Contextuality and the fundamental theorems of quantum mechanics

Döring, Andreas; Frembs, Markus

doi:10.1063/5.0012855

Cited by 6 publications

(9 citation statements)

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“…Only commuting operators can be measured jointly, i.e. those that are contained in a commutative von Neumann subalgebra-a context [12]. 6 The constraints in Gleason's theorem are therefore noncontextuality constraints: a projection p ∈ P(N ) is assigned a probability µ(p) independent of the context that p lies in: µ V (p) = µ Ṽ(p) = µ(p) whenever p ∈ Ṽ, V. Next, we formalise the idea that a measure µ : P(N ) → [0, 1] corresponds to a collection of probability measures over V(N ).…”

Section: Definitionmentioning

confidence: 99%

Gleason’s theorem for composite systems

Frembs,

Döring

2023

J. Phys. A: Math. Theor.

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Gleason’s theorem [A. Gleason, J. Math. Mech., 6, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that ﬁnitely additive measures on the projection lattice P(H) extend to positive linear functionals on the algebra of bounded operators B(H). Over many years, and by the eﬀort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: ﬁrst, we extend its scope to dilations in the sense of Naimark [M. A. Naimark, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 41, 359 (1943)] and Stinespring [W. F. Stinespring, Proc. Am. Math. Soc., 6, 211 (1955)] and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition [E. M. Alfsen and F. W. Shultz, Commun. Math. Phys., 194, 87 (1998)]. We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

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Section: Definitionmentioning

confidence: 99%

Gleason’s theorem for composite systems

Frembs,

Döring

2023

J. Phys. A: Math. Theor.

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“…If J = J (A) for an associative algebra A, then the Jordan algebra is called special, otherwise it is called exceptional. 17 In particular, every C * -(and von Neumann) 18 algebra defines a JB(W) algebra: a JB(W) algebra is a (weakly closed) Jordan algebra that is also a…”

Section: B Entanglement and Time Orientationsmentioning

confidence: 99%

“…Recall that for A = B(H A ) (in particular, for A = M n (C)) every such one-parameter group is given by conjugation with a unitary or anti-unitary operator by Wigner's theorem [7,64]. In fact, Wigner's theorem holds on the level of Jordan algebras [18,40]. In A, we obtain one-parameter groups of the form…”

Section: B Entanglement and Time Orientationsmentioning

confidence: 99%

“…It is interesting to note that Bell's theorem holds for states over C * -algebras as long as one of them is commutative[4]. We discuss the relation with Bell's theorem and Bell nonlocality elsewhere[18].…”

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confidence: 99%

“…For an extensive study of Jordan algebras, see[42] 17. The prototypical exceptional Jordan algebra is the so-called Albert algebra H 3 (O)[1] 18. Recall that a C * -algebra is an involutive Banach algebra (closed in norm) satisfying the defining C *property, x * x = x 2 .…”

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confidence: 99%

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Bipartite entanglement and the arrow of time

Frembs¹

2022

Preprint

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We provide a new perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett., 77, 1413(1996] and in the classification of quantum within more general non-signalling bipartite correlations [M. Frembs and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. We discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.

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Variations on the Choi–Jamiołkowski isomorphism

Frembs,

Cavalcanti

2024

J. Phys. A: Math. Theor.

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The Choi-Jamiolkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗-algebras they arise from, which are distinguished by their order of composition.A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)] further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, cruciallydepends on the dynamical (compositional) structure in C∗-algebras. This in turn lies at the heart of the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiolkowski [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. Here, we overcome this subtle but pervasive issue in a number of ways: first, we prove a versionof Choi’s theorem for Jamiolkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗-algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.

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Contextuality and the fundamental theorems of quantum mechanics

Cited by 6 publications

References 50 publications

Gleason’s theorem for composite systems

Gleason’s theorem for composite systems

Bipartite entanglement and the arrow of time

Variations on the Choi–Jamiołkowski isomorphism

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