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What do entangled states superpose

Art Hobson

Published Feb 25, 2020• 2 citations
OpenAlex

Abstract

There is evidence that superpositions of states of non-composite quantum systems A exist and can be interpreted as the simultaneous existence of two or more physically distinct quantum states of A. This paper shows, however, that entangled superposition states have a quite different character from simple superposition states. Whereas a simple superposition superposes two coherent states of a non-composite object A, entangled states superpose correlations between states of two incoherent subsystems A and B of a composite system AB. Thus an entangled state is best conceptualized as a coherent or phase-dependent superposition of correlations between incoherent or phase-independent subsystem states.

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There is evidence that superpositions of states of non-composite quantum systems A exist and can be interpreted as the simultaneous existence of two or more physically distinct quantum states of A. This paper shows, however, that entangled superposition states have a quite different character from simple superposition states. Whereas a simple superposition superposes two coherent states of a non-composite object A, entangled states superpose correlations between states of two incoherent subsystems A and B of a composite system AB. Thus an entangled state is best conceptualized as a coherent or phase-dependent superposition of correlations between incoherent or phase-independent subsystem states.
The entangled state that results when a detector measures a superposed quantum system has spawned decades of concern about the problem of definite outcomes or "Schrodinger's cat." This state seems to describe a detector in an indefinite or "smeared" situation of indicating two macroscopic configurations simultaneously. This would be paradoxical. Since all entangled states are known to have nonlocal properties, and since measurements have obvious nonlocal characteristics, it's natural to turn to nonlocality experiments for insight into this question. Unlike the measurement situation where the phase is fixed at zero for perfect correlations, nonlocality experiments cover the full range of superposition phases and can thus show precisely what entangled states superpose. For two-state systems, these experiments reveal that the measurement state is not a superposition of two macroscopically different detector states but instead a superposition of two coherent correlations between distinct detector states and corresponding system states. In the measurement situation (i.e. at zero phase), and assuming the Schrodinger's cat scenario, the entangled state can be read as follows: An undecayed nucleus is perfectly correlated with an alive cat, AND a decayed nucleus is perfectly correlated with a dead cat, where "AND" indicates the superposition. This is not paradoxical.
Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.
Continuous variable entangled states and especially entangled coherent states have attracted increasing interest in the Geld of quantum information processing. The characteristic features of the superposition of quantum states can be found in the literature. Because of these significant findings, we introduce and investigate a special superposition of multipartite entangled coherent states. We prove that the free-traveling optical field scheme can generate such a superposed state. Using a geometric measure of entanglement, we then investigate the correlation behavior of the superposed state.
Theory and experiment both demonstrate that an entangled quantum state of two subsystems is neither a superposition of states of its subsystems nor a superposition of composite states but rather a coherent superposition of nonlocal correlations between incoherently mixed local states of the two subsystems. Thus, even if one subsystem happens to be macroscopic as in the entangled "Schrodinger's cat" state resulting from an ideal measurement, this state is not the paradoxical macroscopic superposition it is generally presumed to be. It is, instead, a "macroscopic correlation," a coherent quantum correlation in which one of the two correlated sub-systems happens to be macroscopic. This clarifies the physical meaning of entanglement: When a superposed quantum system A is unitarily entangled with a second quantum system B, the coherence of the original superposition of different states of A is transferred to different correlations between states of A and B, so the entangled state becomes a superposition of correlations rather than a superposition of states. This transfer preserves unitary evolution while permitting B to be macroscopic without entailing a macroscopic superposition. This resolves the "problem of outcomes" but is not a complete resolution of the measurement problem because the entangled state is still reversible.
Schrodinger's famous cat has long been misunderstood. According to quantum theory and experiments with entangled systems, an entangled state such as the Schrodinger's cat state is neither a superposition of states of either subsystem nor a superposition of compound states of the composite system, but rather a nonlocal superposition of correlations between pairs of states of the two subsystems. The entangled post-measurement state that results from an ideal measurement is not paradoxical, but is merely a coherent superposition of two statistical correlations at "zero phase angle," i.e. at 100% positive correlation. Thus the state of the radioactive nucleus and Schrodinger's cat is as follows: an undecayed nucleus is 100% positively correlated with an alive cat, and (i.e. superposed with) a decayed nucleus is 100% positively correlated with a dead cat. The superposition consists merely in the fact that both correlations are simultaneously true. Despite many published statements to the contrary, this superposition is not paradoxical. It is in fact what one expects intuitively.
Given a bipartite quantum state (in arbitrary dimension) and a decomposition of it as a superposition of two others, we find bounds on the entanglement of the superposition state in terms of the entanglement of the states being superposed. In the case that the two states being superposed are biorthogonal, the answer is simple, and, for example, the entanglement of the superposition cannot be more than one ebit more than the average of the entanglement of the two states being superposed. However, for more general states, the situation is very different.