Papers by Karol Zyczkowski
arXiv: Quantum Physics, 2018
Finding the ground state energy of a Hamiltonian $H$, which describes a quantum system of several... more Finding the ground state energy of a Hamiltonian $H$, which describes a quantum system of several interacting subsystems, is crucial as well for many-body physics as for various optimization problems. Variety of algorithms and simulation procedures (either hardware or software based) rely on the separability approximation, in which one seeks for the minimal expectation value of $H$ among all product states. We demonstrate that already for systems with nearest neighbor interactions this approximation is inaccurate, which implies fundamental restrictions for precision of computations performed with near-term quantum annealers. Furthermore, for generic Hamiltonians this approximation leads to significant systematic errors as the minimal expectation value among separable states asymptotically tends to zero. As a result, we introduce an effective entanglement witness based on a generic observable that is applicable for any multipartite quantum system.
Physical Review Research, 2020
Quantum, 2021
We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum syste... more We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical...
Linear Algebra and its Applications, 2018
The joint numerical range W (F) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3) is a con... more The joint numerical range W (F) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3) is a convex and compact subset in R 3. We show that W (F) is generically a three-dimensional oval. Assuming dim(W (F)) = 3, every one-or two-dimensional face of W (F) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F) using random matrices and dual varieties.
Mathematics in Computer Science, 2018
We study a special class of (real or complex) robust Hadamard matrices, distinguished by the prop... more We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even n ≤ 20, and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U , such that B i j = |U i j | 2 , is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order n × n. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. In the case n = 4 we study geometry of the set U 4 of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope B 4 .
Entropy, 2018
We study entanglement properties of generic three-qubit pure states. First, we obtain the distrib... more We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those inva...
Journal of Mathematical Physics, 2018
In this work we analyze properties of generic quantum channels in the case of large system size. ... more In this work we analyze properties of generic quantum channels in the case of large system size. We use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to 1 2 + 2 π , giving also an estimate for the maximum success probability for distinguishing the channels. We also consider the problem of distinguishing two random unitary rotations.
Physical Review Letters, 2020
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum m... more Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled 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.
Quantum Information and Computation, 2013
In the present paper we initiate the study of the product higher rank numerical range. The latter... more In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.
We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum tran... more We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix C is minimized over the set of all bipartite coupling states ρ , such that both of its reduced density matrices ρ and ρ of size m and n are fixed. The value of the quantum optimal transport cost T C (ρ, ρ) can be efficiently computed using semidefinite programming. In the case m = n the cost T C gives a semi-metric if and only if it is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if C satisfies the above
Physical review. E, Statistical, nonlinear, and soft matter physics, 2003
We investigate the correspondence between the decay of correlation in classical systems, governed... more We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tunable resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.
Physics Letters A, 2009
We define a natural ensemble of trace preserving, completely positive quantum maps and present al... more We define a natural ensemble of trace preserving, completely positive quantum maps and present algorithms to generate them at random. Spectral properties of the superoperator Φ associated with a given quantum map are investigated and a quantum analogue of the Frobenius-Perron theorem is proved. We derive a general formula for the density of eigenvalues of Φ and show the connection with the Ginibre ensemble of real non-symmetric random matrices. Numerical investigations of the spectral gap imply that a generic state of the system iterated several times by a fixed generic map converges exponentially to an invariant state.
Physical Review Letters, 1995
We present anh ! 0 approximation for the quasienergy spectrum of a periodically kicked top, valid... more We present anh ! 0 approximation for the quasienergy spectrum of a periodically kicked top, valid under conditions of both regular and chaotic classical motion. In contrast to conventional periodic-orbit theory we implement the semiclassical limit for each matrix element of the Floquet operator rather than for the trace of the propagator. Even though a classical looking action is involved, the approximate matrix elements are specified in terms of complex ghost trajectories instead of real classical orbits. Our mean error for the quasienergies is a surprisingly small 3% of the mean spacing.
Physical Review E, 2010
We analyze a model quantum dynamical system subjected to periodic interaction with an environment... more We analyze a model quantum dynamical system subjected to periodic interaction with an environment, which can describe quantum measurements. Under the condition of strong classical chaos and strong decoherence due to large coupling with the measurement device, the spectra of the evolution operator exhibit an universal behavior. A generic spectrum consists of a single eigenvalue equal to unity, which corresponds to the invariant state of the system, while all other eigenvalues are contained in a disk in the complex plane. Its radius depends on the number of the Kraus measurement operators, and determines the speed with which an arbitrary initial state converges to the unique invariant state. These spectral properties are characteristic of an ensemble of random quantum maps, which in turn can be described by an ensemble of real random Ginibre matrices. This will be proven in the limit of large dimension.
Physical Review A, 2006
Average entanglement of random pure states of an N × N composite system is analyzed. We compute t... more Average entanglement of random pure states of an N × N composite system is analyzed. We compute the average value of the determinant D of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distribution P(D). Similar results are obtained for the rescaled N th root of the determinant, called G-concurrence. We show that in the limit N → ∞ this quantity becomes concentrated at a single point G ⋆ = 1/e. The position of the concentration point changes if one consider an arbitrary N × K bipartite system, in the joint limit N, K → ∞, K/N fixed.
Open Systems & Information Dynamics, 2006
Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an import... more Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,…,16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12, 14 and 16, which we could not find in the existing literature.
For any operator M acting on an N-dimensional Hilbert space HN we introduce its numerical shadow,... more For any operator M acting on an N-dimensional Hilbert space HN we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (M u, u) is equal to z, where u stands for a random complex vector from HN , satisfying ||u|| = 1. In the case of N = 2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R N −1 onto the complex plane. Numerical shadow is found explicitly for Jordan matrices JN , direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.
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Papers by Karol Zyczkowski