Generalized quantum mechanics

Mielnik, Bogdan

doi:10.1007/bf01646346

Cited by 208 publications

(127 citation statements)

References 23 publications

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“…If the propositions of classical mechanics are the subsets of Γ (classical phase space), why cannot we consider the convex subsets of the convex set of states? It seems, after all, that convexity is an important feature of quantum mechanics [15], [16], and [17]. And as will be seen below, this "convexification" of the lattice, allows for an algebraic characterization of entanglement.…”

Section: The Lattice Of Convex Subsetsmentioning

confidence: 93%

“…We think that this approach is suitable in order to consider decoherence or entangled systems from a quantum logical and algebraic point of view. Furthermore, taking the convex set of states as an starting point could be of interest if we take into account that there exists a formulation of QM in terms of convex sets (see [15], [16] and [17]). This is an independent formulation of QM and has the advantage that it can include models of theories which cannot be represented by Hilbert spaces, as is the case of non linear generalizations of quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

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Convex Quantum Logic

2011

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In this work we study the convex set of quantum states from a quantum logical point of view. We consider an algebraic structure based on the convex subsets of this set. The relationship of this algebraic structure with the lattice of propositions of quantum logic is shown. This new structure is suitable for the study of compound systems and shows new differences between quantum and classical mechanics. This differences are linked to the nontrivial correlations which appear when quantum systems interact. They are reflected in the new propositional structure, and do not have a classical analogue. This approach is also suitable for an algebraic characterization of entanglement.

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Section: The Lattice Of Convex Subsetsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Convex Quantum Logic

2011

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“…It is not accidental that the argument is very similar to the one for faster-than-light effects in nonlinear quantum mechanics [14][15][16][17][18][19][20][21][22][23]. As shown by Mielnik [24] the densities ρ(x) = |ψ(x)| α , α = 2, are characteristic of a class of nonlinear Schrödinger equations. Mielnik's equations are 'nonlocal-looking but physically local' in the sense of [25], i.e.…”

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confidence: 89%

Security in Quantum Cryptography vs. Nonlocal Hidden Variables

Aerts¹,

Czachor²,

Pawłowski³

2007

AIP Conference Proceedings

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Abstract. In order to prove equivalence of quantum mechanics with nonlocal hidden-variable theories of a Bohm type one assumes that all the possible measurements belong to a restricted class: (a) we measure only positions of particles and (b) have no access to exact values of initial conditions for Bohm's trajectories. However, in any computer simulation based on Bohm's equations one relaxes the assumption (b) and yet obtains agreement with quantum predictions concerning the results of positional measurements. Therefore a theory where (b) is relaxed, although in principle allowing for measurements of a more general type, cannot be experimentally falsified within the current experimental paradigm. Such generalized measurements have not been invented, or have been invented but the information is qualified, but we cannot exclude their possibility on the basis of known experimental data. Since the measurements would simultaneously allow for eavesdropping in standard quantum cryptosystems, the arguments for security of quantum cryptography become logically circular: Bohm-type theories do not allow for eavesdropping because they are fully equivalent to quantum mechanics, but the equivalence follows from the assumption that we cannot measure hidden variables, which would be equivalent to the possibility of eavesdropping... Here we break the vicious circle by a simple modification of entangled-state protocols that makes them secure even if our enemies have more imagination and know how to measure hidden-variable initial conditions with arbitrary precision. Keywords THE FUNDAMENTAL PROBLEM OF QUANTUM CRYPTOGRAPHY?There is a general agreement that quantum mechanics can be replaced by a nonlocal hidden-variable theory [1]. Such a theory involves additional degrees of freedom that we assume to be beyond our reach. However, statistics of the standard measurements would not be changed even if we could access the hidden variables. The proof is trivial: In any computer simulation based on Bohm trajectories we know the initial conditions in question, and yet obtain the correct quantum probabilities [2] satisfying no-eavesdropping criteria. Paradoxically, the criteria are fulfilled even though the eavesdropping is possible [3][4][5][6][7]. There is no contradiction with Bell's theorem because Bohm's hidden variables are nonlocal. This nonlocality has to be taken into account in the computer simulation

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“…The COM approach has its roots in operational theories and has been shown to be useful to generalize many quantum mechanical notions mentioned above, such as teleportation protocols, no broadcasting, and no cloning theorems [8,22,23]. The geometrical approach based on convex sets can also be seen as a framework in which non-linear theories which generalize quantum mechanics, can be included, studied, and compared with it [24][25][26]. It is also important to remark that an axiomatization independent (and equivalent to) the von Neumann formalism can be given using the geometrical-operational approach [24][25][26].…”

Section: Introductionmentioning

confidence: 99%

“…The geometrical approach based on convex sets can also be seen as a framework in which non-linear theories which generalize quantum mechanics, can be included, studied, and compared with it [24][25][26]. It is also important to remark that an axiomatization independent (and equivalent to) the von Neumann formalism can be given using the geometrical-operational approach [24][25][26]. The importance of entanglement as a resource for measuring classicality of a state has been highlighted in [27].…”

Section: Introductionmentioning

confidence: 99%

Generalizing Entanglement via Informational Invariance for Arbitrary Statistical Theories

Holik¹,

Massri

Plastino

2014

TPHY

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Given an arbitrary statistical theory, different from quantum mechanics, how to decide which are the nonclassical correlations? We present a formal framework which allows for a definition of nonclassical correlations in such theories, alternative to the current one. This enables one to formulate extrapolations of some important quantum mechanical features via adequate extensions of "reciprocal" maps relating states of a system with states of its subsystems. These extended maps permit one to generalize i) separability measures to any arbitrary statistical model as well as ii) previous entanglement criteria. The standard definition of entanglement becomes just a particular case of the ensuing, more general notion.

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Generalized quantum mechanics

Cited by 208 publications

References 23 publications

Convex Quantum Logic

Convex Quantum Logic

Security in Quantum Cryptography vs. Nonlocal Hidden Variables

Generalizing Entanglement via Informational Invariance for Arbitrary Statistical Theories

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