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.
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.
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.
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.
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.
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.
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.
Symmetric informationally complete measurements (SICs) are elegant, celebrated, and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC compound is defined to be a collection of d3 vectors in d-dimensional Hilbert space that can be partitioned in two different ways: into d SICs and into d2 orthonormal bases. While a priori their existence may appear unlikely when d > 2, we surprisingly find an explicit construction for d = 4. Remarkably this SIC compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalization of the six-state protocol.