arXiv subjects

Zbigniew Puchała

Publications and source records attributed to Zbigniew Puchała.

At least 37 records · Page 2Linked to original sources

arXiv2022

Accessible maps in a group of classical or quantum channels

We study the problem of accessibility in a set of classical and quantum channels admitting a group structure. Group properties of the set of channels, and the structure of the closure of the analyzed group $G$ plays a pivotal role in this regard. The set of all convex combinations of the group elements contains a subset of channels that are accessible by a dynamical semigroup. We demonstrate that accessible channels are determined by probability vectors of weights of a convex combination of the group elements, which depend neither on the dimension of the space on which the channels act, nor on the specific representation of the group. Investigating geometric properties of the set $\mathcal{A}$ of accessible maps we show that this set is non-convex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set $\mathcal{A}$ covers a positive volume in the polytope of all convex combinations of the elements of the group.

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arXiv2021

Dephasing superchannels

We characterise a class of environmental noises that decrease coherent properties of quantum channels by introducing and analysing the properties of dephasing superchannels. These are defined as superchannels that affect only non-classical properties of a quantum channel $\mathcal{E}$, i.e., they leave invariant the transition probabilities induced by $\mathcal{E}$ in the distinguished basis. We prove that such superchannels $Ξ_C$ form a particular subclass of Schur-product supermaps that act on the Jamiolkowski state $J(\mathcal{E})$ of a channel $\mathcal{E}$ via a Schur product, $J'=J\circ C$. We also find physical realizations of general $Ξ_C$ through a pre- and post-processing employing dephasing channels with memory, and show that memory plays a non-trivial role for quantum systems of dimension $d>2$. Moreover, we prove that coherence generating power of a general quantum channel is a monotone under dephasing superchannels. Finally, we analyse the effect dephasing noise can have on a quantum channel $\mathcal{E}$ by investigating the number of distinguishable channels that $\mathcal{E}$ can be mapped to by a family of dephasing superchannels. More precisely, we upper bound this number in terms of hypothesis testing channel divergence between $\mathcal{E}$ and its fully dephased version, and also relate it to the robustness of coherence of $\mathcal{E}$.

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arXiv2021

Relating entropies of quantum channels

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

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arXiv2021

Generating random quantum channels

Several techniques of generating random quantum channels, which act on the set of $d$-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations. We compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the $2$-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector.

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arXiv2021

Multiple-shot and unambiguous discrimination of von Neumann measurements

We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the minimal number of queries needed for perfect discrimination. We also show that in this scenario querying the measurements gives the optimal strategy, and hence any possible adaptive methods do not offer any advantage over the parallel scheme. In the unambiguous discrimination scenario, we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.

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arXiv2021

Algebraic and geometric structures inside the Birkhoff polytope

The Birkhoff polytope $\mathcal{B}_d$ consisting of all bistochastic matrices of order $d$ assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset $\mathcal{U}_d$ of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the set $\mathcal{L}_d$ of bracelet matrices that forms a subset of $\mathcal{B}_d$, but a superset of $\mathcal{U}_d$. We prove that for every dimension $d$ this set contains the set of factorisable bistochastic matrices $\mathcal{F}_d$ and is closed under matrix multiplication by elements of $\mathcal{F}_d$. Moreover, we prove that both $\mathcal{L}_d$ and $\mathcal{F}_d$ are star-shaped with respect to the flat matrix. We also analyse the set of $d\times d$ unistochastic matrices arising from circulant unitary matrices, and show that their spectra lie inside $d$-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterise the set of circulant unistochastic matrices of order $d\leq 4$, and prove that such matrices form a monoid for $d=3$.

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arXiv2020

On the optimal certification of von Neumann measurements

In this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of $q$-numerical range.

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arXiv2020

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

We analyze the set ${\cal A}_N^Q$ 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 ${\cal A}_N^Q$ 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 ${\cal A}_N^C$ 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 ${\cal A}_3^Q$ is not included in the set ${\cal U}^Q_3$ 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 $N\ge 3$ 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.

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arXiv2020

Discrimination of POVMs with rank-one effects

The main goal of this work is to provide an insight into the problem of discrimination of positive operator valued measures with rank-one effects. It is our intention to study multiple shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end we construct a pair of symmetric, information complete positive operator valued measures which can be perfectly discriminated in a two-shot adaptive scheme. On top of this we provide an explicit algorithm which allows us to find this adaptive scheme.

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arXiv2019

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.

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arXiv2019

Simulating all quantum measurements using only projective measurements and postselection

We report an alternative scheme for implementing generalized quantum measurements that does not require the usage of auxiliary system. Our method utilizes solely: (a) classical randomness and post-processing, (b) projective measurements on a relevant quantum system and (c) postselection on non-observing certain outcomes. The scheme implements arbitrary quantum measurement in dimension $d$ with the optimal success probability $1/d$. We apply our results to bound the relative power of projective and generalised measurements for unambiguous state discrimination. Finally, we test our scheme experimentally on IBM's quantum processor. Interestingly, due to noise involved in the implementation of entangling gates, the quality with which our scheme implements generalized qubit measurements outperforms the standard construction using the auxiliary system.

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arXiv2019

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, $Φ= e^{{\cal L}t}$,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.

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arXiv2018

Strategies for optimal single-shot discrimination of quantum measurements

In this work we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.

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arXiv2018

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 $Φ$, i.e., we look for channels $Φ^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $Φ$ we measure the coherence of the corresponding Jamiołkowski state $J_Φ$. 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 $J_Φ^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process $Φ^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_Φ^{\mathcal{C}}$, and relate the latter to the unitarity of $Φ^{\mathcal{C}}$. 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 $Φ$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.

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arXiv2018

Conditional entropic uncertainty relations for Tsallis entropies

The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem~\ref{th:bound}. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.

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arXiv2018

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.

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arXiv2018

Vertices cannot be hidden from quantum spatial search for almost all random graphs

In this paper we show that all nodes can be found optimally for almost all random Erdős-Rényi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $p=ω(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log (n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty $ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.

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arXiv2018

Unconditional Security of a $K$-State Quantum Key Distribution Protocol

Quantum key distribution protocols constitute an important part of quantum cryptography, where the security of sensitive information arises from the laws of physics. In this paper we introduce a new family of key distribution protocols and we compare its key with the well-known protocols such as BB84, PBC0 and generation rate to the well-known protocols such as BB84, PBC0 and R04. We also state the security analysis of these protocols based on the entanglement distillation and CSS codes techniques.

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