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arXiv:1111.3965 · Physical Review Letters

Quantum measurement occurrence is undecidable

Jens Eisert, Markus P. Müller, Christian Gogolin

Published Jun 25, 2012• 50 citations
arXivHTMLPDF

Abstract

In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable.

DOI
10.1103/physrevlett.108.260501

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In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable.
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The problem of measurement taken at face value shows clearly that there is an inconsistency inside the quantum formalism.The phenomenon of decoherence is often presented as a solution to it.A widely debated question is to decide between two different interpretations.The first one is to consider that the decoherence process has the effect to actually project a superposed state into one of its classically interpretable component, hence doing the same job as the reduction postulate.For the second one, decoherence is only a way to show why no macroscopic superposed state can be observed and so, to explain the classical appearance of the macroscopic world, while the quantum entanglement between the system, the apparatus and the environment never disappears.In this case, explaining why only one single definite outcome is observed remains to do.In this paper, we examine arguments for and against both interpretations and defend a position according to which the outcome that is observed is relative to the observer in close parallel to the Everett interpretation.
Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories. The apparent paradoxical situation gets resolved once it is perceived that the reducibility of Halting problem changes with varying resources available for computations in different theories. In the end, we propose a multi-agent game where winnability of the player having access to only classical resources is undecidable while quantum resources provide a perfect winning strategy.
A discussion of quantum measurement theory is presented. It is argued that the quantum measurement problem consists in the failure of any formalism based on unitary evolution to describe fully the observed behaviour in measurement-like processes. It is shown that the problem is essentially concerned with the phase of the coherence between orthogonal eigenstates of an object/measuring device system. It is then proposed that the observed behaviour is fully described if the value of this phase is formally undecidable.