We consider sets of trace-normalized non-negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity-limited classical or quantum states on a finite-dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so-called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.
We consider sets of trace-normalized non-negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity-limited classical or quantum states on a finite-dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so-called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.
Despite being pursued with a lot of dedication in frame theory and quantum information theory, maximal equiangular tight frames have only been confirmed to exist for lowest dimensions. This is especially puzzling since maximal sets of mutually unbiased bases, which are also known to be optimal packings but have a slightly larger number of vectors, are known to exist in complex Hilbert spaces of any prime power dimension. This talk presents a method by which we can construct a set of M2+1 vectors in M complex dimensions that form a tight complex orthoplectic Grassmannian frame that is in addition related to a weighted complex projective 2-design with optimality properties for quantum state tomography.
This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).
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.
We consider a quantum version of the famous low-rank approximation problem. Specifically, we consider the distance D(ρ,σ) between two normalized quantum states, ρ and σ, where the rank of σ is constrained to be at most R. For both the trace distance and Hilbert-Schmidt distance, we analytically solve for the optimal state σ that minimizes this distance. For the Hilbert-Schmidt distance, the unique optimal state is σ=τR+NR, where τR=ΠRρΠR is given by projecting ρ onto its R principal components with projector ΠR, and NR is a normalization factor given by NR=R1−Tr(τR)ΠR. For the trace distance, this state is also optimal but not uniquely optimal, and we provide the full set of states that are optimal. We briefly discuss how our results have application for performing principal component analysis (PCA) via variational optimization on quantum computers.
We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).
Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. The optimal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, the optimal encoding is not unique. That is, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.