Distribution functions in physics: Fundamentals

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

The Theory of Quantum Information

We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

/pdf/symmetric-informationally-complete-quantum-measurements-4v18eptkry.pdf

Symmetric informationally complete quantum measurements

The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (ie, not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems

The statistical interpretation of quantum mechanics

Quantum mechanics in Hilbert space

We present criteria for comparing measurements on a given system from the point of view of the information they provide. These criteria lead to a concept ofinformational completeness of a set of observables, which generalizes the conventional concept of completeness. The entropy of a state with respect to an arbitrary sample space of potential measurement outcomes is defined, and then studied in the context of configuration space and fuzzy stochastic phase space.

https://link.springer.com/content/pdf/10.1007/BF01807146.pdf

Information-Theoretical Aspects of Quantum Measurement

Stochastic quantum mechanics and quantum spacetime

Quantum Mechanics in Hilbert Space.

It is shown that the extremal phase-space representations of quantum mechanics can be expressed in terms of wave-functions on L2-spaces which are embedded in L2(Γ). In L2(Γ) all these representations are restrictions of a globally defined representation of the canonical commutation relations. The master Liouville space B 2(Γ) over L2(Γ) can accommodate representations of both classical and quantum statistical mechanics, and serves as a medium for their comparison. As a specific example, a Boltzmann-type equation on B 2(Γ) is considered in the classical as well as quantum context.

Eduard Prugovečki

Papers

Quantum mechanics in Hilbert space

Information-Theoretical Aspects of Quantum Measurement

Stochastic quantum mechanics and quantum spacetime

Quantum Mechanics in Hilbert Space.

Classical and quantum statistical mechanics in a common Liouville space