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arXiv:1612.03234 · The European Physical Journal D

Introducing the Qplex: a novel arena for quantum theory

Marcus Appleby, Christopher A. Fuchs, Blake C. Stacey, Huangjun Zhu

Published Jul 1, 2017• 54 citations
arXivHTMLPDF

Abstract

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of 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 constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC.

DOI
10.1140/epjd/e2017-80024-y

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Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

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We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of 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 constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
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 potential. In this talk, I introduce the “symmetric informationally complete” (SIC) sets of quantum states as a candidate for that structure. By their aid, one can rewrite quantum states so that they become simply probability distributions, unitary transformations so that they become doubly stochastic matrices, and the Born rule so that it becomes a rather simple variant of the classical law of total probability. These representations hold the potential for entirely new ways of analyzing quantum communication channels and algorithms. Surprisingly however, despite the way they can be used to make quantum theory look formally close to classical information theory, there is also a sense in which the SIC states are as far from classical as possible: For instance, by some measures these states are as sensitive to quantum eavesdropping as any alphabet of quantum states can be. Time permitting, I will show off some of the latest things known about the SICs.
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 system may be worth more than a hundred blue‐in‐the‐face arguments over the meaning of quantum states. The philosophy here is that if quantum theory really is a theory about how to structure one’s degrees of belief, rather than a theory of ontic quantum states, then the theory should become conceptually clearest when written in its native terms—i.e., in a purely probabilistic language, with no quantum states at all as intermediaries. This paper will chronicle the benefits of one particularly nice class of “coordinate systems” (i.e., operator bases) for enacting a map from quantum states on a finite dimensional Hilbert space to an appropriately chosen finite dimensional probability simplex. This is the so‐called Symmetric Informationally Complete measurements or SICs.
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 any attention in our great series of meetings: It is the phenomenon first demonstrated by Kochen and Specker, quantum contextuality. Recently there has been a rapid growth of activity aimed toward better understanding this aspect of quantum mechanics, which Asher Peres sloganized by the phrase, “unperformed experiments have no results.” Below is a sampling of some important papers on the topic for the reader not yet familiar with the subject. What is the source of this phenomenon? Does it depend only on high level features of quantum mechanics, or is it deep in the conceptual framework on which the theory rests? Might it, for instance, arise from the way quantum mechanics amends the classic laws of probability? What are the mathematically simplest ways contextuality can be demonstrated? How might the known results be made amenable to experimental tests? These were the sorts of discussions we hoped the session would foster.
Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.