Criteria to detect macroscopic quantum coherence, macroscopic quantum entanglement, and an Einstein-Podolsky-Rosen paradox for macroscopic superposition states

Abstract

According to classical theory, a system with two or more macroscopically distinct states available to it is in one of those states at all times. Quantum mechanics gives a different interpretation where the system can be in a superposition of such states. In this paper, we derive criteria in the form of inequalities to detect this effect, referred to as mesoscopic quantum coherence, where the states are (at least) mesoscopically distinct. Such criteria are also signatures of a mesoscopic Schrödinger cat paradox. We extend the treatment to consider definitions and criteria for mesoscopically entangled states, which are a subset of the states exhibiting mesoscopic quantum coherence. It is shown how the criteria (referred to as type I criteria) can be applied to detect the mesoscopic entanglement of systems prepared in NOON states, Greenberger-Horne-Zeilinger states, and entangled cat states involving coherent states. Here, a larger system C is entangled with a second system S, which can be small. The proposed definition of mesoscopic (macroscopic) entanglement involves the Schmidt decomposition, and can be applied to systems in a superposition of many states, only some pairs of which are mesoscopically (macroscopically) distinguishable. We prove that a higher-order Hillery-Zubairy entanglement criterion will detect mesoscopic entanglement of this type, and use this to demonstrate the mesoscopic entanglement of the ground state of a two-mode Bose-Einstein condensate. Finally, we explain how a subset of the type I criteria (called type II criteria) are EPR steering inequalities allowing realization of a mesoscopic or macroscopic version of the Einstein-Podolsky-Rosen paradox for macroscopic superposition states. Where the two systems C and S are spatially separated, we then use results from the literature to point out it is not possible to complete quantum mechanics using hidden-variable theories compatible with the assumption of locality between the two systems. The violation of certain Bell inequalities is sufficient to prove this result, which gives a stricter signature of the Schrödinger cat paradox and of mesoscopic quantum coherence. We call such signatures type III criteria.

Macroscopic delayed-choice and retrocausality: quantum eraser, Leggett-Garg and dimension witness tests with cat states

“…The results can be viewed consistently with weak macroscopic realism. Nonetheless, in Section VI we show that at the microscopic level of , EPR-type paradoxes can be constructed (similar to those discussed in [32,36]), based on the fringe pattern.…”

“…However, these paradoxes give inconsistencies for local causality, or local realism. Less has been done on paradoxes that illustrate the inconsistency between weak macroscopic realism and the incompleteness of quantum mechanics, although related examples were given for number-state superpositions in [36]. We expect such paradoxes may also be possible for NOON and GHZ states, and for the higher dimensional GHZ extensions with multiple particles at each site [80,81].…”

“…With the assumption that ϕ + and ϕ − are to be quantum states, the Heisenberg uncertainty relation (∆X A )(∆P A ) ≥ 1/2 applies to each state. Then, as explained in [36], for the ensemble of systems in a classical mixture of states ϕ + and ϕ − , it is readily proved that (∆X A ) ave (∆P A ) ≥ 1/2, where…”

“…The second plot (right) shows the full calculation for E 2 given by eqs. (35) and (36) based on the proposed method to measure β using XB, which assumes β to be sufficiently large. Here, we show E 2 versus α for β = 0.5, β = 1 and β = 2.…”

Retrocausal model of reality for quantum fields

“…Hence, one has an interpretation in which the system is not paradoxically in "both states at once," which gives a partial resolution of the measurement or Schrödinger cat paradox. A paradox remains, however, at the microscopic level (of orderh), in that the system cannot be viewed as being in one or other of two quantum states, that determine the system to be q 1 or q 2 [99][100][101].…”