Classification of all alternatives to the Born rule in terms of informational properties

Galley, Thomas D.; Masanes, Lluís

doi:10.22331/q-2017-07-14-15

Cited by 17 publications

(29 citation statements)

References 28 publications

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“…All these restricted representations were classified by some of the authors in ref. 35 . This amounts to a classification of all alternatives to the measurement postulate for single systems, that is, when the consistency constraints related to composite systems (9–14) are ignored.…”

Section: Methodsmentioning

confidence: 99%

The measurement postulates of quantum mechanics are operationally redundant

2019

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Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born’s rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as “unitary quantum mechanics”, and the assumption that ensembles on finite-dimensional Hilbert spaces are characterized by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability.

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Section: Methodsmentioning

confidence: 99%

The measurement postulates of quantum mechanics are operationally redundant

2019

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“…In fact, it is so strong that it is currently not clear whether there are any theories other than standard complex quantum theory and some of its subtheories [70] that satisfy it. A potential alternative can be found in the work by Galley and Masanes [57,58] who have pioneered an approach to construct composite state spaces directly in terms of group representations, without assuming tomographic locality.…”

Section: Discussionmentioning

confidence: 99%

Quantum computation is the unique reversible circuit model for which bits are balls

Krumm

Müller

2019

npj Quantum Inf

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The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible transformation in a connected matrix group; single wires carry quantum bits, i.e. states of a threedimensional Bloch ball; states on two or more wires are uniquely determined by local measurement statistics and their correlations. In this paper, we ask whether other types of computation are possible if we relax one of those characteristics (and keep all others), namely, if we allow wires to be described by d-dimensional Bloch balls, where d is different from three. Theories of this kind have previously been proposed as possible generalizations of quantum physics, and it has been conjectured that some of them allow for interesting multipartite reversible transformations that cannot be realized within quantum theory. However, here we show that all such potential beyond-quantum models of computation are trivial: if d is not three, then the set of reversible transformations consists entirely of single-bit gates, and not even classical computation is possible. In this sense, qubit quantum computation is an island in theoryspace.that tomographic locality forces computations to be contained in a class called AWPP [15,16], and that in some theories (satisfying additional axioms) higher-order interference does not lead to a speed-up in Grover's algorithm [17]. Further examples can be found e.g. in [18][19][20][21].

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“…Examples of GPTs (excepting classical and quantum theory) include Boxworld [15,[39][40][41][42], quantum theory over the field of real numbers [43][44][45] or quaternions [46], theories based on Euclidean Jordan algebras [35], quartic quantum theory [47], d-balls [16,48,49], density cubes [50] and quantum systems with modified measurements [51]. Amongst these, only Boxworld, quantum theory over real or quaternionic fields and theories based on Euclidean Jordan algebras are full theories, in that they have non-trivial composites.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is to provide tools to systematically explore the space of non-classical systems. Rather than generating examples of non-classical systems we can give full classifications of families of non-classical systems which share a common dynamical structure (pure states and reversible dynamics) but different probabilistic structures (measurements and measurement outcome probabilities); as done in [51] for systems which share the dynamical structure of quantum systems. We can thus obtain a richer picture of the space of non-classical systems, of which quantum systems are just one example.…”

Section: Introductionmentioning

confidence: 99%

How dynamics constrains probabilities in general probabilistic theories

Galley

Masanes

2021

Quantum

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We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.

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Classification of all alternatives to the Born rule in terms of informational properties

Cited by 17 publications

References 28 publications

The measurement postulates of quantum mechanics are operationally redundant

The measurement postulates of quantum mechanics are operationally redundant

Quantum computation is the unique reversible circuit model for which bits are balls

How dynamics constrains probabilities in general probabilistic theories

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