Derivation of quantum theory with superselection rules

Nakahira, Kenji

doi:10.1103/physreva.101.022104

Cited by 10 publications

(8 citation statements)

References 51 publications

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“…The postulate (LT) can also be applied to complete other reconstructions of quantum mechanics that first derive the formally real Jordan algebras [1][2][3][4][5][6][7][8][9][10][11]. Some of these reconstructions already use local tomography, but a stronger version (see §5) which could be replaced by the less restrictive (LT).…”

Section: Discussionmentioning

confidence: 99%

“…Several approaches [1][2][3][4][5][6][7][8][9][10][11] succeed in deriving the need for formally real Jordan algebras. A further approach [12,13] also results in the formally real Jordan algebras, when it is restricted to the finite-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

“…The main result can be applied to complete reconstructions of quantum mechanics that first derive the formally real Jordan algebras [1][2][3][4][5][6][7][8][9][10][11]. In some of these approaches, local tomography has already been used.…”

Section: Introductionmentioning

confidence: 99%

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Local tomography and the role of the complex numbers in quantum mechanics

Niestegge¹

2020

Proc. R. Soc. A.

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Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but it is less restrictive than the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local tomography and the role of the complex numbers in quantum mechanics

Niestegge¹

2020

Proc. R. Soc. A.

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“…More recently, a third wave of reconstruction attempts took off [14,15,116], which can be seen as resurrecting the mathematical spirit of the first wave, while still embracing the principled underpinning that guides the second wave. Recently, there have been various other reconstructions of quantum theory from a variety of perspectives, for example [24,66,73,74,92,109,113].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing quantum theory from diagrammatic postulates

Selby

Scandolo

Coecke

2021

Quantum

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A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

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“…The maximum violation of this inequality that can be reached by any quantum state is 2 √ 2 [4]. A straightforward testing of Bell's inequality may not be possible for systems of indistinguishable particles, since rotations away from fixed particle number bases may not be allowed due to the restriction imposed by a superselection rule on indistinguishable systems [5][6][7][8][9][10][11]. Interestingly, Heaney, Lee, and Jaksch [12] derived a method for testing Bell inequalities for states of indistinguishable particles even within the superselection-rule restrictions.…”

Section: Introductionmentioning

confidence: 99%

Steering inequality for pairs of particle-number-superselection-rule restricted states

Kumari¹,

Sen²

2021

Preprint

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We consider violations of a Clauser-Horne-Shimony-Holt-type steering inequality for quantum states of systems of indistinguishable particles restricted by a particle-number-superselection rule. We check for violations in non-interacting Bose-Einstein condensate and N00N states, by using twocopies of the states for bypassing the superselection rule. The superselection rule prevents the states from maximally violating the steering inequality. But the steering inequality violations are higher than Bell inequality violations for the same states. This in particular implies, in certain cases, that visibilities of the steering inequality violations are higher than the same for Bell inequality violations, for admixtures with white noise. We also found that an increase in the number of particles in the non-interacting condensate states results in a decrease of the violation amount of the steering inequality.

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Derivation of quantum theory with superselection rules

Cited by 10 publications

References 51 publications

Local tomography and the role of the complex numbers in quantum mechanics

Local tomography and the role of the complex numbers in quantum mechanics

Reconstructing quantum theory from diagrammatic postulates

Steering inequality for pairs of particle-number-superselection-rule restricted states

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