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arXiv:quant-ph/0310025

Least paradoxical states of the Schrödinger cat

Andrey Pereverzev

Published Oct 3, 2003• 0 citations
arXivHTMLPDF

Abstract

Modeling the Schrödinger cat by a two state system and assuming that the cat is coupled to the environment we look for the least paradoxical states of the Schrödinger cat in the following way. We require the reduced density matrix of the cat for one of the two states in the superposition to be the same as the one for the total state while distinct from the reduced density matrix of the cat for the other state in the superposition. We then look for the reduced density matrices for which the cat is as alive as possible for the first state (and as dead as possible for the second state). The resulting states are those in which the probability for the cat to be alive (or dead) is $1/2+\sqrt 2/4\approx 0.854$

DOI
10.48550/arxiv.quant-ph/0310025

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Modeling the Schrödinger cat by a two state system and assuming that the cat is coupled to the environment we look for the least paradoxical states of the Schrödinger cat in the following way. We require the reduced density matrix of the cat for one of the two states in the superposition to be the same as the one for the total state while distinct from the reduced density matrix of the cat for the other state in the superposition. We then look for the reduced density matrices for which the cat is as alive as possible for the first state (and as dead as possible for the second state). The resulting states are those in which the probability for the cat to be alive (or dead) is 1/2+2/4≈0.8541/2+\sqrt 2/4\approx 0.8541/2+2​/4≈0.854
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.
The Schrödinger's Cat paradox was proposed in 1935 by Edwin Schrodinger, one of the founders of quantum mechanics, as an attempt to visualize the macroscopic realization of a quantum superposition state. A cat is placed in a sealed box together with a vial of poison. A two-state particle (e.g. an electron) is sent into a detector in the box resulting either in a broken or an intact vial and a dead or live cat, respectively. The main problem consists in whether the superposition state of a microscopic particle can be transferred upon the macroscopic cat, that is, whether the cat can exist in a superposition state, being simultaneously dead and alive. Since the standard Copenhagen interpretation is unable to assign any reality to the quantum superposition state, the paradox finds no resolution within the regime of this interpretation. Von Neumann's insistence on the uniform treatment of both microscopic (quantum) and macroscopic (classical) objects according to the laws of quantum mechanics provides a more consistent framework for the resolution of the paradox. In particular, the discovery of the phenomenon of decoherence, whereby the disappearance of the quantum interferences at the macro level is accounted for, suggests the onset of an extremely efficient interference relaxation process (10-23 s) upon the interaction of the two state particle with the detector. As a result, Schrodinger's cat can exist macroscopically either as dead or alive and never as a combination of both. Decoherence not only aids the resolution of the Schrodinger's Cat paradox but also sheds light upon the mechanisms by which the macro-world emerges from the microscopic quantum realm.
Abstract In this paper we re-investigate the core of Schrödinger’s “cat paradox”. We argue that one has to distinguish clearly between superpositions of macroscopic cat states |☺〉 + |☹〉 and superpositions of entangled states |☺, ↑〉 + |☹, ↓〉 which comprise both the state of the cat (☺=alive, ☹=dead) and the radioactive substance (↑=not decayed, ↓=decayed). It is shown, that in the case of the cat experiment recourse to decoherence or other mechanisms is not necessary in order to explain the absence of macroscopic superpositions. Additionally, we present modified versions of two quantum optical experiments as experimenta crucis. Applied rigorously, quantum mechanical formalism reduces the problem to a mere pseudo-paradox.
This paper reviews and suggests a resolution of the problem of definite outcomes of measurement. This problem, also known as ‘Schrodinger’s cat’, has long posed an apparent paradox because the state resulting from a measurement appears to be a quantum superposition in which the detector is in two macroscopically distinct states (alive and dead in the case of the cat) simultaneously. Many alternative interpretations of the quantum mathematical formalism, and several alternative modifications of the theory, have been proposed to resolve this problem, but no consensus has formed supporting any one of them. Applying standard quantum theory to the measurement state, together with the analysis and results of decades of nonlocality experiments with pairs of entangled systems, this paper shows the entangled measurement state is not a paradoxical macroscopic superposition of states. It is instead a phase-dependent superposition of correlations between states of the subsystems. Thus Schrodinger’s cat is a non-paradoxical ‘macroscopic correlation’ in which one of the two correlated systems happens to be a detector. This insight resolves the problem of definite outcomes but it does not entirely resolve the measurement problem because the entangled state is still reversible.
Abstract Complementarity tells us we cannot know precisely the values of all the properties of a quantum object at the same time: the precise determination of one property implies that the value of some other (complementary) property is undefined. E.g. the precise knowledge of the position of a particle implies that its momentum is undefined. Here we show that a Schrödinger cat has a well defined value of a property that is complementary to its “being dead or alive” property. Then, thanks to complementarity, it has an undefined value of the property “being dead or alive”. In other words, the cat paradox is explained through quantum complementarity: of its many complementary properties, any quantum system, such as a cat, can have a well defined value only of one at a time. Schrödinger’s cat has a definite value of a property which is complementary to “being dead or alive”, so it is neither dead nor alive. Figuratively one can say it is both dead and alive. While this interpretation only uses textbook concepts (the Copenhagen interpretation), apparently it has never explicitly appeared in the literature. We detail how to build an Arduino based simulation of Schrödinger’s experiment based on these concepts for science outreach events.