We analyze entropic uncertainty relations in a finite-dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained projective measurements with respect to any orthogonal bases. improve recent by Coles Piani [P. M. Piani, Phys. Rev. A 89, 022112 (2014)], which are known be stronger than well-known result Maassen Uffink [H. J. B. Uffink, Lett. 60, 1103 (1988)]. Furthermore, we find bound based on majorization techniques, also happens results involving...

Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for sum R\'enyi entropies describing probability distributions associated with given pure state expanded eigenbases two observables. Obtained expressed terms largest singular values submatrices unitary rotation matrix. Numerical simulations show that generic matrix size N = 5 our bound is stronger than well known result Maassen and...

Abstract We present IntU package for Mathematica computer algebra system. The presented performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. describe number special cases which can be used optimize calculation speed some classes integrals. also provide examples usage package.

Several techniques of generating random quantum channels, which act on the set d-dimensional states, are investigated. We present three approaches to problem sampling channels and show that they mathematically equivalent. discuss under conditions give uniform Lebesgue measure convex operations compare their advantages computational complexity demonstrate them is particularly suitable for numerical investigations. Additional results focus spectral gap other properties invariant states....

Properties of random mixed states order $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$, due concentration measure, trace distance between two tends a fixed number ${\tilde D}=1/4+1/\pi$, which yields Helstrom bound on their distinguishability. To arrive at this result we apply free calculus and derive symmetrized Marchenko--Pastur distribution, is shown describe numerical data model coupled quantum kicked tops. Asymptotic...

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution system? If not, then what is minimal amount randomness required by theory given process? Here, we address this problem studying coherifications channel $\Phi$, i.e., look for channels $\Phi^{\mathcal{C}}$ that induce same classical $T$, but are "more coherent". To quantify coherence $\Phi$ measure corresponding Jamio{\l}kowski state...

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 postprocessing, (b) projective on a relevant system, (c) postselection nonobserving certain outcomes. The implements arbitrary measurement in dimension $d$ with optimal success probability $1/d$. apply our results to bound relative power unambiguous state discrimination. Finally, we test experimentally...

We introduce and analyse the problem of encoding classical information into different resources a quantum state. More precisely, we consider general class communication scenarios characterised by operations that commute with unique resource destroying map leave free states invariant. Our motivating example is given coherences system respect to fixed basis (with unitaries diagonal in as encodings decoherence channel map), but generality framework allows us explore applications ranging from...

We analyze the problem of finding sets quantum states that can be deterministically discriminated. From a geometric point view, this is equivalent to embedding simplex points whose distances are maximal with respect Bures distance (or trace distance). derive upper and lower bounds for fidelity between two states, which imply unitary orbits both states. thus show that, when analyzing minimal fixed spectra, it sufficient consider diagonal only. Hence optimal discrimination considered, given...

We derive several bounds on fidelity between quantum states. In particular we show thatfidelity is bounded from above by a simple to compute quantity call super-fidelity.It analogous another called sub-fidelity. For any two states of two-dimensional system (N = 2) all three quantities coincide. demonstratethat sub- and super-fidelity are concave functions. also that issuper-multiplicative while sub-fidelity sub-multiplicative design feasible schemesto measure these in an experiment....

In this work we provide an efficiency analysis of the problem discrimination two randomly chosen unknown quantum operations in single-shot regime. We tight bounds for success probability such a protocol arbitrary channels and generalized measurements.

We derive several bounds on fidelity between quantum states. In particular we show that is bounded from above by a simple to compute quantity call super--fidelity. It analogous another called sub--fidelity. For any two states of two--dimensional system (N=2) all three quantities coincide. demonstrate sub-- and super--fidelity are concave functions. also super--multiplicative while sub--fidelity sub--multiplicative design feasible schemes measure these in an experiment.Super--fidelity can be...

Numerical range of a Hermitian operator X is defined as the set all possible expectation values this observable among normalized quantum state. We analyze modification definition in which value taken certain subset states. One considers for instance real states, product separable or maximally entangled show exemplary applications these algebraic tools theory information: analysis k-positive maps and entanglement witnesses, well study minimal output entropy channel. Product numerical unitary...

Nonlocal properties of an ensemble diagonal random unitary matrices order ${N}^{2}$ are investigated. The average Schmidt strength such a bipartite quantum gate is shown to scale as $lnN$, in contrast the $ln{N}^{2}$ behavior characteristic gates. Entangling power $U$ related von Neumann entropy auxiliary state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, where square matrix $A$ obtained by reshaping vector elements length into $N$. This fact provides motivation...

In this work, we study the problem of single-shot discrimination von Neumann measurements, which associate with measure-and-prepare channels. There are two possible approaches to problem. The first one is simple and does not utilize entanglement. We focus only on classical probability distributions, outputs find necessary sufficient criterion for perfect in case. A more advanced approach requires usage quantify distance between measurements terms diamond norm (called sometimes completely...

In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based predominantly on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of that studies engines operating two baths at different temperatures. Here, investigate performance engines, which replace access temperatures arbitrary constraints state transformations. The idea is imitate action two–stroke...

The totality of normalised density matrices order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by Hilbert-Schmidt distance we consider images orthogonal projections onto two-plane and show that they are similar to numerical ranges N. For matrix A one defines its shadow as probability distribution supported on range W(A), unitarily invariant Fubini-Study measure complex projective manifold CP^(N-1). We define generalized, mixed-states shadows demonstrate their...

We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state size $N$ in $L$ orthogonal bases. Lower lead to novel entropic uncertainty relations, while upper allow us formulate universal certainty relations. For $L=2$ maximal saturates at $logN$ because there exists mutually coherent state, but relations are shown be nontrivial $L\ensuremath{\ge}3$ measurements. In case prime power dimension, $N={p}^{k}$, and number $L=N+1$, bound becomes minimal...

We provide a bound for the trace distance between two quantum states. The lower is based on superfidelity, which provides upper fidelity. One of advantages presented that it can be estimated using simple measurement procedure. also compare this with one provided in terms

Spectral properties of an arbitrary matrix can be characterized by the entropy its rescaled singular values. Any quantum operation described associated dynamical or corresponding superoperator. The describes degree decoherence introduced map, while superoperator characterizes a priori knowledge receiver outcome channel Phi. We prove that for any map acting on N--dimensional system sum both entropies is not smaller than ln N. For bistochastic this lower bound reads 2 investigate also R\'enyi...

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

We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) output statistics of protocols which objects are intertwined with random circuits measured standard basis. show that for forming approximate 4-designs, TVDs can be approximated by simple explicit functions underlying – average-case distances (ACDs). apply them to effects...