NCN Research Grant Maestro 7
Uncertainty Relations and Quantum Entanglement
Number 2015/18/A/ST2/00274 realised in 2016-2021

National Science Center Jagiellonian University National Science Center


(in reverse chronological order)

[29] T. Linowski, G. Rajchel-Mieldzioć, K. Życzkowski

Entangling power of multipartite unitary gates

We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an $n$-partite gate as an explicit function of the gate, linking the entangling power of gates acting on n-partite Hilbert space of dimension $d1\hdots dn$ to the entanglement of pure states in the Hilbert space of dimension $(d1\hdots dn)^2$. Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with $2n$ parties. Finally, we provide a detailed analysis of the entangling properties of three-qubit unitary and orthogonal gates.

[28] J. Czartowski, D. Goyeneche, M. Grassl and K. Życzkowski

Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs

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 matrices of the $20$ pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius $\sqrt{3/20}$ 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. These peculiar constellations of density matrices constitute a class of generalized measurements with additional symmetries useful for the reconstruction of unknown quantum states. Furthermore, we show 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.

[27] K. Szymański and K. Życzkowski

Geometric and algebraic origins of additive uncertainty relations

Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number $j$ and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.

[26] G. M. Quinta, R. André, A. Burchardt, K. Życzkowski

Cut-resistant links and multipartite entanglement resistant to particle loss

In this work, we explore the space of quantum states composed of $N$ particles. To investigate the entanglement resistant to particles loss, we introduce the notion of $m$-resistant states. A quantum state is $m$-resistant if it remains entangled after losing an arbitrary subset of m particles, but becomes separable after losing a number of particles larger than $m$. We establish an analogy to the problem of designing a topological link consisting of $N$ rings such that, after cutting any $(m+1)$ of them, the remaining rings become disconnected. We present a constructive solution to this problem, which allows us to exhibit several distinguished N-particles states with the desired property of entanglement resistance to a particle loss.

[25] J. Czartowski, D. Braun, K. Życzkowski

Trade-off relations for operation entropy of complementary quantum channels

The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi-Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel $\Phi$ acting on a state of size $N$ one defines the complementary channel $\tilde{\Phi}$, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy we show that for any dimension $N$ the sum of both entropies, $S(\Phi)+S(\tilde{\Phi})$, is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps, $N=2$, we describe the interpolating family of depolarising maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.

[24] K. Korzekwa, S. Czachórski, Z. Puchała, K. Życzkowski

Distinguishing classically indistinguishable states and channels

We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between $M$ quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of $M$ quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of $M$ perfectly distinguishable states (channels) that are classically indistinguishable.

[23] J. Czartowski, K. Szymański, B. Gardas, Y. Fyodorov, K. Życzkowski

Separability gap and large-deviation entanglement criterion

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.

[22] S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruściński, K. Życzkowski

Universal spectra of random Lindblad operators

To understand typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markovian completely positive evolution in the space of density matrices. Spectral properties of these operators, including the shape of the spectrum in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate universality of the spectral features. The notion of ensemble of random generators of Markovian qauntum evolution constitutes a step towards categorization of dissipative quantum chaos.

[21] W. Kłobus, A. Burchardt, A. Kołodziejski, M. Pandit, T. Vertesi K. Życzkowski, W. Laskowski

On $k$-uniform mixed states

We investigate the maximum purity that can be achieved by k-uniform mixed states of N parties. Such N-party states have the property that all their k-party reduced states are maximally mixed. A scheme to construct explicitly k-uniform states using a set of specific N-qubit Pauli matrices is proposed. We provide several different examples of such states and demonstrate that in some cases the state corresponds to a particular orthogonal array. The obtained states, despite being mixed, reveal strong non-classical properties such as genuine multipartite entanglement or violation of Bell inequalities.

[20] Z. Puchała, Ł. Rudnicki, K. Życzkowski,

Pauli semigroups and unistochastic quantum channels

We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, $\Phi=e^{\mathcal{L}t}$,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together with the maximally depolarizing channel. Consequently, every member of the Pauli semigroup is unitarily equivalent to a unistochastic map, describing a coupling with one-qubit environment initially in the maximally mixed state, determined by a unitary matrix of order four.

[19] M. Enriquez, F. Delgado, K. Życzkowski

Entanglement of three-qubit random pure states

We study non-local 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 Acìn 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 Rènyi-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 and the SLOCC classes in terms of the corresponding entanglement polytope.

[18] G. Rajchel, A. Gąsiorowski, K. Życzkowski

Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces

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 of this size. This is the case for any even dimension $n\leq 20$, 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 $B_{ij}=|U_{ij}|^2$, 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 $n\times n$. 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.

[17] M. Białończyk, A. Jamiołkowski, K. Życzkowski

Application of Shemesh theorem to analysis of spectral properties of quantum channels

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,…A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,…A_K)$ one can predict the asymptotic dynamics for a class of operations.

[16] J. Czartowski, D. Goyeneche, K. Życzkowski

Multipartite entanglement for informationally complete quantum measurements

We present a comprehensible introduction to tight informationally complete measurements for arbitrarily large multipartite systems and study their allowed entanglement configurations, focusing on those requiring simplest possible quantum resources. We demonstrate that tight measurements cannot be exclusively composed neither by fully separable nor maximally entangled states, according to a selected measure of entanglement. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and we study the distinguished case in which every measurement operator has 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.

[15] A. Mandarino, T. Linowski, K. Życzkowski

Bipartite unitary gates and billiard dynamics in the Weyl chamber

Long time behavior of a unitary quantum gate $U$, acting sequentially on two subsystems of dimension $N$ each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a $3D$ billiard. Due to ergodicity of such a dynamics an average along a trajectory $V^t$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber - the minimal $3D$ set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension $N$ the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on $U(N^2)$.

[14] K. Korzekwa, S. Czachórski, Z. Puchała and K. Życzkowski

Coherifying quantum channels

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels $\Phi^C$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamiołkowski state $J_{\Phi}$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamiołkowski state $J^C_{\Phi}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To asses the extent to which an optimal process $\Phi^C$ is indeterministic we find explicit bounds on the entropy and purity of $J^C_{\Phi}$, and relate the latter to the unitarity of $\Phi^C$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.

[13] Z. Puchała, Ł. Rudnicki, A. Krawiec and K. Życzkowski

Majorization uncertainty relations for mixed quantum states

Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.

[12] L. Seveso, D. Goyeneche, K. Życzkowski

Coarse-grained entanglement classification through orthogonal arrays

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 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.

[11] Ł. Rudnicki, Z. Puchała, K. Życzkowski

Gauge invariant information concerning quantum channels

Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

[10] D. Goyeneche, Z. Raissi, S. Di Martino, K. Życzkowski

Entanglement and quantum combinatorial designs

We introduce the notion of quantum orthogonal arrays as a generalization of orthogonal arrays. These quantum combinatorial designs naturally induce the concepts of quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. Furthermore, quantum orthogonal arrays are one-to-one related to k-uniform states, i.e., pure states such that every reduction to k parties is maximally mixed. We derive quantum orthogonal arrays having an arbitrary large number of columns and, consequently, infinitely many classes of mutually orthogonal quantum Latin arrangements and absolutely maximally entangled states.

[9] W. Bruzda, D. Goyeneche, K. Życzkowski

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.

[8] W. Słomczyński and A. Szczepanek

Quantum dynamical entropy, chaotic unitaries and complex Hadamard matrices

We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies. They quantify the randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy or, equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.

[7] B. Jonnadula, P. Mandayam, K. Życzkowski and A. Lakshminarayan

Can local dynamics enhance entangling power?

It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate typicality and study its properties. Analyzing multiple, say $n$, applications of any entangling operator, $U$, interlaced with random local gates we prove that both investigated quantities approach their asymptotic values in a simple exponential form. These values coincide with the averages for random nonlocal operators on the full composite space and could be significantly larger than that of $U^n$. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.

[6] P. Lipka-Bartosik and K. Życzkowski

Nuclear numerical range and quantum error correction codes for non-unitary noise models

We introduce a notion of nuclear numerical range defined as the set of expectation values of a given operator A among normalized pure states, which belong to the nucleus of an auxiliary operator Z. This notion proves to be applicable to inves- tigate models of quantum noise with block-diagonal structure of the corresponding Kraus operators. The problem of constructing a suitable quantum error correction code for this model can be restated as a geometric problem of finding intersection points of certain sets in the complex plane. This technique, worked out in the case of two-qubit systems, can be generalized for larger dimensions.

[5] A. Szymusiak and W. Słomczyński

Informational power of the Hoggar SIC-POVM

We compute the informational power for the Hoggar symmetric informationally complete positive operator- valued measure (SIC-POVM) in dimension eight, i.e., the classical capacity of a quantum-classical channel generated by this measurement. We show that the states constituting a maximally informative ensemble form a twin Hoggar SIC-POVM being the image of the original one under a conjugation.

[4] Ł. Skowronek

There is no direct generalization of positive partial transpose criterion to the three-by-three case

We show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to three-by-three systems. We call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested two-partite state. In particular, we show that a family of extreme positive maps discussed in a paper by Ha and Kye [Open Syst. Inf. Dyn. 18, 323–337 (2011)], cannot be replaced by a finite set of witnesses in the task of entanglement detection in three-by-three systems. In a more mathematically elegant parlance, our result says that the convex cone of positive maps of the set of three-dimensional matrices into itself is not finitely generated as a mapping cone.

[3] M. Smaczyński, W. Roga and K. Życzkowski

Selfcomplementary quantum channels

Selfcomplementary quantum channels are characterized by such an interac- tion between the principal quantum system and the environment that leads to the same output states of both interacting systems. These maps can describe approximate quantum copy machines, as perfect copying of an unknown quantum state is not possible due to the celebrated no-cloning theorem. We provide here a parametrization of a large class of selfcomplementary channels and analyze their properties. Selfcomplementary channels pre- serve some residual coherences and residual entanglement. Investigating some measures of non-Markovianity, we show that time evolution under selfcomplementary channels is highly non-Markovian.

[2] D. Alsina, A. Cervera, D. Goyeneche, J. I. Latorre and K. Życzkowski

Operational approach to Bell inequalities: applications to qutrits

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.

[1] A. E. Rastegin and K. Życzkowski

Majorization entropic uncertainty relations for quantum operations

Majorization uncertainty relations are derived for arbitrary quantum operations acting on a finite-dimensional space. The basic idea is to consider submatrices of block matrices comprised of the corresponding Kraus operators. This is an extension of the previous formulation, which deals with submatrices of a unitary matrix relating orthogonal bases in which measurements are per- formed. Two classes of majorization relations are considered: one related to the tensor product of probability vectors and another one related to their direct sum. We explicitly discuss an example of a pair of one-qubit operations, each of them represented by two Kraus operators. In the particular case of quantum maps describing orthogonal measurements the presented formulation reduces to earlier results derived for measurements in orthogonal bases. The presented approach allows us also to bound the entropy characterizing results of a single generalized measurement.