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Quantum coherence constitutes a foundational characteristic of quantum mechanics and is integral to emerging quantum resource theories. However, quantum coherence is severely restricted by environmental noise in general quantum processing, indicated by the loss of information of a quantum system. Such processing can be described by the trade-offs between the coherence and the mixedness. Based on the l 2 norm coherence, conditional von Neumann entropy and Wigner–Yanase skew information, we derive basis-independent constraints on the attainable quantum coherence imposed by the mixedness of a quantum state, which generalize the prior basis-dependent relations, provide fundamental insights into the latent coherence resources present within arbitrary quantum systems that undergo decoherence and quantify the inherent limits on extractable coherence imposed by environmental noise. [ABSTRACT FROM AUTHOR]

Thermodynamics as a fundamental branch of physics examines the relationships between heat, work, and energy. The maximum energy extraction can be characterized by using the passive states that have no extracted energy under any cyclic unitary process. In this paper, the focus is on the concept of marginal passive state energy and derive polygon inequalities for multi‐qubit entangled pure states. It is shown that the marginal passive state energies collectively form a convex polytope for each class of quantum states that are equivalent under SLOCC. Multipartite passive state energy criteria is introduced to classify multipartite entanglement under SLOCC. The present result provides a thermodynamic method to witness multipartite entanglement. [ABSTRACT FROM AUTHOR]

We study the separability of bipartite quantum systems in arbitrary dimensions based on the Bloch representation of density matrices. We present two separability criteria for quantum states in terms of the matrices T α β (ρ) and W a b , α β (ρ) constructed from the correlation tensors in the Bloch representation. These separability criteria can be simplified and detect more entanglement than the previous separability criteria. Detailed examples are given to illustrate the advantages of results. [ABSTRACT FROM AUTHOR]

The entanglement in multipartite quantum system is hard to characterize and quantify, although it has been intensively studied in bipartite systems. The monogamy of entanglement, as a special property of multipartite systems, shows the distribution of entanglement in the system. In this paper, we investigate the monogamy relations for multi-qubit systems. By using two entangled measures, namely the concurrence C and the negativity N c , we establish tighter monogamy inequalities for their α -th power than those in all the existing ones. We also illustrate the tightness of our results for some classes of quantum states. [ABSTRACT FROM AUTHOR]

Noise exists inherently in realistic quantum systems and affects the evolution of quantum systems. We investigate the dynamics of quantum networks in noisy environments by using the fidelity of the quantum evolved states and the classical percolation theory. We propose an analytical framework that allows us to characterize the stability of quantum networks in terms of quantum noises and network topologies. The calculation results of the framework determine the maximal time that quantum networks with different network topologies can maintain the ability to communicate under noise. We demonstrate the results of the framework through examples of specific graphs under amplitude damping and phase damping noises. We further consider the capacity of the quantum network in a noisy environment according to the proposed framework. The analytical framework helps us better understand the evolution time of a quantum network and provides a reference for designing large quantum networks. [ABSTRACT FROM AUTHOR]

Finding the ground state of a Hamiltonian system is of great significance in many‐body quantum physics and quantum chemistry. An improved iterative quantum algorithm to prepare the ground state of a Hamiltonian is proposed. The crucial point is to optimize a cost function on the state space via the quantum gradient descent (QGD) implemented on quantum devices. Practical guideline on the selection of the learning rate in QGD are provided by finding a fundamental upper bound and establishing a relationship between the algorithm and the first‐order approximation of the imaginary time evolution. Furthermore, a variational quantum state preparation method is adapted as a subroutine to generate an ancillary state by utilizing only polylogarithmic quantum resources. The performance of the algorithm is demonstrated by numerical calculations of the deuteron molecule and Heisenberg model without and with noises. Compared with the existing algorithms, the approach has advantages including the higher success probability at each iteration, the measurement precision‐independent sampling complexity, the lower gate complexity, and only quantum resources are required when the ancillary state is well prepared. [ABSTRACT FROM AUTHOR]

Quantum entanglement is the core resource in quantum information processing and quantum computing. It is a significant challenge to effectively characterize the entanglement of quantum states. Recently, elegant separability criterion is presented by Elben et al. (Phys Rev Lett 125:200501, 2020) based on the first three partially transposed (PT) moments of density matrices. Then, Yu et al. (Phys Rev Lett 127:060504, 2021) proposed two general powerful criteria based on the PT moments. In this paper, based on the realignment operations of matrices we propose entanglement detection criteria in terms of such realignment moments. We show by detailed example that the realignment moments can also be used to identify quantum entanglement. [ABSTRACT FROM AUTHOR]

Iteration method is commonly used in solving linear systems of equations. We present quantum algorithms for the relaxed row and column iteration methods by constructing unitary matrices in the iterative processes, which generalize row and column iteration methods to solve linear systems on a quantum computer. Comparing with the conventional row and column iteration methods, the convergence accelerates when appropriate parameters are chosen. Once the quantum states are efficiently prepared, the complexity of our relaxed row and column methods is improved exponentially and is linear with the number of the iteration steps. In addition, phase estimations and Hamiltonian simulations are not required in these algorithms. [ABSTRACT FROM AUTHOR]

The conventional coherence is defined with respect to a fixed orthonormal basis, i.e., to a von Neumann measurement. Recently, generalized quantum coherence with respect to general positive operator-valued measurements has been presented. Several well-defined coherence measures, such as the relative entropy of coherence C r , the l 1 norm of coherence C l 1 and the coherence C T , α based on Tsallis relative entropy with respect to general POVMs have been obtained. In this work, we investigate the properties of C r , C l 1 and C T , α . We estimate the upper bounds of C l 1 ; we show that the minimal error probability of the least square measurement state discrimination is given by C T , 1 / 2 ; we derive the uncertainty relations given by C r , and calculate the average values of C r , C T , α and C l 1 over random pure quantum states. All these results include the corresponding results of the conventional coherence as special cases. [ABSTRACT FROM AUTHOR]

The optimal state-independent lower bounds for the sum of variances or deviations of observables are of significance for the growing number of experiments that reach the uncertainty limited regime. We present a framework for computing the tight uncertainty relations of variance or deviation via determining the uncertainty regions, which are formed by the tuples of two or more of quantum observables in random quantum states induced from the uniform Haar measure on the purified states. From the analytical formulae of these uncertainty regions, we present state-independent uncertainty inequalities satisfied by the sum of variances or deviations of two, three and arbitrary many observables, from which experimentally friend entanglement detection criteria are derived for bipartite and tripartite systems. [ABSTRACT FROM AUTHOR]

Quantum catalytic transformations play important roles in the transformation of quantum entangled states under local operations and classical communications (LOCC). The key problems in catalytic transformations are the existence and the bounds on the catalytic states. We present the necessary conditions of catalytic states based on a set of points given by the Schmidt coefficients of the entangled source and target states. The lower bounds on the dimensions of the catalytic states are also investigated. Moreover, we give a detailed protocol of quantum mixed state transformation under entanglement-assisted LOCC. [ABSTRACT FROM AUTHOR]

We study quantum information masking of arbitrary dimensional states. Given a set of fixed reducing pure states, we study the linear combinations of them, such that they all have the same marginal states with the given ones. We define the so-called Hadamard set of quantum states whose Gram–Schmidt matrix can be diagonalized by Hadamard unitary matrices. We show that any Hadamard set can be deterministically masked by a unitary operation. We analyze the states which can be masked together with the given Hadamard set using the result about the linear combinations of fixed reducing states. Detailed examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

To distill quantum coherence, in [Phys. Rev. A 99, 012321 (2019)], authors proposed a strictly incoherent operation that produces one of a family of maximally coherent states of variable dimension from any pure quantum state. For a d-dimensional pure state, an incoherent state may be obtained with a nonzero probability, which results in a complete waste of resource, namely the probability of transforming a given pure state to the incoherent state is not zero. Here, we give a specific method to avoid a complete waste of resource with the maximal probability to transform the pure state to a d-dimensional maximally coherent state and study the range of average coherence of the corresponding output state in this case. We also give a method to transform a mixed state to a maximally coherent state with probability. [ABSTRACT FROM AUTHOR]

We study the detection of multipartite entanglement based on the generalized local uncertainty relations. A sufficient criterion for the entanglement of four-partite quantum systems is presented in terms of the local uncertainty relations. Detailed examples are given to illustrate the advantages of our criterion. The approach is generalized to general multipartite entanglement cases. [ABSTRACT FROM AUTHOR]

Quantum multipartite entangled states play significant roles in quantum information processing. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant mixed orthogonal arrays (IrMOAs) and thus provide positive answers to two open problems. The first is the extension of the method for constructing homogeneous systems from orthogonal arrays to heterogeneous multipartite systems with different individual levels. The second is the existence of k-uniform states in heterogeneous quantum systems. We present explicit constructions of two- and three-uniform states for arbitrary heterogeneous multipartite systems with coprime individual levels and characterize the entangled states in heterogeneous systems consisting of subsystems with nonprime power dimensions as well. Moreover, we obtain infinite classes of k-uniform states for heterogeneous multipartite systems for any k ≥ 2 . The non-existence of a class of IrMOAs is also proved. [ABSTRACT FROM AUTHOR]

Coherence and entanglement are fundamental concepts in resource theory. The coherence (entanglement) of assistance is the coherence (entanglement) that can be extracted assisted by another party with local measurement and classical communication. We introduce and study the general coherence of assistance. First, in terms of real symmetric concave functions on the probability simplex, the coherence of assistance and the entanglement of assistance are shown to be in one-to-one correspondence. We then introduce two classes of quantum states: the assisted maximally coherent states and the assisted maximally entangled states. They can be transformed into maximally coherent or entangled pure states with the help of another party using local measurement and classical communication. We give necessary conditions for states to be assisted maximally coherent or assisted maximally entangled. Based on these, a unified framework between coherence and entanglement including coherence (entanglement) measures, coherence (entanglement) of assistance, coherence (entanglement) resources is proposed. Then we show that the coherence of assistance as well as entanglement of assistance are strictly larger than the coherence of convex roof and entanglement of convex roof for all full rank density matrices. So all full rank quantum states are distillable in the assisted coherence distillation. [ABSTRACT FROM AUTHOR]

Quantum coherence is the basic characteristic of quantum mechanics. It is an important task to order states with various coherence measures. We investigate the order relations between the l 1 norm of coherence and the α -affinity of coherence for quantum states. We find that the l 1 norm of coherence and any α -affinity of coherence provide the same ordering for qubit pure states and maximally coherent mixed qubit states. We also show that there exist some qubit states which give rise to different ordering under the l 1 norm of coherence and any α -affinity of coherence. In addition, the l 1 norm of coherence and α -affinity of coherence do not generate the same ordering for some high-dimensional pure states, but they do for some special X states of two-qubit systems. [ABSTRACT FROM AUTHOR]

The complementarity (upper bound) and uncertainty relations (lower bound) of the coherence via skew information under mutually unbiased bases (MUBs) are studied. The complementarity relation for the geometric measure of coherence is also obtained based on the relation between the coherence via skew information and the geometric measure of coherence. As applications, two tighter upper bounds are presented on the minimum error probabilities that discriminate a set of pure states with the least square measurement, which improve the results of Xiong et al. (Phys Rev A 98:032324, 2018). [ABSTRACT FROM AUTHOR]

We study the average skew information-based coherence for both random pure and mixed states. The explicit formulae of the average skew information-based coherence are derived and shown to be the functions of the dimension N of the state space. We demonstrate that as N approaches to infinity, the average coherence is 1 for random pure states, and a positive constant less than 1/2 for random mixed states. We also explore the typicality of average skew information-based coherence of random quantum states. Furthermore, we identify a coherent subspace such that the amount of the skew information-based coherence for each pure state in this subspace can be bounded from below almost always by a fixed number that is arbitrarily close to the typical value of coherence. [ABSTRACT FROM AUTHOR]

We study the degree to which the coherence of quantum states is affected by noise. We give the definition of the n-th decay rate and investigate the coherence of Bell-diagonal states under n iterations of channels. We derive explicit formulas of the n-th decay rates based on l1 norm of coherence, relative entropy of coherence and skew information-based coherence. It is found that the larger n is, the faster the n-th decay rate decreases as the parameter p of Bell-diagonal states increases. Moreover, for any fixed n, with the increase of p, Bell-diagonal states can be completely incoherent under generalized amplitude damping (GAD) channels, depolarization (DEP) channels and phase flip (PF) channels, while this is not the case for bit flip (BF) channels and bit-phase flip (BPF) channels. We also investigate the geometry of the relative entropy of coherence and skew information-based coherence of Bell-diagonal states under different channels when the n-th decay rate is one, i.e., the coherence is frozen. It is shown that compared with BF and BPF channels, when n is large enough, the coherence of Bell-diagonal states will not be frozen under GAD, DEP and PF channels. For skew information-based coherence, similar properties of coherence freezing are found. [ABSTRACT FROM AUTHOR]

Genuine multipartite entanglement plays important roles in quantum information processing. The detection of genuine multipartite entanglement has been long time a challenging problem in the theory of quantum entanglement. We propose a criterion for detecting genuine tripartite entanglement of arbitrary dimensional tripartite states based on quantum Fisher information. We show that this criterion is more effective for some states in detecting genuine tripartite entanglement by detailed example. [ABSTRACT FROM AUTHOR]

We study the concurrence of arbitrary-dimensional multipartite quantum states. Analytical lower bounds of concurrence for tripartite quantum states are derived by projecting high-dimensional states to 2 ⊗ 2 ⊗ 2 substates. The results are then generalized to arbitrary multipartite quantum systems. Furthermore, the scheme enables us obtain lower bounds of concurrence for arbitrary four-partite quantum states by projecting high-dimensional states to arbitrary given lower dimensional substates. By detailed examples we show that our results improve the existing lower bounds of concurrence. [ABSTRACT FROM AUTHOR]

Entanglement of high-dimensional and multipartite quantum systems offer promising perspectives in quantum information processing. However, the characterization and measure of such kind of entanglement is of great challenge. Here, we consider the overlaps between the maximal quantum mean values and the classical bound of the CHSH inequalities for pairwise-qubit states in two-dimensional subspaces. We show that the concurrence of a pure state in any high-dimensional multipartite system can be equivalently represented by these overlaps. Here, we consider the projections of an arbitrary high-dimensional multipartite state to two-qubit states. We investigate the non-localities of these projected two-qubit sub-states by their violations of CHSH inequalities. From these violations, the overlaps between the maximal quantum mean values and the classical bound of the CHSH inequality, we show that the concurrence of a high-dimensional multipartite pure state can be exactly expressed by these overlaps. We further derive a lower bound of the concurrence for any quantum states, which is tight for pure states. The lower bound not only imposes restriction on the non-locality distributions among the pairwise-qubit states, but also supplies a sufficient condition for distillation of bipartite entanglement. Effective criteria for detecting genuine tripartite entanglement and the lower bound of concurrence for genuine tripartite entanglement are also presented based on such non-localities. [ABSTRACT FROM AUTHOR]

Monogamy and polygamy are the most striking features of the quantum world. We investigate the monogamy and polygamy relations satisfied by all quantum correlation measures for arbitrary multipartite quantum states. By introducing residual quantum correlations, analytical polygamy inequalities are presented, which are shown to be tighter than the existing ones. Then, similar to polygamy relations, we obtain strong monogamy relations that are better than all the existing ones. Typical examples are presented for illustration. [ABSTRACT FROM AUTHOR]

The monogamy relations of quantum correlation restrict the sharability of quantum correlations in multipartite quantum states. We show that all measures of quantum correlations satisfy some kind of monogamy relations for arbitrary multipartite quantum states. Moreover, by introducing residual quantum correlations, we present tighter monogamy inequalities that are better than all the existing ones. In particular, for multi-qubit pure states, we also establish new monogamous relations based on the concurrence and concurrence of assistance under the partition of the first two qubits and the remaining ones. [ABSTRACT FROM AUTHOR]

Entanglement polygamy, like entanglement monogamy, is a fundamental property of multipartite quantum states. We investigate the polygamy relations related to the concurrence C and the entanglement of formation E for general n-qubit states. We extend the results in [Phys. Rev. A 90, 024304 (2014)] from the parameter region α ≤ 0 to α ≤ α0, where 0 < α0 ≤ 2 for C, and 0 < α 0 ≤ 2 for E. [ABSTRACT FROM AUTHOR]

The geometry of the structure of entanglement and discord for Bell-diagonal states is depicted by Lang and Caves (Phys. Rev. Lett. 105, 150501, 2010). In this paper, we investigate the geometry with respect to several distance-based quantifiers of coherence for Bell-diagonal states. We find that as both l1 norm and relative entropy of coherence vary continuously from zero to one, their related geometric surfaces move from the region of separable states to the region of entangled states, a fact illustrating intuitively that quantum states with nonzero coherence can be used for entanglement creation. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equals to its relative entropy of coherence, and depict the surfaces related to the equality. We give surfaces of relative entropy of coherence for X states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherences under local nondissipative channels decrease. [ABSTRACT FROM AUTHOR]

We present uncertainty relations based onWigner-Yanase-Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states. [ABSTRACT FROM AUTHOR]

We study the perfectly local indistinguishability of multipartite product states. Firstly, we follow the method of Zhang et al. (Phys Rev A 93:012314, 2016) to give another more concise set of $$2n-1$$ orthogonal product states in $${\mathbb {C}}^m\otimes {\mathbb {C}}^n$$ $$(4\le m\le n)$$ which can not be distinguished by local operations and classical communication. Then we use the three-dimensional cubes to present some product states which give us an intuitive view on how to construct locally indistinguishable product states in tripartite quantum systems. At last, we give an explicit construction of locally indistinguishable orthogonal product states for general multipartite systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the concurrence of arbitrary-dimensional tripartite quantum states. An explicit operational lower bound of concurrence is obtained in terms of the concurrence of substates. A given example shows that our lower bound may improve the well-known existing lower bounds of concurrence. The significance of our result is to get a lower bound when we study the concurrence of arbitrary $$m\otimes n\otimes l$$ -dimensional tripartite quantum states. [ABSTRACT FROM AUTHOR]

We investigate one-way quantum deficit for 2 ⊗ d systems. Analytical expressions of one-way quantum deficit under both von Neumann measurement and weak measurement are presented. As an illustration, qubit-qutrit systems are studied in detail. It is shown that there exists non-zero one-way quantum deficit even quantum entanglement vanishes. Moreover, one-way quantum deficit via weak measurement turns out to be weaker than that via von Neumann measurement. The dynamics of entanglement and one-way quantum deficit under dephasing channels is also investigated. [ABSTRACT FROM AUTHOR]

We study sum uncertainty relations for arbitrary finite N quantum mechanical observables. Some uncertainty inequalities are presented by using skew information introduced by Wigner and Yanase. These uncertainty inequalities are nontrivial as long as the observables are mutually noncommutative. The relations among these new and existing uncertainty inequalities have been investigated. Detailed examples are presented. [ABSTRACT FROM AUTHOR]

We study one-way quantum deficit of two-qubit X states systematically from analytical derivations. An effective approach to compute one-way quantum deficit of two-qubit X states has been provided. Analytical results are presented as for detailed examples. Moreover, we demonstrate the decoherence of one-way quantum deficit under phase damping channel. [ABSTRACT FROM AUTHOR]

In this paper, we study the one-way local operations and classical communication (LOCC) problem. In $$\mathbb {C}^d\otimes \mathbb {C}^d$$ with $$d\ge 4$$ , we construct a set of $$3\lceil \sqrt{d}\rceil -1$$ one-way LOCC indistinguishable maximally entangled states which are generalized Bell states. Moreover, we show that there are four maximally entangled states which cannot be perfectly distinguished by one-way LOCC measurements for any dimension $$d\ge 4$$ . [ABSTRACT FROM AUTHOR]

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)$ are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in $\mathbb {C}^{2}\otimes \mathbb {C}^{4}$ and $\mathbb {C}^{2}\otimes \mathbb {C}^{8}$, respectively. [ABSTRACT FROM AUTHOR]

We revisit the quantum discord, quantum entanglement and local hidden variable models in quantum mechanics, and present a kind of understanding of quantum states from the view of correlations given by the probability distributions of local measurements outcomes. [ABSTRACT FROM AUTHOR]

We study the quantum separability problem by using general symmetric informationally complete measurements and present separability criteria for both $$d$$ -dimensional bipartite and multipartite systems. The criterion for bipartite quantum states is effective in detecting several well-known classes of quantum states. For isotropic states, it becomes both necessary and sufficient. Furthermore, our criteria can be experimentally implemented, and the criterion for two-qudit states requires less local measurements than the one based on mutually unbiased measurements. [ABSTRACT FROM AUTHOR]

In this paper, we give a method for the local unitary equivalent problem which is more efficient than that was proposed by Bin Liu et al []. [ABSTRACT FROM AUTHOR]

We study the concurrence of four-qubit quantum states and provide analytical lower bounds of concurrence in terms of the monogamy inequality of concurrence for qubit systems. It is shown that these lower bounds are able to improve the existing bounds and detect entanglement better. The approach is generalized to arbitrary qubit systems. [ABSTRACT FROM AUTHOR]

We study certain quantum states for which the PPT criterion is both sufficient and necessary for separability. A class of n× n bipartite mixed states is presented and the conditions of PPT for these states are derived. The separable pure state decompositions of these states are explicitly constructed when they are PPT. [ABSTRACT FROM AUTHOR]

We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full rank mixed states with non-degenerate eigenvalues, this set of invariants is also the set of the necessary and sufficient conditions for the local unitary equivalence of such two states. [ABSTRACT FROM AUTHOR]

Abstract: In terms of the analysis of fixed point subgroup and tensor decomposability of certain matrices, we study the equivalence of quantum bipartite mixed states under local unitary transformations. For non-degenerate case an operational criterion for the equivalence of two such mixed bipartite states under local unitary transformations is presented. [Copyright &y& Elsevier]

Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the l1-norm measure of coherence for convex combination of arbitrary n pure states of a quantum state ρ. Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in Dai et al. (Phys. Rev. A 96:062308, 2017). Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities. [ABSTRACT FROM AUTHOR]

We study the quantum separability problem by using general symmetric informationally complete measurements and present a separability criterion for arbitrary dimensional bipartite systems. We show by detailed examples that our criterion is more powerful than the existing ones in entanglement detection. [ABSTRACT FROM AUTHOR]

The construction of unextendible maximally entangled bases is tightly related to quantum information processing like local state discrimination. We put forward two constructions of UMEBs in Cpd⊗Cqd(p≤q) based on the constructions of UMEBs in Cd⊗Cd and in Cp⊗Cq, which generalizes the results in Guo (Phys Rev A 94:052302, 2016) by two approaches. Two different 48-member UMEBs in C6⊗C9 have been constructed in detail. [ABSTRACT FROM AUTHOR]

We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented. [ABSTRACT FROM AUTHOR]

Abstract: In this Letter, two counterexamples show that the superadditivity inequality of relative entropy is not true even for the full-ranked quantum states. Thus, an inequality of quantum channels and complementary channels is not also true. Finally, a conjecture of weak superadditivity inequality is presented. [Copyright &y& Elsevier]