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)

8) D. Goyeneche, W. Bruzda, O. Turek, D. Alsina, K. Życzkowski

The powerfulness of classical correlations

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.

7) F. Shahbeigi, D. Amaro-Alcala, Z. Puchała and K. Życzkowski

Log-Convex set of Lindblad semigroups acting on N-level system

We analyze the set $\mathcal{A}^Q_N$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set $\mathcal{A}^Q_N$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels the hyper-decoherence commutes with the dynamics, so that decohering a quantum accessible channel we obtain a bistochastic matrix form the set $\mathcal{A}^C_N$ of classical maps accessible by a semigroup. Focusing on 3-level systems we investigate the geometry of the sets of quantum accessible maps, its classical counterpart and the support of their spectra. We demonstrate that the set $\mathcal{A}^Q_3$ is not included in the set $\mathcal{U}^Q_3$ of quantum unistochastic channels, although an analogous relation holds for $N=2$. The set of transition matrices obtained by hyper-decoherence of unistochastic channels of order $N \geq 3$ is shown to be larger than the set of unistochastic matrices of this order, and yields a motivation to introduce the larger sets of $k$-unistochastic matrices.

6) P. Horodecki, Ł. Rudnicki and K. Życzkowski

Five selected problems in the theory of quantum information are presented. The first four concern existence of certain objects relevant for quantum information, namely mutually unbiased bases in dimension six, an infinite family of symmetric informationally complete generalized measurements, absolutely maximally entangled states for four subsystems with six levels each and bound entangled states with negative partial transpose. The last problem requires checking whether a certain state of a two-ququart system is 2-copy distillable. Finding a correct answer to any of them will be rewarded by the Golden KCIK Award established by the National Quantum Information Centre (KCIK) in Poland. A detailed description of the problems in question, the motivation to analyze them, as well as the rules for the open competition are provided.

5) W. Bruzda, S. Friedland, K. Życzkowski

Tensor rank and entanglement of pure quantum states

The rank of a tensor is analyzed in context of description of entanglement of pure states of multipartite quantum systems. We discuss the notions of the generic rank of a tensor with $d$ indices and $n$ levels in each mode and the maximal rank of a tensor of these dimensions. Other variant of this notion, called border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. As entanglement of a given quantum state depends on the way the total system is divided into subsystems, we introduce a notion of `partitioning rank' of a tensor, which depends on a way how the entries forming the tensor are treated. In particular, we analyze the tensor product of several copies of the $n$-qubit state $|W_n>$ and analyze its partitioning rank for various splittings of the entire space. Some results concerning the generic rank of a tensor are also provided.

4) K. Korzekwa, Z. Puchała, M. Tomamichel, K. Życzkowski

Encoding classical information into quantum resources

We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper-bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is a twirling channel over some unitary group, we find matching one-shot lower-bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.

3) B. Jonnadula, P. Mandayam, K. Życzkowski, A. Lakshminarayan

Thermalization of entangling power with arbitrarily weak interactions

The change of the entangling power of n fixed bipartite unitary gates, describing interactions, when interlaced with local unitary operators describing monopartite evolutions, is studied as a model of the entangling power of generic Hamiltonian dynamics. A generalization of the local unitary averaged entangling power for arbitrary subsystem dimensions is derived. This quantity shows an exponential saturation to the random matrix theory (RMT) average of the bipartite space, indicating thermalization of quantum gates that could otherwise be very non-generic and have arbitrarily small, but nonzero, entanglement. The rate of approach is determined by the entangling power of the fixed bipartite unitary, which is invariant with respect to local unitaries. The thermalization is also studied numerically via the spectrum of the reshuffled and partially transposed unitary matrices, which is shown to tend to the Girko circle law expected for random Ginibre matrices. As a prelude, the entangling power $e_p$ is analyzed along with the gate typicality $g_t$ for bipartite unitary gates acting on two qubits and some higher dimensional systems. We study the structure of the set representing all unitaries projected into the plane $(e_p,g_t)$ and characterize its boundaries which contains distinguished gates including Fourier gate, CNOT and its generalizations, swap and its fractional powers. In this way, a family of gates with extreme properties is identified and analyzed. We remark on the use of these operators as building blocks for many-body quantum systems.

2) I. Bengtsson and K. Życzkowski

On discrete structures in finite Hilbert spaces

We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner function, symmetric informationally complete measurements, projective and unitary t--designs are discussed. Some recent results in the field are covered and several important open questions are formulated. We advocate a geometric approach to the subject and emphasize numerous links to various mathematical problems.

1) I. Bengtsson and K. Życzkowski

A brief introduction to multipartite entanglement

A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.