance away cannot induce the macroscopic change, ``dead'' to ``alive'' and vice versa, to the ``cat''). The predictions of quantum mechanics for certain states, generated using states exhibiting continuous variable entanglement, are shown to be incompatible with the predictions of all theories based on this dual premise. Our proof is along the lines of Bell's theorem, but where all relevant measurements give macroscopically distinct results.
Schrödinger's famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger's cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger's language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of 10810^8108 photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.
Recent experiments claiming the formation of quantum superposition states in near-macroscopic systems raise the question of how the sizes of general quantum superposition states in an interacting system are to be quantified. We propose here a measure of size for such superposition states that is based on what measurements can be performed to probe and distinguish the different branches of the state. The measure allows the comparison of the effective size for superposition states in very different physical systems. It can be applied to a very general class of superposition states and reproduces known results for near-ideal cases. Comparison with a prior measure based on analysis of coherence between branches indicates that significantly smaller effective superposition sizes result from our measurement-based measure. Application to a system of interacting bosons in a double-well trapping potential shows that the effective superposition size is strongly dependent on the relative magnitude of the barrier height and interparticle interaction.