Quartic quantum theory: an extension of the standard quantum mechanics

Życzkowski, Karol

doi:10.1088/1751-8113/41/35/355302

Cited by 64 publications

(75 citation statements)

References 61 publications

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“…Of course, in all the papers mentioned the authors are somehow able to overcome this difficulty and find reasons for this particular dimension, but the derivation at this point always loses some of its compelling elegance. It is then only natural that toy theories with higher dimensional Bloch spheres as state spaces have also been studied [4][5][6][7]. In such theories the formula for the probabilities of experimental outcomes stays the same.…”

Section: Introduction: Hyperbitsmentioning

confidence: 99%

“Hyperbits”: The information quasiparticles

Pawłowski¹,

Winter²

2012

Phys. Rev. A

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Information theory has its particles, bits and qubits, just as physics has electrons and photons. However, in physics we have a special category of objects with no clear counterparts in information theory: quasiparticles. They are introduced to simplify complex emergent phenomena making otherwise very difficult calculations possible and providing additional insight into the inner workings of the system. We show that we can adopt a similar approach in information theory. We introduce the hyperbits, information quasiparticles which we prove to be a resource equivalent to entanglement and classical communication, and we give examples of how they can be used to simplify calculations and get more insight into communication protocols.

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Section: Introduction: Hyperbitsmentioning

confidence: 99%

“Hyperbits”: The information quasiparticles

Pawłowski¹,

Winter²

2012

Phys. Rev. A

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“…[32]). Also, Dunkl has raised the issue of whether or not nonnegativity of the determinant of the partial transpose is equivalent to separability, as it is known to be in the two-rebit and two-qubit cases [17].)…”

Section: Estimation Of Separability Probabilities Using Conjecmentioning

confidence: 99%

Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities

Slater

Dunkl

2012

J. Phys. A: Math. Theor.

122

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Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |ρ| k |ρ P T | n , k, n = 0, 1, 2, 3, . . ., P T denoting partial transpose, for both generic (9-dimensional) two-rebit (α = 1 2 ) and generic (15-dimensional) two-qubit (α = 1) density matrices ρ. The results are, then, incorporated by Dunkl into a general formula (Appendix D 6), parameterized by k, n and α, with the case α = 2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter α fixed, the induced univariate moments (|ρ||ρ P T |) n and |ρ P T | n are inputted into a Legendre-polynomialbased (least-squares) probability-distribution reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding α-specific separability probability estimates. Since, as the number of inputted moments grows, estimates based on the variable |ρ||ρ P T | strongly decrease, while ones employing |ρ P T | strongly increase (and converge faster), the gaps between upper and lower estimates diminish, yielding sharper and sharper bounds. Remarkably, for α = 2, with the use of 2,325 moments, a separability-probability lower-bound 0.999999987 as large as 43, 195302 (2010)).

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“…We introduce in this section a family of GPTs that we call hypersphere theories (HSTs). These theories have been studied before [13,16,19,[33][34][35][36][37][38]. The state space of single systems in HSTs is defined as the unit ball of dimension n, r r 1…”

Section: Hypersphere Theoriesmentioning

confidence: 99%

“…We consider a particular class of GPTs in which the state space of the communicating systems corresponds to an Euclidean ball of arbitrary dimension n .  Î These hypersphere theories (HSTs), as we call them here, have been previously investigated [13,16,19,[33][34][35][36][37][38]. They have important physical motivations.…”

Section: Introductionmentioning

confidence: 99%

Hyperdense coding and superadditivity of classical capacities in hypersphere theories

2015

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In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit. We study this feature in the broader framework of general probabilistic theories. We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidian hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory). We show that a single n-ball can encode at most one bit of information, independently of n. We introduce a bipartite extension of such theories for which there exist dense coding protocols such that n log 1 2 + ( ) bits are communicated if entanglement is previously shared by the communicating parties. For n 3, > these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit. We call this phenomenon hyperdense coding (HDC). Our HDC protocols imply superadditive classical capacities: two entangled systems can encode n log 1 2 2 + > ( ) bits, even though each system individually encodes at most one bit. In our examples, HDC and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility.

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Quartic quantum theory: an extension of the standard quantum mechanics

Cited by 64 publications

References 61 publications

“Hyperbits”: The information quasiparticles

“Hyperbits”: The information quasiparticles

Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities

Hyperdense coding and superadditivity of classical capacities in hypersphere theories

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