Fano type quantum inequalities in terms of qq-entropies

Generalizations of the quantum Fano inequality are considered. The notion of $q$-entropy exchange is introduced. This quantity is concave in each of its two arguments. For $q\geq0$, the inequality of Fano type with $q$-entropic functionals is established. The notion of coherent information and the perfect reversibility of a quantum operation are discussed in the context of $q$-entropies. By the monotonicity property, the lower bound of Pinsker type in terms of the trace norm distance is obtained for the Tsallis relative $q$-entropy of order $q=1/2$. For $0\leq{q}\leq2$, Fano type quantum inequalities with freely variable parameters are obtained.Comment: 10 pages, no figures. A partly incorrect statement of Section III is replaced with the valid one. Detected typos are corrected. The bibliography is extended and update

Coherence quantifiers from the viewpoint of their decreases in the measurement process

Measurements can be considered as a genuine example of processes that crush quantum coherence. In the case of an observable with degeneracy, the formulations of L\"{u}ders and von Neumann are known. These pictures postulate the two different states of a system immediately following the act of measurement. Hence, they are associated with divers variants of coherence losses during the measurement. Recent studies have focused on several ways to characterize quantum coherence appropriately. One of the existing types of quantifier is based on quantum $\alpha$-divergences of the Tsallis type. In this paper, we introduce coherence quantifiers associated with the L\"{u}ders picture of quantum measurements. The are shown to satisfy the same properties as coherence $\alpha$-quantifiers related to some orthonormal basis. Further, we consider losses of quantum coherence during a generalized measurement. The proposed approach is exemplified with unambiguous state discrimination; extreme properties of the states to be discriminated are clearly shown.Comment: 15 pages, three figures. Published as a contribution to Journal of Physics A Special Issue on Quantum Coherenc

On the role of dealing with quantum coherence in amplitude amplification

Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique generalizes key ideas of the Grover search algorithm. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. As measures of coherence, the geometric coherence and the relative entropy of coherence are considered. In terms of the relative entropy of coherence, complementarity relations with the success probability seem to be the most expository. The general relations presented are illustrated within several model scenarios of amplitude amplification processes.Comment: 15 pages, 2 figures. Major revision in v2. The bibliography is extended and updated. To appear in Quantum Information Processin

Quantum work fluctuations versus macrorealism in terms of non-extensive entropies

Fluctuations of the work performed on a driven quantum system can be characterized by the so-called fluctuation theorems. The Jarzynski relation and the Crooks theorem are famous examples of exact equalities characterizing non-equilibrium dynamics. Such statistical theorems are typically formulated in a similar manner in both classical and quantum physics. Leggett-Garg inequalities are inspired by the two assumptions referred to as the macroscopic realism and the non-invasive measurability. Together, these assumptions are known as the macrorealism in the broad sense. Quantum mechanics is provably incompatible with restrictions of the Leggett-Garg type. It turned out that Leggett-Garg inequalities can be used to distinguish quantum and classical work fluctuations. We develop this issue with the use of entropic functions of the Tsallis type. Varying the entropic parameter, we are often able to reach more robust detection of violations of the corresponding Leggett-Garg inequalities. In reality, all measurement devices suffer from losses. Within the entropic formulation, detection inefficiencies can naturally be incorporated into the consideration. This question also shows advantages that are provided due to the use of generalized entropies.Comment: 10 pages, 3 figures. Major revision in v2, matches the journal versio

R\'{e}nyi formulation of uncertainty relations for POVMs assigned to a quantum design

Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and R\'{e}nyi entropies for POVMs assigned to a quantum design. Relations of the Landau--Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.Comment: 14 pages, four figures. [v3] Final corrections, will be published in J. Phys. A: Mathematical and Theoretica

Optimal cloning with respect to the relative error

The relative error of cloning of quantum states with arbitrary prior probabilities is considered. It is assumed that the ancilla may contain some a priori information about the input state to be cloned. The lower bound on the relative error for general cloning scenario is derived. Both the case of two-state set and case of multi-state set are analyzed in details. The treated figure of merit is compared with other optimality criteria. The quantum circuit for optimal cloning of a pair of pure states is constructed.Comment: 9 pages, 2 figures, 1 table. Major changes. New section in which different criteria are compared is added. The bibliography is recas

Bounds on Shannon distinguishability in terms of partitioned measures

A family of quantum measures like the Shannon distinguishability is presented. These measures are defined over the two classes of POVM measurements and related to separate parts in the expression for mutual information. Changes of Ky Fan's norms and the partitioned trace distances under the operation of partial trace are discussed. Upper and lower bounds on the introduced quantities are obtained in terms of partitioned trace distances and Uhlmann's partial fidelities. These inequalities provide a kind of generalization of the well-known bounds on the Shannon distinguishability. The notion of cryptographic exponential indistinguishability for quantum states is revisited. When exponentially fast convergence is required, all the metrics induced by unitarily invariant norms are shown to be equivalent.Comment: 11 pages, no figures. The paper is reorganized. Many parts are recast, detected errors are corrected. Some results are improved. More explanations. To appear in QI

Unified-entropy trade-off relations for a single quantum channel

Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamio{\l}kowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver $(q,s)$-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.Comment: 8 pages, no figures. The title is slightly changes. Minor grammatical improvements are made. The bibliography is update

Uncertainty and certainty relations for successive projective measurements of a qubit in terms of Tsallis' entropies

We study uncertainty and certainty relations for two successive measurements of two-dimensional observables. Uncertainties in successive measurement are considered within the following two scenarios. In the first scenario, the second measurement is performed on the quantum state generated after the first measurement with completely erased information. In the second scenario, the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome. Induced quantum uncertainties are characterized by means of the Tsallis entropies. For two successive projective measurement of a qubit, we obtain minimal and maximal values of related entropic measures of induced uncertainties. Some conclusions found in the second scenario are extended to arbitrary finite dimensionality. In particular, a connection with mutual unbiasedness is emphasized.Comment: 8 pages, no figures. Minor improvements in the version

Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis' entropies

Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $\alpha$-entropies. For all real $\alpha\in(0;1]$ and integer $\alpha\geq2$, lower bounds on the sum of three $\alpha$-entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate of one of the Pauli observables. Using concavity with respect to the parameter $\alpha$, we derive approximate lower bounds for non-integer $\alpha\in(1;+\infty)$. In the case of pure states, the developed method also allows to obtain upper bounds on the entropic sum for real $\alpha\in(0;1]$ and integer $\alpha\geq2$. For applied purposes, entropic bounds are often used with averaging over the individual entropies. Combining the obtained bounds leads to a band, in which the rescaled average $\alpha$-entropy ranges in the pure-state case. A width of this band is essentially dependent on $\alpha$. It can be interpreted as an evidence for sensitivity in quantifying the complementarity.Comment: 11 pages, one figure. Typos are fixed. Grammatical improvements are made. The bibliography is updated. To appear in Quantum Information Processin