A Quantum-Bayesian Route to Quantum-State Space

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent's personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.Comment: 7 pages, 1 figure, to appear in Foundations of Physics; this is a condensation of the argument in arXiv:0906.2187v1 [quant-ph], with special attention paid to making all assumptions explici

Efficient measurements, purification, and bounds on the mutual information

When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers average Shannon entropy for the encoding ensemble. This reduction, being the {\em mutual information}, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by re-writing a bound derived by Hall [Phys. Rev. A {\bf 55}, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. {\bf 76}, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy.Comment: 4 pages, Revtex4. v3: rewritten and reinterpreted somewha

Continuous variable quantum cryptography

We propose a quantum cryptographic scheme in which small phase and amplitude modulations of CW light beams carry the key information. The presence of EPR type correlations provides the quantum protection.Comment: 8 pages, 3 figure

Quantum-Limited Measurement and Information in Mesoscopic Detectors

We formulate general conditions necessary for a linear-response detector to reach the quantum limit of measurement efficiency, where the measurement-induced dephasing rate takes on its minimum possible value. These conditions are applicable to both non-interacting and interacting systems. We assess the status of these requirements in an arbitrary non-interacting scattering based detector, identifying the symmetries of the scattering matrix needed to reach the quantum limit. We show that these conditions are necessary to prevent the existence of information in the detector which is not extracted in the measurement process.Comment: 13 pages, 1 figur

Quantum copying: Fundamental inequalities

How well one can copy an arbitrary qubit? To answer this question we consider two arbitrary vectors in a two-dimensional state space and an abstract copying transformation which will copy these two vectors. If the vectors are orthogonal, then perfect copies can be made. If they are not, then errors will be introduced. The size of the error depends on the inner product of the two original vectors. We derive a lower bound for the amount of noise induced by quantum copying. We examine both copying transformations which produce one copy and transformations which produce many, and show that the quality of each copy decreases as the number of copies increases.Comment: 5 pages + 1 figure, LaTeX with revtex, epsfig submitted to Phys. Rev.

D-Branes on K3-Fibrations

B-type D-branes are constructed on two different K3-fibrations over IP_1 using boundary conformal field theory at the rational Gepner points of these models. The microscopic CFT charges are compared with the Ramond charges of D-branes wrapped on holomorphic cycles of the corresponding Calabi-Yau manifold. We study in particular D4-branes and bundles localized on the K3 fibers, and find from CFT that each irreducible component of a bundle on K3 gains one modulus upon fibration over IP_1. This is in agreement with expectations and so provides a further test of the boundary CFT.Comment: 16p, harvmac, tables.tex; typos corrected, refs added, discussion about moduli spaces improve

Universality of optimal measurements

We present optimal and minimal measurements on identical copies of an unknown state of a qubit when the quality of measuring strategies is quantified with the gain of information (Kullback of probability distributions). We also show that the maximal gain of information occurs, among isotropic priors, when the state is known to be pure. Universality of optimal measurements follows from our results: using the fidelity or the gain of information, two different figures of merits, leads to exactly the same conclusions. We finally investigate the optimal capacity of $N$ copies of an unknown state as a quantum channel of information.Comment: Revtex, 5 pages, no figure

Minimal Informationally Complete Measurements for Pure States

We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PSI-complete) POVM. We show that a measurement with 2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D outcomes. We also consider PSI-complete POVMs that have only rank-one POVM elements and construct an example with 3D-2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PSI-complete POVM is left open.Comment: 2 figures, submitted for the Asher Peres festschrif

Glass transition in the quenched and annealed version of the frustrated lattice gas model

In this paper we study the 3d frustrated lattice gas model in the annealed version, where the disorder is allowed to evolve in time with a suitable kinetic constraint. Although the model does not exhibit any thermodynamic transition it shows a diverging peak at some characteristic time in the dynamical non-linear susceptibility, similar to the results on the p-spin model in mean field and Lennard-Jones mixture recently found by Donati et al. [cond-mat/9905433]. Comparing these results to those obtained in the model with quenched interactions, we conclude that the critical behavior of the dynamical susceptibility is reminiscent of the thermodynamic transition present in the quenched model, and signaled by the divergence of the static non-linear susceptibility, suggesting therefore a similar mechanism also in supercooled glass-forming liquids.Comment: 8 pages, 14 figure

A toy model for quantum mechanics

The toy model used by Spekkens [R. Spekkens, Phys. Rev. A 75, 032110 (2007)] to argue in favor of an epistemic view of quantum mechanics is extended by generalizing his definition of pure states (i.e. states of maximal knowledge) and by associating measurements with all pure states. The new toy model does not allow signaling but, in contrast to the Spekkens model, does violate Bell-CHSH inequalities. Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations.Comment: in which the author breaks his vow to never use the words "ontic" and "epistemic" in publi