Foundations of Quantum Mechanics and Ordered Linear Spaces

Hartkämper, A.; Neumann, Holger

doi:10.1007/3-540-06725-6

Cited by 27 publications

(6 citation statements)

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“…A natural way to extend the domain of this function is to symmetrise the set under consideration, and consider the gauge Γ D∪(−D) instead-since the origin is now contained in the convex hull of the set, the function takes finite values for all Hermitian matrices. In fact, Γ D∪(−D) is simply equal to the trace norm itself [54], with the domain extended to H. One can go a step further and, instead of limiting ourselves to the real vector space of Hermitian matrices, define the set S = {|a b|} where |a , |b are any normalised vectors. The gauge Γ S is then precisely the trace norm • 1 in its most general formulation, which has full domain in C d×d .…”

Section: Selection Of Gaugesmentioning

confidence: 99%

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Convex geometry of quantum resource quantification

Regula

2017

J. Phys. A: Math. Theor.

108

166

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We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resourcefree states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that several of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, characterising the dual sets of resource witnesses.We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof-extended negativity, a measure of k-coherence which generalises the 1 norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.

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Section: Selection Of Gaugesmentioning

confidence: 99%

“…If the set S + spans the whole space of Hermitian matrices, the gauge Γ S+∪(−S+) defines a valid norm for H called the base norm [54]. Base norms have found a variety of uses in state and channel discrimination [49,[55][56][57].…”

Section: Base Norms and Robustnessmentioning

confidence: 99%

Convex geometry of quantum resource quantification

Regula

2017

J. Phys. A: Math. Theor.

108

166

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“…The only requirement for GPTs is the convexity for primitive notions of states and effects, and there are in general not assumed any Hilbert space structures or operator algebraic properties. In this sense, GPTs are a more general framework than quantum theory and classical theory, and play an active role in the study of quantum foundations [22,23,24,25,26,27,28,29,30,31] 1 after their initial proposition and development in the 1960s and 1970s [32,33,34,35,36,37]. 2 In this chapter, we explore the mathematical foundations of GPTs in detail to show how they give the most intuitive and fundamental description of nature.…”

Section: Generalized Probabilistic Theoriesmentioning

confidence: 99%

Convexity and uncertainty in operational quantum foundations

Takakura¹

2022

Preprint

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List of papersThis thesis is based on the following papers:1. (Reproduced from [1], with the permission of AIP Publishing) Ryo Takakura, Takayuki Miyadera, "Preparation Uncertainty Implies Measurement Uncertainty in a Class of Generalized Probabilistic Theories", Journal of Mathematical Physics, 61, 082203 (2020); 2. ([2]) Ryo Takakura, Takayuki Miyadera, "Entropic uncertainty relations in a class of generalized probabilistic theories", Journal of Physics A: Mathematical and Theoretical, 54, 315302 (2021); 3. ([3]) Teiko Heinosaari, Takayuki Miyadera, Ryo Takakura, "Testing incompatibility of quantum devices with few states", Physical Review A, 104, 032228 (2021); 4. ([4]) Ryo Takakura, "Entropy of mixing exists only for classical and quantumlike theories among the regular polygon theories",

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“…In this family, it is possible iff the set of monomials is symmetric under a permutation of indices. Observe that in the rows (1,3,6,8) the monomials in f have this feature, and in the rows (2,4,5,7) this applies to the monomials in g. To kill the part f or g of a row, one has to take monomials related by permutation of indices with opposite coefficients, so to consider combinations of functions of the form:…”

Section: Lemma 2 the Following Statements Holdsmentioning

confidence: 99%

“…It is therefore clear that the knowledge of positive maps B(K)→B(H) allows one to classify states of a composed quantum system living in H⊗K. Unfortunately, in spite of the considerable effort, the structure of positive maps is rather poorly understood [5][6][7][8][9][10][11][12][13]. For recent papers about positive maps in entanglement theory, see e.g.…”

Section: Introductionmentioning

confidence: 99%