Christopher A. Fuchs

University of Massachusetts Boston | UMB · Department of Physics

Never has there been such a friend to the open and liberal discussion of quantum foundations as Andrei Khrennikov. At his yearly meeting in Växjö—the longest running series of conferences on the subject in history!—one will find all kinds, from the crackpot to the philosopher to the Nobel prizewinner. Andrei’s philosophy has always been that everyo...

This paper expresses what a breath of fresh air it has been since a few phenomenological philosophers have started to engage with QBism. In service of the newfound discussion, the aim of this exposition is to lay out the structure of QBism as clearly as possible for that audience. In the process, we arrive at eight tenets for QBism: 1) A quantum st...

From the Text: How shall I tribute Andrei Khrennikov in this volume? With an email collection of course! But with what theme? It ought to be something big. One of the troubles of QBism's ontological program is that it is so sideways to the ways most run-of-the-mill philosophers of physics think, they don't even have the tools to parse its sentences...

The subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the s...

The subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the s...

We remark on John Earman's paper ``Quantum Bayesianism Assessed'' [The Monist 102 (2019), 403--423], illustrating with a number of examples that the quantum ``interpretation'' Earman critiques and the interpretation known as QBism have almost nothing to do with each other.

According to QBism, quantum states, unitary evolutions, and measurement operators are all understood as personal judgments of the agent using the formalism. Meanwhile, quantum measurement outcomes are understood as the personal experiences of the same agent. Wigner’s conundrum of the friend, in which two agents ostensibly have different accounts of...

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about th...

According to QBism, quantum states, unitary evolutions, and measurement operators are all understood as personal judgments of the agent using the formalism. Meanwhile, quantum measurement outcomes are understood as the personal experiences of the same agent. Wigner's conundrum of the friend, in which two agents ostensibly have different accounts of...

We describe a general procedure for associating a minimal informationally complete quantum measurement with a purely probabilistic representation of the Born rule. Such representations provide a way to understand the Born rule as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities a...

In the last five years of his life Itamar Pitowsky developed the idea that the formal structure of quantum theory should be thought of as a Bayesian probability theory adapted to the empirical situation that Nature's events just so happen to conform to a non-Boolean algebra. QBism too takes a Bayesian stance on the probabilities of quantum theory,...

It's sometimes said that the field of quantum information and computing ought to be called applied quantum foundations. That's because so many of the ideas that first arose when scientists began thinking deeply about the mysteries of quantum theory-entanglement, Bell inequality violations, parallel worlds, interference of probabilities, and quantum...

We study a class of quantum measurements that furnish probabilistic representations of finite-dimensional quantum theory. The Gram matrices associated with these Minimal Informationally Complete quantum measurements (MICs) exhibit a rich structure. They are "positive" matrices in three different senses, and conditions expressed in terms of them hav...

This is a slightly extended version of a review of Adam Becker's book What is Real? The Unfinished Quest for the Meaning of Quantum Physics, to appear in American Journal of Physics. The key addition is the reference list.

We describe a general procedure for associating a minimal informationally-complete quantum measurement (or MIC) and a set of linearly independent post-measurement quantum states with a purely probabilistic representation of the Born Rule. Such representations are motivated by QBism, where the Born Rule is understood as a consistency condition betwe...

In the Philosophical Investigations, Ludwig Wittgenstein wrote, “‘I’ is not the name of a person, nor ‘here’ of a place, .... But they are connected with names. ... [And] it is characteristic of physics not to use these words.” This statement expresses the dominant way of thinking in physics: Physics is about the impersonal laws of nature; the “I”...

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability repres...

Without Niels Bohr, QBism would be nothing. But QBism is not Bohr. This paper attempts to show that, despite a popular misconception, QBism is no minor tweak to Bohr's interpretation of quantum mechanics. It is something quite distinct. Along the way, we lay out three tenets of QBism in some detail: 1) The Born Rule---the foundation of what quantum...

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now fur...

This paper represents an elaboration of the lectures delivered by one of us (CAF) during "Course 197 -- Foundations of Quantum Physics" at the International School of Physics "Enrico Fermi" in Varenna, Italy, July 2016. Much of the material for it is drawn from arXiv:1003.5209, arXiv:1401.7254, and arXiv:1405.2390. However there are substantial add...

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, "Unperformed experiments have no results." The tools of modern quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres's dictum holds true. That expression is a con...

In the Philosophical Investigations, Ludwig Wittgenstein wrote, " 'I' is not the name of a person, nor 'here' of a place, .... But they are connected with names. ... [And] it is characteristic of physics not to use these words." This statement expresses the dominant way of thinking in physics: Physics is about the impersonal laws of nature; the "I"...

We compare the standard 1/2-efficient single beam splitter method for Bell-state measurement to a proposed 3/4-efficient extension [W. P. Grice, Phys. Rev. A 84, 042331 (2011)] in light of realistic experimental conditions. Our comparison is two-fold. First, we analyze the performance when one replaces number-resolving detectors with standard non-i...

We analyze an entanglement-based quantum key distribution (QKD) architecture that uses a linear chain of quantum repeaters employing photon-pair sources, spectral-multiplexing, linear-optic Bell-state measurements, multimode quantum memories, and classical-only error correction. Assuming perfect sources, we find an exact expression for the secret-k...

This is a reply to Michael Nauenberg's arXiv:1502.00123, to be published in the American Journal of Physics, in which he comments critically on our paper "An introduction to QBism with an application to the locality of quantum mechanics", Am. J. Phys. 82, 749--754 (2014) and arXiv:1311.5253.

In QBism (or Quantum Bayesianism) a quantum state does not represent an element of physical reality but an agent's personal probability assignments, reflecting his subjective degrees of belief about the future content of his experience. In this paper, we contrast QBism with hidden-variable accounts of quantum mechanics and show the sense in which Q...

In the quantum Bayesian (or QBist) conception of quantum theory, "quantum measurement" is understood not as a comparison of something pre-existent with a standard, but instead indicative of the creation of something new in the universe: Namely, the fresh experience any agent receives upon taking an action on the world. We explore the implications o...

We present simulations of rates for a quantum key distribution scheme using a frequency multiplexed repeater architecture with broadband solid-state quantum memories. We find that key can be generated over 1000 km with eight elementary links.

This document is the second installment of three in the Cerro Grande Fire Series. Like its predecessor arXiv:quant-ph/0105039, "Notes on a Paulian Idea," it is a collection of letters written to various friends and colleagues, most of whom regularly circuit this archive. The unifying theme of all the letters is that each has something to do with th...

We analyze a long-distance entanglement based quantum key distribution (QKD) architecture, which uses multiple linear-optic quantum repeaters, frequency-multiplexed entanglement sources, and classical error correction. We find an exact expression for the secret-key rate, and an analytical characterization of how errors propagate through noisy repea...

The end of the last decade saw a media frenzy over possibility of an H1N1 flu pandemic. The frenzy turned out to be misplaced, but it did serve to remind us of a basic truth: that a healthy body can be stricken with a fatal disease which to outward appearances is nearly identical to a common yearly annoyance. There are lessons here for quantum mech...

Over the last 10 years there has been an explosion of "operational reconstructions" of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind...

Although symmetric informationally complete positive operator valued measures (SIC-POVMs, or SICs for short) have been constructed in every dimension $\le 67$, a general existence proof remains elusive. The purpose of this paper is to show that the SIC-existence problem is equivalent to three other, on the face of it quite different problems. We ho...

This review explores some of the consequences and features of the quantum-Bayesian approach to quantum theory. This approach contends that the difficulties in the foundations of quantum theory arise from the difficulties in understanding the nature of probabilities. “Dutch-book” wager games are explored to illustrate the Bayesian view on probabilit...

We give an introduction to the QBist interpretation of quantum mechanics. We note that it removes the paradoxes, conundra, and pseudo-problems that have plagued quantum foundations for the past nine decades. As an example, we show in detail how it eliminates quantum "nonlocality".

Some time ago, Steven Weinberg wrote an article for the New York Review of Books with the title, “Symmetry: A `Key to Nature’s Secrets’.” So too, I would like to say of quantum information: Only by identifying Hilbert space’s most stringent and hard-to-attain symmetries will we be able to unlock quantum information’s deepest secrets and greatest po...

Some time ago, Steven Weinberg wrote an article for the New York Review of Books with the title, “Symmetry: A `Key to Nature’s Secrets’.” So too, I would like to say of quantum information: Only by identifying Hilbert space’s most stringent and hard-to-attain symmetries will we be able to unlock quantum information’s deepest secrets and greatest po...

In the Quantum-Bayesian interpretation of quantum theory (or QBism), 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...

A passionate and personal account of the early days of quantum information and quantum computing, this unique book is a collection of more than 500 letters between the author and many of the founders of these intriguing fields. Christopher A. Fuchs is one of the most penetrating modern thinkers on the philosophical foundations of quantum mechanics....

This paper collects into one place (most of) my answers to the questions Maximilian Schlosshauer posed in his interview volume, "Elegance and Enigma: The Quantum Interviews" (Springer, Frontiers Collection, 2011).

A symmetric informationally complete positive operator valued measure (SIC-POVM) is usually thought of as a highly symmetric structure in quantum state space. However Appleby, Flammia and Fuchs (J. Math. Phys. 52, 022202, 2011) have shown that the existence of a SIC-POVM in dimension d is equivalent to a proposition concerning the Lie Algebra gl(d,...

Providing quantum theory, and thus quantum information, with some meaning, can only be the result of adopting a well-defined approach to the notion of probability. From a subjectivistic Bayesian approach to the latter, it follows that not only quantum states, but also quantum operations, are structural elements of a subjective nature. Furthermore,...

Physicists have become accustomed to the idea that a theory’s content is most transparent when written in coordinate‐free language. But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues. This is particularly so for the Quantum Bayesian, or QBist, approach to quantum theory: One good coordinate...

This article reviews Coming of Age with Quantum Information: Notes on a Paulian Idea. by Christopher A. Fuchs 543 pp. Cambridge U. P., Cambridge, UK, 2011. Price: $70.00 (hardcover) ISBN 978-0-521-19926-1.

Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting P...

The probabilities a Bayesian agent assigns to a set of events typically change with time, for instance when the agent updates them in the light of new data. In this paper we address the question of how an agent's probabilities at different times are constrained by Dutch-book coherence. We review and attempt to clarify the argument that, although an...

The Weyl-Heisenberg (WH) group was used by Hermann Weyl to construct finite-dimensional quantum mechanics in the earliest days of the theory and, through its ubiquitous use in quantum information theory, is even more important today. While investigating properties of symmetric informationally-complete (SIC) measurements, we found a linear dependenc...

With the help of a certain mathematical structure in quantum information theory, there is a particularly elegant way to rewrite the quantum mechanical Born rule as an expression purely in terms of probabilities. In this way, one can in principle get rid of complex Hilbert spaces and operators as fundamental entities in the theory. In the place of a...

1. Letters to David Baker; 2. Letters to Howard Baker; 3. Letters to Howard Barnum; 4. Letters to Paul Benioff; 5. Letters to Charlie Bennett; 6. Letters to Herb Bernstein; 7. Letters to Doug Bilodeau; 8. Letters to Gilles Brassard; 9. Letters to Jeffrey Bub; 10. Letters to Carlton Caves; 11. Letters to Greg Comer; 12. Letters to Charles Enz; 13. L...

Generalized quantum measurements (also known as POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the SIC-POVM, is the most compact...

Imagine...a world without density matrices! A scheme to perform arbitrary POVMs is proposed and a SIC-POVM is implemented on a qutrit. The Quantum Law of Total Probability is verified.

This article summarizes the Quantum Bayesian point of view of quantum mechanics, with special emphasis on the view's outer edges---dubbed QBism. QBism has its roots in personalist Bayesian probability theory, is crucially dependent upon the tools of quantum information theory, and most recently, has set out to investigate whether the physical world...

Symmetric informationally-complete (SIC) sets of quantum states have received growing attention due to their many nice properties. For prime dimensions, we add another property to the list: Weyl-Heisenberg covariant SIC states achieve minimum uncertainty (in a sense defined independently by the authors [1] and Wootters and Sussman [2]) with respect...

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 elemen...

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 constra...

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication [1] advocates such a...

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...

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 prior...

In the quantum Bayesian understanding of quantum states being developed by the authors and collaborators, 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 Ba...

This conference was devoted to fundamental questions raised by quantum mechanics, especially in quantum information theory. As has become customary in our series of conference in Vxj, we were glad to welcome a fruitful assembly of theoretical physicists, experimentalists, mathematicians and even philosophers interested in the foundations of probabi...

This year in Växjö we thought we would try an experiment-it felt high time for a new result. Much of the foundations discussion of previous years has focussed on EPR-style arguments and the meaning and experimental validity of various Bell inequality violations. Yet, there is another pillar of the quantum foundations puzzle that has hardly received...

Abstract unavailable.

Since Renes et al. [J. Math. Phys. 45, 2171 (2004)], there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: 1) by a nat...

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 quan...

Recently there has been much interest in the quantum information community to prove (or find a counterexample to) the existence of so-called symmetric informationally complete measurements (SICs). In this talk we show that there should be even more interest. For, under a robust measure of orthonormality for operator bases (one that does not build i...

Once again, I take advantage of the wonderfully liberal and tolerant mood Andrei Khrennikov sets at his yearly conferences by submitting a nonstandard paper for the proceedings. This pseudo‐paper consists of excerpts drawn from two of my samizdats [Quantum States: What the Hell Are They? and Darwinism All the Way Down (and Probabilism All the Way B...

All papers have been peer reviewed. This was the 4th conference arranged by ICMM on probabilistic foundations of classical and quantum physics. The first three conferences took place in 2000, 2002, and 2004. Some closely related conferences are Bohmian Mechanics 2000 and Quantum Theory: Reconsideration of Foundations 2001, 2003, and 2005. The main...

In the neo-Bayesian view of quantum mechanics that Appleby, Caves, Pitowsky, Schack, the author, and others are developing, quantum states are taken to be compendia of partial beliefs about potential measurement outcomes, rather than objective properties of quantum systems. Different observers may validly have different quantum states for a single...

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 necessar...

This is a collection of statements gathered on the occasion of the Quantum Physics of Nature meeting in Vienna.

Two measures of sensitivity to eavesdropping for alphabets of quantum states were recently introduced by C. A. Fuchs and M. Sasaki [Quantum Inf. Comput. 3, No. 5, 377–404 (2003; Zbl 1152.81714), see http://arxiv.org/abs/quant-ph/0302092]. These are the accessible fidelity and quantumness. In this paper we prove an important property of both measure...

The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. In this paper, we motivate and review two results that generalize de Finetti's theorem to the quantum mechanical set...

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 ensem...

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 ab...

There was a time when it was not easy to get a postdoctoral position in quantum information theory. With the field only getting off the ground in the early 1990s, there were almost no positions to be had when I finished my PhD in 1995. Essentially, the only schools with enough nerve to invest in this untried field were the best of the very best – t...

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...

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...

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...

June 2000, "Greece" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Carl's Preply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 07 September 2000, "Critical Letters and Reply" . . . . . . . . . . . . . . . . . . . . 133 13 November 2000, "Bush vs. Bohr" . . . . . . . . . . . . . . . . . . . . ....

In this paper, I try to cause some good-natured trouble. The issue is, 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, axiomati...

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 alphabe...

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...

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.

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti's classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potenti...

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 co...

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...

In this paper, I try to cause some good-natured trouble. The issue at stake is when will we ever stop burdening the taxpayer with conferences and workshops devoted--- explicitly or implicitly---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 statemen...

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 t...

This document is the first installment of three in the Cerro Grande Fire Series. It is a collection of letters written to various colleagues, most of whom regularly circuit this archive, including Howard Barnum, Paul Benioff, Charles Bennett, Herbert Bernstein, Doug Bilodeau, Gilles Brassard, Jeffrey Bub, Carlton Caves, Gregory Comer, Robert Griffi...

A way to quantify the notion that quantum measurements bring a disturbance to the system being measured is highlighted. It is shown how one observer among two observers can increase by measurement the purity of his density operator without affecting the purity of the other observer's.

In this paper we describe a way to quantify the folkloric notion that quantum measurements bring a disturbance to the system being measured. We consider two observers who initially assign identical mixed-state density operators to a two-state quantum system. The question we address is to what extent one observer can, by measurement, increase the pu...

Fidelity F{classical} = 1/2 has been established as setting the boundary between classical and quantum domains in the teleportation of coherent states of the electromagnetic field (S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt. 47, 267 (2000)). Two recent papers by P. Grangier and F. Grosshans (quant-ph/0009079 and quant-ph/0010107)...