We study a natural generalization of the additivity problem in quantum information theory: given a pair of quantum channels, then what is the set of convex trace functions that attain their maximum on unentangled inputs, if they are applied to the corresponding output state?
We prove several results on the structure of the set of those convex functions that are "additive" in this more general sense. In particular, we show that all operator convex functions are additive for the Werner-Holevo channel in 3x3 dimensions, which contains the well-known additivity results for this channel as special cases.
The continuity properties of the convex closure of the output entropy of infinite dimensional channels and their applications to the additivity problem are considered. The main result of this paper is the statement that the superadditivity of the convex closure of the output entropy for all finite dimensional channels implies the superadditivity of the convex closure of the output entropy for all infinite dimensional channels, which provides the analogous statements for the strong superadditivity of the EoF and for the additivity of the minimal output entropy. The above result also provides in…
We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This prov…
We study quantum channels with respect to their image, i.e., the image of the set of density operators under the action of the channel. We first characterize the set of quantum channels having polytopic images and show that additivity of the minimal output entropy can be violated in this class. We then provide a complete characterization of quantum channels $T$ that are universally image additive in the sense that for any quantum channel $S$, the image of $T \otimes S$ is the convex hull of the tensor product of the images of $T$ and $S$. These channels turn out to form a strict subset of enta…
We simplify some conjectures in quantum information theory; the additivity of minimal output entropy, the multiplicativity of maximal output p-norm and the superadditivity of convex closure of output entropy. We construct a unital channel for a given channel so that they share the above additivity properties; we can reduce the conjectures for all channels to those for unital channels.
We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification …
Quantum resource theories offer a powerful framework for studying various phenomena in quantum physics. Despite considerable effort has been devoted to developing a unified framework of resource theories, there are few common properties that hold for all quantum resources. In this paper, we fill this gap by introducing the flag additivity based on the tensor product structure and the flag basis for the general quantum resources. To illustrate the usefulness of flag additivity, we show that flag additivity can be used to derive other nontrivial properties in quantum resource theories, e.g., str…
The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden-Winter disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).
Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environmen…
A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output $p$-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is …
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property…
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-n…
Quantum monotone metric was introduced by Petz,and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always one based on completely positive, trace preserving (CPTP) maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators,and did not have some ideal properties such as monotonicity and convexity with …
We consider convex trace functions $Φ_{p,q,s} = Trace[ (A^{q/2}B^p A^{q/2})^s]$ where $A$ and $B$ are positive $n\times n$ matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of $A^{q/2}B^p A^{q/2}$ and convexity/concavity of the closely related trace functional $Trace[ A^{q/2}B^p A^{q/2} C^r]$. For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta Rényi entropy conjectures are proved for some cases.
We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for the map (A,B) --> Tr K^* A^p K B^{1-p} Lieb's joint concavity for 0 < p < 1 and Ando's joint convexity for 1 < p < 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix ineq…
Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Bar-Ilan University（巴伊兰大学）；Technion - Israel Institute of Technology（以色列理工学院）
Quantum conditional entropies from convex trace functionals
Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
We study quantum channels that are close to another channel with weakly additive Holevo information and derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above that measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as…
We consider the following trace function on n-tuples of positive operators: Φ_p(A_1,A_2,...,A_n) = Trace (\sum_{j=1}^n A_j^p)^{1/p} and prove that it is jointly concave for 0<p\le 1 and convex for p=2. We then derive from this a Minkowski type inequality for operators on a tensor product of three Hilbert spaces, and show how this implies the strong subadditivity of quantum mechanical entropy. For p>2, Φ_p is neither convex nor concave. We conjecture that Φ_p is convex for 1<p<2, but our methods do not show this.
In this paper we consider the $χ$-function (the Holevo capacity of constrained channel) and the convex closure of the output entropy for arbitrary infinite dimensional channel. It is shown that the $χ$-function of an arbitrary channel is a concave lower semicontinuous function on the whole state space, having continuous restriction to any set of continuity of the output entropy. The explicit representation for the convex closure of the output entropy is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the outp…
In this article the operator trace function $ Λ_{r,s}(A)[K, M] := {\operatorname{tr}}(K^*A^r M A^r K)^s$ is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data processing inequalities of various relative entropies. In the paper the interplay between $Λ_ {r,s}$ and the well-known operator functions $Γ_{p,s}$ and $Ψ_{p,q,s}$ is used to study the stability of their convexity (concavity) properties. This interplay may…
Quantum channels, which are completely positive and trace preserving mappings, can alter the dimension of a system; e.g., a quantum channel from a qubit to a qutrit. We study the convex set properties of dimension-altering quantum channels, and particularly the channel decomposition problem in terms of convex sum of extreme channels. We provide various quantum circuit representations of extreme and generalized extreme channels, which can be employed in an optimization to approximately decompose an arbitrary channel. Numerical simulations of low-dimensional channels are performed to demonstrate…
New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum …
We develop an approximation approach to infinite dimensional quantum channels based on detailed investigation of the continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair ``channel, input state''. The obtained results are then applied to the following problems: continuity of the $χ$-capacity as function of a channel; strong additivity of the $χ$-capacity for infinite dimensional channels; the analytical expression for the convex closure of the output entropy of arbitrary quantum channel.
We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this conditio…
In classical information theory, the information bottleneck method (IBM) can be regarded as a method of lossy data compression which focusses on preserving meaningful (or relevant) information. As such it has recently gained a lot of attention, primarily for its applications in machine learning and neural networks. A quantum analogue of the IBM has recently been defined, and an attempt at providing an operational interpretation of the so-called quantum IB function as an optimal rate of an information-theoretic task, has recently been made by Salek et al. However, the interpretation given in th…
We studied the change of the nonlocal correlation of the entanglement in Rindler spacetime by showing that the Unruh effect can be interpreted as a noisy quantum channel having a complete positive and trace preserving map with an operator sum representation. It is shown that the entanglement fidelity is obtained in analytic form from the operator sum representation, which agrees well numerically with the entanglement monotone and the entanglement measure obtained previously. Non-zero entropy exchange between the system Q and the region II of the Rindler wedge indicates the nonlocal correlation…
We consider properties of quantum channels with use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map $(q,s)$-entropy is naturally defined as the unified $(q,s)$-entropy of rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. T…
Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ense…
