Publications
A crucial task for secure communication networks is to determine the minimum of physical requirements to certify a cryptographic protocol. A widely accepted candidate for certification is the principle of relativistic causality which is equivalent to the disallowance of causal loops. Contrary to expectations, we demonstrate how correlations allowed by relativistic causality could be exploited to break security for a broad class of multi-party protocols (all modern protocols belong to this class). As we show, deep roots of this dramatic lack of security lies in the fact that unlike in previous (quantum or no-signaling) scenarios the new theory “decouples” the property of extremality and that of statistical independence on environment variables. Finally, we find out, that the lack of security is accompanied by some advantage: the new correlations can reduce communication complexity better than the no-signaling ones. As a tool for analysis of this advantage, we characterize relativistic causal polytope by its extremal points in the simplest multi-party scenario that goes beyond the no-signaling paradigm.
Determination of classical and quantum values of bipartite Bell inequalities plays a central role in quantum nonlocality. In this work, we characterize in a simple way bipartite Bell inequalities, free of marginal terms, for which the quantum value can be achieved by considering a classical strategy, for any number of measurement settings and outcomes. These findings naturally generalize known results about nonlocal computation and quantum XOR games. Additionally, our technique allows us to determine the classical value for a wide class of Bell inequalities, having quantum advantage or not, in any bipartite scenario.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius √3/20 located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design—a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.
It is known that a real symmetric circulant matrix with diagonal entries d≥0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d+2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d≥0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n=2d+2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings Zm. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.
We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary number of projective measurements in any finite dimension d. The algorithm is flexible enough to allow us to impose any desired rank r to the state to be reconstructed, ranging from pure (r=1) to full rank (r=d) quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that the method outperforms semidefinite programming quantum state tomography for some sets of physically relevant quantum measurements in every finite dimension.
We design a series of quantum circuits that generate absolute maximally entangled (AME) states to benchmark a quantum computer. A relation between graph states and AME states can be exploited to optimize the structure of the circuits and minimize their depth. Furthermore, we find that most of the provided circuits obey majorization relations for every partition of the system and every step of the algorithm. The main goal of the work consists in testing efficiency of quantum computers when requiring the maximal amount of genuine multipartite entanglement allowed by quantum mechanics, which can be used to efficiently implement multipartite quantum protocols.
We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and complex Hadamard matrix in dimension d2, and a pair of ETFs (equiangular tight frames) in dimensions d(d±1)/2. We also show that a WH (Weyl–Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with dbetween 3 and 16, suggesting it may be true for all d≥3.
We study the existence and construction of circulant matrices C of order N≥2 with diagonal entries d≥0, off-diagonal entries ±1 and mutually orthogonal rows. These matrices generalize circulant conference (d=0) and circulant Hadamard (d=1) matrices. We demonstrate that matrices C exist for every order n and for d chosen such that n=2d+2, and we find all solutions C with this property. Furthermore, we prove that if C is symmetric, or n-1 is prime, or d is not an odd integer, then necessarily n=2d+2. Finally, we conjecture that the relation n=2d+2 holds for every matrix C, which generalizes the circulant Hadamard conjecture. We support the proposed conjecture by computing all the existing solutions up to n=50.
A physical theory is called non-local when observers can produce instantaneous effects over distant systems. Non-local theories rely on two fundamental effects: local uncertainty relations and steering of physical states at a distance. In quantum mechanics, the former one dominates the other in a well-known class of non-local games known as XOR games. In particular, optimal quantum strategies for XOR games are completely determined by the uncertainty principle alone. This breakthrough result has yielded the fundamental open question whether optimal quantum strategies are always restricted by local uncertainty principles, with entanglement-based steering playing no role. In this work, we provide a negative answer to the question, showing that both steering and uncertainty relations play a fundamental role in determining optimal quantum strategies for non-local games. Our theoretical findings are supported by an experimental implementation with entangled photons.
We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes, and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called k-uniform, i.e., multipartite pure states such that every reduction to k parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite k-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.
Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on the properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.
We analyze tight informationally complete measurements for arbitrarily large multipartite systems and study their configurations of entanglement. We demonstrate that tight measurements cannot be exclusively composed neither of fully separable nor maximally entangled states. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and investigate the distinguished case in which every measurement operator carries the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.
We introduce a method to determine whether a given generalized quantum measurement is isolated or if 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 a consequence, we provide a simple derivation of the maximal family of symmetric informationally complete positive operator-valued measure 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) contextual Kochen-Specker sets in dimension 3, 4, and 6, composed of 13, 18, and 21 vectors, respectively.
We demonstrate that a complex equiangular tight frame composed of N vectors in dimension d, denoted ETF (d, N), exists if and only if a certain bistochastic matrix, univocally determined by N and d, belongs to a special class of unistochastic matrices. This connection allows us to find new complex ETFs in infinitely many dimensions and to derive a method to introduce non-trivial free parameters in ETFs. We present an explicit six-parametric family of complex ETF(6,16), which defines a family of symmetric POVMs. Minimal and maximal possible average entanglement of the vectors within this qubit–qutrit family are described. Furthermore, we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix, which we apply to find all existing classes of complex ETFs containing up to 20 vectors.
In this work we develop two methods to construct Bell inequalities for multipartite systems. By considering non-Hermitian operators we study Bell inequalities for the cases of three settings, three outcomes, and three to six parties. The maximal value achieved in the framework of quantum theory is computed for subsystems with three levels each. The other technique, based on a mapping from pure entangled states to Bell operators, allows us to construct further multipartite Bell inequalities. As a consequence, we reproduce some known results in a different way and find some multipartite Bell inequalities for systems having three settings and three outcomes per party.
Heterogeneous bipartite quantum pure states, composed of two subsystems with a different number of levels, cannot have both reductions maximally mixed. In this work, we demonstrate the existence of a wide range of highly entangled states of heterogeneous multipartite systems consisting of N>2 parties such that every reduction to one and two parties is maximally mixed. Two constructions of generating genuinely multipartite maximally entangled states of heterogeneous systems for an arbitrary number of subsystems are presented. Such states are related to quantum error correction codes over mixed alphabets and mixed orthogonal arrays. Additionally, we show the advantages of considering heterogeneous systems in practical implementations of multipartite steering.
We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of m real mutually unbiased bases existing in dimension N>2 admits the introduction of (m−1)N/2 free parameters that cannot be absorbed by a global unitary operation. As consequence, there are m=k+1 mutually unbiased bases in every dimension N=k2with k3/2 free parameters, where k is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.
Absolutely maximally entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible bipartitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing, and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME states, namely, their relation to tensors, which can be understood as unitary transformations in all of their bipartitions. We call this property multiunitarity.
A long-standing problem in quantum mechanics is the minimum number of observables required for the characterization of unknown pure quantum states. The solution to this problem is especially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring five observables, that is, via projective measurements onto the states of five orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple postprocessing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of eight-dimensional quantum states, encoded in the momentum of single photons.
A pure quantum state of N subsystems with d levels each is called k-multipartite maximally entangled state, which we call a k-uniform state, if all its reductions to k qudits are maximally mixed. These states form a natural generalization of N-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and k-uniform states and prove the existence of several classes of such states for N-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of k-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
A powerful tool for studying geometrical problems in Hilbert spaces is developed. We demonstrate the convergence and robustness of our method in every dimension by considering dynamical systems theory. This method provides numerical solutions to hard problems involving many coupled nonlinear equations in low and high dimensions (e.g., quantum tomography problem, existence and classification of Pauli partners, mutually unbiased bases, complex Hadamard matrices, equiangular tight frames, etc.). Additionally, this tool can be used to find analytical solutions and also to implicitly prove the existence of solutions. Here, we develop the theory for the quantum pure state tomography problem in finite dimensions but this approach is straightforwardly extended to the rest of the problems. We prove that solutions are always attractive fixed points of a nonlinear operator explicitly given. As an application, we show that the statistics collected from three random orthonormal bases is enough to reconstruct pure states from experimental (noisy) data in every dimension d ⩽ 32.
We present a new parametrization of families of complex Hadamard matrices stemming from the Fourier matrices in every prime power dimension. We connect continuous Abelian groups with families of complex Hadamard matrices and conjecture that the constructed families are maximal. Also, we derive new relations for complex Hadamard matrices in every prime power dimension and prove that some real Hadamard matrices can be written as a product of an arbitrarily large number of real Hadamard matrices.
