arXiv subjects

Christopher A. Fuchs

Publications and source records attributed to Christopher A. Fuchs.

At least 55 records · Page 3Linked to original sources

arXiv2009

The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.

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arXiv2009

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.

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arXiv2009

Quantum-Bayesian Coherence

In a quantum-Bayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics. It seems to get quite far.

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arXiv2009

Delirium Quantum

This pseudo-paper consists of excerpts drawn from two of my quantum-email samizdats. Section 1 draws a picture of a physical world whose essence is ``Darwinism all the way down.'' Section 2 outlines how quantum theory should be viewed in light of this, i.e., as being an expression of probabilism (in Bruno de Finetti or Richard Jeffrey's sense) all the way back up. Section 3 describes how the idea of ``identical'' quantum measurement outcomes, though sounding atomistic in character, nonetheless meshes well with a Jamesian style ``radical pluralism.'' Sections 4 and 5 further detail how quantum theory should not be viewed so much as a ``theory of the world,'' but rather as a theory of decision-making for agents immersed within a world of a particular character--the quantum world. Finally, Sections 6 and 7 attempt to sketch the very positive sense in which quantum theory is incomplete, but still just as complete is it can be. In total, I hope these heady speculations convey some of the excitement and potential I see for the malleable world quantum mechanics hints of.

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arXiv2009

Priors in quantum Bayesian inference

In quantum Bayesian inference problems, any conclusions drawn from a finite number of measurements depend not only on the outcomes of the measurements but also on a prior. Here we show that, in general, the prior remains important even in the limit of an infinite number of measurements. We illustrate this point with several examples where two priors lead to very different conclusions given the same measurement data.

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arXiv2007

Subjective probability and quantum certainty

In the Bayesian approach to quantum mechanics, probabilities--and thus quantum states--represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [A. Stairs, Phil. Sci. 50, 578 (1983)], which applies the Kochen-Specker theorem to an entangled bipartite system.

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arXiv2005

Influence-free states on compound quantum systems

Let Alice and Bob be able to make local quantum measurements and communicate classically. The set of mathematically consistent joint probability assignments (``states'') for such measurements is properly larger than the set of quantum-mechanical mixed states for the Alice-Bob system. It is canonically isomorphic to the set of positive (not necessarily completely positive) linear maps Phi from the bounded linear operators on Alice's space to those on Bob's, for which Tr Phi(I)=1. We review the fact that allowing classical communication is equivalent to enforcing ``no-instantaneous-signalling'' (``no--influence'') in the direction opposite the communication. We establish that in the subclass of ``decomposable'' states, i.e. convex combinations of positive states with "PTP" ones whose partial transpose is positive, the extremal states are just the extremal positive and extremal PTP states. We show that two such states, shared by the same pair of parties, cannot necessarily be combined as independent states (their tensor product) if the full set of quantum operations is allowed locally to each party. We use a framework of ``test spaces'' and states on these, suited for exhibiting the analogies and deviations of empirical probabilistic theories from classical probability theory. This leads to a deeper understanding of analogies between quantum mechanics and Bayesian probability theory. The existence of a ``most Bayesian'' quantum rule for updating states after measurement, and its association with the situation when information on one system is gained by measuring another, is a case of a general proposition holding for test spaces combined subject to the no-signalling requirement.

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arXiv2004

On the Quantumness of a Hilbert Space

We derive an exact expression for the quantumness of a Hilbert space (defined in quant-ph/0302092), and show that in composite Hilbert spaces the signal states must contain at least some entangled states in order to achieve such a sensitivity. Furthermore, we establish that the accessible fidelity for symmetric informationally complete signal ensembles is equal to the quantumness. Though spelling the most trouble for an eavesdropper because of this, it turns out that the accessible fidelity is nevertheless easy for her to achieve in this case: Any measurement consisting of rank-one POVM elements is an optimal measurement, and the simple procedure of reproducing the projector associated with the measurement outcome is an optimal output strategy. Two and epsilon elevator stories are added for entertainment.

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arXiv2003

Squeezing Quantum Information through a Classical Channel: Measuring the "Quantumness" of a Set of Quantum States

In this paper we propose a general method to quantify how "quantum" a set of quantum states is. The idea is to gauge the quantumness of the set by the worst-case difficulty of transmitting the states through a purely classical communication channel. Potential applications of this notion arise in quantum cryptography, where one might like to use an alphabet of states that promises to be the most sensitive to quantum eavesdropping, and in laboratory demonstrations of quantum teleportation, where it is necessary to check that quantum entanglement has actually been used in the protocol.

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arXiv2003

A de Finetti Representation Theorem for Quantum Process Tomography

In quantum process tomography, it is possible to express the experimenter's prior information as a sequence of quantum operations, i.e., trace-preserving completely positive maps. In analogy to de Finetti's concept of exchangeability for probability distributions, we give a definition of exchangeability for sequences of quantum operations. We then state and prove a representation theorem for such exchangeable sequences. The theorem leads to a simple characterization of admissible priors for quantum process tomography and solves to a Bayesian's satisfaction the problem of an unknown quantum operation.

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arXiv2003

Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.

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arXiv2003

The Quantumness of a Set of Quantum States

This article is an introduction to quant-ph/0302092. We propose to quantify how "quantum" a set of quantum states is. The quantumness of a set is the worst-case difficulty of transmitting the states through a classical communication channel. Potential applications of this measure arise in quantum cryptography, where one might like to use an alphabet of states most sensitive to quantum eavesdropping, and in lab demonstrations of quantum teleportation, where it is necessary to check that entanglement has indeed been used.

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arXiv2002

Conditions for compatibility of quantum state assignments

Suppose N parties describe the state of a quantum system by N possibly different density operators. These N state assignments represent the beliefs of the parties about the system. We examine conditions for determining whether the N state assignments are compatible. We distinguish two kinds of procedures for assessing compatibility, the first based on the compatibility of the prior beliefs on which the N state assignments are based and the second based on the compatibility of predictive measurement probabilities they define. The first procedure leads to a compatibility criterion proposed by Brun, Finkelstein, and Mermin [BFM, Phys. Rev. A 65, 032315 (2002)]. The second procedure leads to a hierarchy of measurement-based compatibility criteria which is fundamentally different from the corresponding classical situation. Quantum mechanically none of the measurement-based compatibility criteria is equivalent to the BFM criterion.

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arXiv2002

Quantum Mechanics as Quantum Information (and only a little more)

In this paper, I try once again to cause some good-natured trouble. The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or three statements of crisp physical (rather than abstract, axiomatic) significance. In this regard, no tool appears better calibrated for a direct assault than quantum information theory. Far from a strained application of the latest fad to a time-honored problem, this method holds promise precisely because a large part--but not all--of the structure of quantum theory has always concerned information. It is just that the physics community needs reminding. This paper, though taking quant-ph/0106166 as its core, corrects one mistake and offers several observations beyond the previous version. In particular, I identify one element of quantum mechanics that I would not label a subjective term in the theory--it is the integer parameter D traditionally ascribed to a quantum system via its Hilbert-space dimension.

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arXiv2002

The Anti-Vaxjo Interpretation of Quantum Mechanics

In this note, I try to accomplish two things. First, I fulfill Andrei Khrennikov's request that I comment on his "Vaxjo Interpretation of Quantum Mechanics," contrasting it with my own present view of the subject matter. Second, I try to paint an image of the hopeful vistas an information-based conception of quantum mechanics indicates.

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arXiv2001

The Quantum State of an Ideal Propagating Laser Field

We give a quantum information-theoretic description of an ideal propagating CW laser field and reinterpret typical quantum-optical experiments in light of this. In particular we show that contrary to recent claims [T. Rudolph and B. C. Sanders, Phys. Rev. Lett. 87, 077903 (2001)], a conventional laser can be used for quantum teleportation with continuous variables and for generating continuous-variable entanglement. Optical coherence is not required, but phase coherence is. We also show that coherent states play a priveleged role in the description of laser light.

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arXiv2001

The Quantum State of a Propagating Laser Field

Optical implementations of quantum communication protocols typically involve laser fields. However, the standard description of the quantum state of a laser field is surprisingly insufficient to understand the quantum nature of such implementations. In this paper, we give a quantum information-theoretic description of a propagating continuous-wave laser field and reinterpret various quantum-optical experiments in light of this. A timely example is found in a recent controversy about the quantum teleportation of continuous variables. We show that contrary to the claims of T. Rudolph and B. C. Sanders [Phys. Rev. Lett. 87, 077903 (2001)], a conventional laser can be used for quantum teleportation with continuous variables and for generating continuous-variable quantum entanglement. Furthermore, we show that optical coherent states do play a privileged role in the description of propagating laser fields even though they cannot be ascribed such a role for the intracavity field.

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arXiv2001

Quantum probabilities as Bayesian probabilities

In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally we give a Bayesian formulation of quantum-state tomography.

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