Prime Dimension Research Articles

Quantum measurements with prescribed symmetry

We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively.

Spekkens’ toy model in all dimensions and its relationship with stabiliser quantum mechanics

Spekkens’ toy model is a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. Many aspects of quantum mechanics and protocols from quantum information can be reproduced in the model. In spite of its significance, a number of aspects of Spekkens’ model remained incomplete. Formal rules for the update of states after measurement had not been written down, and the theory had only been constructed for prime-dimensional and infinite dimensional systems. In this work, we remedy this, by deriving measurement update rules and extending the framework to derive models in all dimensions, both prime and non-prime. Stabiliser quantum mechanics (SQM) is a sub-theory of quantum mechanics with restricted states, transformations and measurements. First derived for the purpose of constructing error correcting codes, it now plays a role in many areas of quantum information theory. Previously, it had been shown that Spekkens’ model was operationally equivalent to SQM in the case of odd prime dimensions. Here, exploiting known results on Wigner functions, we extend this to show that Spekkens’ model is equivalent to SQM in all odd dimensions, prime and non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional SQM.

On the Brauer–Picard groups of fusion categories

We develop methods of computation of the Brauer–Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac–Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer–Picard groups of pointed fusion categories.

Algebraic Methods for Quantum Codes on Lattices

This is a note from a series of lectures at Encuentro Colombiano de Computacion Cuantica, Universidad de los Andes, Bogota, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Clifford quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Clifford circuits.

Investigating quantum coherence from discrete Wigner function

Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.

Classification of locally distinguishable and indistinguishable sets of maximally entangled states

We present a necessary condition for two sets of maximally entangled states to be locally equivalent in terms of local invariants. With the help of these invariants, first, we are able to completely classify all the distinguishable and indistinguishable sets of four generalized Bell states in the $4\ensuremath{\bigotimes}4$ quantum system: there are ten inequivalent classes including three locally indistinguishable and seven locally distinguishable quadruples. All distinguishable quadruples in this case can be distinguished by using only one-way classical communication, implying that the two-way classical communication has no advantage over one-way classical communication, similar to the case of qubit lattice states. Second, we are able to completely classify all the triplets of generalized Bell states in the quantum systems of all prime dimensions as well as in some composite dimensions.

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

Abstract The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear C * {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid C * {\mathrm{C}^{*}} -algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

Permutation Symmetry Determines the Discrete Wigner Function.

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

The Birkhoff theorem for unitary matrices of prime dimension

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. We prove that any unitary matrix with equal line sums can also be written as a sum of permutation matrices (with sum of weights equal 1). Furthermore, when the matrix dimension is prime, we prove that the unitary matrix lies inside a convex complex Birkhoff polytope.

Type II universal spacetimes

We study type II universal metrics of the Lorentzian signature. These metrics simultaneously solve vacuum field equations of all theories of gravitation with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its covariant derivatives of an arbitrary order. We provide examples of type II universal metrics for all composite number dimensions. On the other hand, we have no examples for prime number dimensions and we prove the non-existence of type II universal spacetimes in five dimensions. We also present type II vacuum solutions of selected classes of gravitational theories, such as Lovelock, quadratic and gravities.

Galois unitaries, mutually unbiased bases, and mub-balanced states

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in all dimensions d = p^n where p is an odd prime and the exponent n is odd. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.

Certainty relations, mutual entanglement, and nondisplaceable manifolds

We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $\log N$ as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L \ge 3$ measurements. In the case of a prime power dimension, $N=p^k$, and the number of measurements $L=N+1$, the upper bound for the average entropy becomes minimal for a collection of mutually unbiased bases. Analogous approach is used to study entanglement with respect to $L$ different splittings of a composite system, linked by bi-partite quantum gates. We show that for any two-qubit unitary gate $U\in \mathcal{U}(4)$ there exist states being mutually separable or mutually entangled with respect to both splittings (related by $U$) of the composite system. The latter statement follows from the fact that the real projective space $\mathbb{R}P^{3}\subset\mathbb{C}P^{3}$ is non-displacable. For $L=3$ splittings the maximal sum of $L$ entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.

Nonexistence of sharply covariant mutually unbiased bases in odd prime dimensions

Mutually unbiased bases (MUB) are useful in a number of research areas. The symmetry of MUB is an elusive and interesting subject. A (complete set of) MUB in dimension $d$ is sharply covariant if it can be generated by a group of order $d(d+1)$ from a basis state. Such MUB, if they exist, would be most appealing to theoretical studies and practical applications. Unfortunately, they seem to be quite rare. Here we prove that no MUB in odd prime dimensions is sharply covariant, by virtue of clever applications of Mersenne primes, Galois fields, and Frobenius groups. This conclusion provides valuable insight about the symmetry of MUB and the geometry of quantum state space. It complements and strengthens the earlier result of the author that only two stabilizer MUB are sharply covariant. Our study leads to the conjecture that no MUB other than those in dimensions 2 and 4 is sharply covariant.

States that are far from being stabilizer states

Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of minimum uncertainty states (MUSs) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of ‘far from being a stabilizer state’ can be quantified. Two theorems valid for all prime dimensions are given, as well as detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all of the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.

Mutually unbiased bases as minimal Clifford covariant 2-designs

Mutually unbiased bases (MUB) are interesting for various reasons. The most attractive example of (a complete set of) MUB is the one constructed by Ivanovi\'c as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the one with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUB provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.

Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applications, we reprove easily several main results of Cusick et al. on the number of equivalence classes under permutations for MRS in prime power dimensions, as well as give a count for the number of classes in pq number of variables, where p,q are prime numbers with p<q<p2. Also, we find a connection between the generalized inverse of a circulant matrix and the invertibility of its generating polynomial over F2, modulo a product of cyclotomic polynomials, thus generalizing a known result on nonsingular circulant matrices.

Counting equivalence classes for monomial rotation symmetric Boolean functions with prime dimension

Recently much progress has been made on the old problem of determining the equivalence classes of Boolean functions under permutation of the variables. In this paper we prove an asymptotic formula for the number of equivalence classes under permutation for degree d monomial rotation symmetric (MRS) functions, in the cases where d ? 3 is arbitrary and the number of variables n is a prime. Our counting formula has two main terms and an error term; this is the first instance of such a detailed result for Boolean function equivalence classes which is valid for arbitrary degree and infinitely many n. We also prove an exact formula for the count of the equivalence classes when d = 5; this extends previous work for d = 3 and 4.

Maximum nonlocality and minimum uncertainty using magic states

We prove that magic states from the Clifford hierarchy give optimal solutions for tasks involving nonlocality and entropic uncertainty with respect to Pauli measurements. For both the nonlocality and uncertainty tasks, stabilizer states are the worst possible pure states so our solutions have an operational interpretation as being highly non-stabilizer. The optimal strategy for a qudit version of the Clauser-Horne-Shimony-Holt (CHSH) game in prime dimensions is achieved by measuring maximally entangled states that are isomorphic to single-qudit magic states. These magic states have an appealingly simple form and our proof shows that they are "balanced" with respect to all but one of the mutually unbiased stabilizer bases. Of all equatorial qudit states, magic states minimize the average entropic uncertainties for collision entropy and also, for small prime dimensions, min-entropy -- a fact that may have implications for cryptography.

Affine equivalence of quartic monomial rotation symmetric Boolean functions in prime power dimension

In this paper we analyze and exactly compute the number of affine equivalence classes under permutations for quartic monomial rotation symmetric functions in prime and prime power dimensions.

Graphs attached to simple Frobenius-Perron dimensions of an integral fusion category

Let \({\mathcal C}\) be an integral fusion category. We study some graphs, called the prime graph and the common divisor graph, related to the Frobenius-Perron dimensions of simple objects in the category \({\mathcal C}\), that extend the corresponding graphs associated to the irreducible character degrees and the conjugacy class sizes of a finite group. We describe these graphs in several cases, among others, when \({\mathcal C}\) is an equivariantization under the action of a finite group, a \(2\)-step nilpotent fusion category, and the representation category of a twisted quantum double. We prove generalizations of known results on the number of connected components of the corresponding graphs for finite groups in the context of braided fusion categories. In particular, we show that if \({\mathcal C}\) is any integral non-degenerate braided fusion category, then the prime graph of \({\mathcal C}\) has at most \(3\) connected components, and it has at most \(2\) connected components if \({\mathcal C}\) is in addition solvable. As an application we prove a classification result for weakly integral braided fusion categories all of whose simple objects have prime power Frobenius-Perron dimension.