We derive an extremal equation for optimal completely-positive map which most closely approximates a given transformation between pure quantum states. Moreover, we also obtain an upper bound on the maximal mean fidelity that can be attained by the optimal approximate transformation. The developed formalism is applied to universal-NOT gate, quantum cloning machines, quantum entanglers, and qubit theta-shifter.
We define an entanglement witness in a composite quantum system as an observable having nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are as nonnegative biquadratic forms or as positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\times 3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not…
Convex sets of completely positive maps and positive semidefinite kernels are considered in the most general context of modules over $C^*$-algebras and a complete charaterization of their extreme points is obtained. As a byproduct, we determine extreme quantum instruments, preparations, channels, and extreme autocorrelation functions. Various applications to quantum information and measurement theories are given. The structure of quantum instruments is analyzed thoroughly.
In this paper we describe a new connection between UPB (unextendable product bases) and P (positive) maps which are not CP (completely positive). We show that inner automorphisms of the set of P maps which are not CP, produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth PPT (positive under partial transpose) entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orth…
We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also…
Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level system, the extremal channel can be characterized in terms of $N^2$-$N$ real parameters coupled with rotations. We give a representation for $N$= 2, 3, 4.
The paper is devoted to the problem of classification of extremal positive maps acting between $B(K)$ and $B(H)$ where $K$ and $H$ are Hilbert spaces. It is shown that every positive map with the property that $\rank ϕ(P)\leq 1$ for any one-dimensional projection $P$ is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that $\rank ϕ(P)\leq 1…
The diamond and completely bounded norms for linear maps play an increasingly important role in quantum information science, providing fundamental stabilized distance measures for differences of quantum operations. Based on the theory of completely bounded maps, we formulate an algorithm to compute the norm of an arbitrary linear map. We present an implementation of the algorithm via Maple, discuss its efficiency, and consider the case of differences of unitary maps.
The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix representation of Hermitian linear map, we show that every positive map which is not completely positive is a {\em difference} of two completely positive maps. A necessary and sufficient condition for a positive map which is not completely positive is also presented, which is illustrated by some examples.
A dynamical map is a map which takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) using a spectral decomposition. The method of the construction applies to more general maps which need not be completely positive. The OSR not unique; there is a freedom to choose the set of operators in the OSR differently, yet still obtain the same map. Here we identify all maps which are equivalent to a given map. Whereas the freedom for completely positive maps is unitary, the freedom for maps which are not necessarily completely positive is pseudo-unitary…
We investigate completely positive maps for an open system interacting with its environment. The families of the initial states for which the reduced dynamics can be described by a completely positive map are identified within the framework of direct-sum decomposition of state space. They includes not only separable states with vanishing or nonvanishing quantum discord but also entangled states. A general expression of the families as well as the Kraus operators for the completely positive maps are explicitly given. It significantly extends the previous results.
A new method of analysing positive bistochastic maps on the algebra of complex matrices $M_{3}$ has been proposed. By identifying the set of such maps with a convex set of linear operators on $\mathbb{R}^{8}$, one can employ techniques from the theory of compact affine semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been show that all positive bistochastic maps, extremal in the set of…
We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full positivity domain. The definition is motivated by specific examples. We prove that for valid NCP maps, the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Based on this, we argue that in general the volume measure of qubit maps, including NCP maps, is not well defined.
We present an "uncertainty principle" for quantum channels, showing a relationship between the dimensions of the range of a channel and the range of its complement. We examine some interesting specific cases, and discuss consequences for privacy of quantum channels.
A theorem of Blackwell about comparison between information structures in classical statistics is given an analogue in the quantum probabilistic setup. The theorem provides an operational interpretation for trace-preserving completely positive maps, which are the natural quantum analogue of classical stochastic maps. The proof of the theorem relies on the separation theorem for convex sets and on quantum teleportation.
By introducing an operator sum representation for arbitrary linear maps, we develop a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory of "linear quantum error correction" is applicable in cases where the standard and restrictive assumption of a factorized initial system-bath state does not apply.
The notion of potential output purity of a completely positive map is introduced as a generalization of the regularized output purity. An upper bound is derived for this quantity, and for several classes of maps (including CQ, QC and Hadamard channels) it is shown that potential purity does not exceed the standard output purity. As an application the potential purity is used to bound the logarithmic Sobolev constant of a product of depolarizing channel semigroups.
The time evolution of an initially uncorrelated system is governed by a completely positive (CP) map. More generally, the system may contain initial (quantum) correlations with an environment, in which case the system evolves according to a not-completely positive (NCP) map. It is an interesting question what the relative measure is for these two types of maps within the set of positive maps. After indicating the scope of the full problem of computing the true volume for generic maps acting on a qubit, we study the case of Pauli channels in an abstract space whose elements represent an equival…
The phenomenological dissipation of the Bloch equations is reexamined in the context of completely positive maps. Such maps occur if the dissipation arises from a reduction of a unitary evolution of a system coupled to a reservoir. In such a case the reduced dynamics for the system alone will always yield completely positive maps of the density operator. We show that, for Markovian Bloch maps, the requirement of complete positivity imposes some Bloch inequalities on the phenomenological damping constants. For non-Markovian Bloch maps some kind of Bloch inequalities involving eigenvalues of the…
In this work we examine quantum states which have non-negative amplitudes (in a fixed basis) and the channels which preserve them. These states include the ground states of stoquastic Hamiltonians and they are of interest since they avoid the Sign Problem and can thus be efficiently simulated. In optimization theory, the convex cone generated by such states is called the set of completely positive (CP) matrices (not be confused with completely positive superoperators). We introduce quantum channels which preserve these states and call them completely positive completely positive. To study thes…
It is widely believed that the Choi map generates an extremal ray in the cone $\mathcal P(M_3)$ of all positive linear maps between $C^*$-algebra $M_3$ of all $n\times n$ matrices over the complex field. But the only proven fact is that the Choi map generates the extremal ray in the cone of all positive linear map preserving all real symmetric $3\times 3$ matrices. In this note, we show that the Choi map is indeed extremal in the cone $\mathcal P(M_3)$. We also clarify some misclaims about the correspondence between positive semi-definite biquadratic real forms and postive linear maps, and dis…
We generalize the notion of relative phase to completely positive maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map.
We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We outline t…
A class of quantum channels and completely positive maps (CPMs) are introduced and investigated. These, which we call subspace preserving (SP) CPMs has, in the case of trace preserving CPMs, a simple interpretation as those which preserve probability weights on a given orthogonal sum decomposition of the Hilbert space of a quantum system. Several equivalent characterizations of SP CPMs are proved and an explicit construction of all SP CPMs, is provided. For a subclass of the SP channels a construction in terms of joint unitary evolution with an ancilla system, is presented.
Axiomatic approach to measurement theory is developed. All the possible statistical properties of apparatuses measuring an observable with nondegenerate spectrum allowed in standard quantum mechanics are characterized.
The description of the dynamics of a system that may be correlated with its environment is only meaningful within the context of a specific framework. Different frameworks rely upon different assumptions about the initial system-environment state. We reexamine the connections between complete-positivity and quantum discord within two different sets of assumptions about the relevant family of initial states. We present an example of a system-environment state with non-vanishing quantum discord that leads to a completely-positive map. This invalidates an earlier claim on the necessity of vanishi…
We introduce a framework for the construction of completely positive maps for subsystems of indistinguishable fermionic particles. In this scenario, the initial global state is always correlated, and it is not possible to tell system and environment apart. Nonetheless, a reduced map in the operator sum representation is possible for some sets of states where the only non-classical correlation present is exchange.
Two long standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (1) a map between the initial and final system states, and (2) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is "all", with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from th…
Einstein introduced the locality principle which states that all physical effect in some finite space-time region does not influence its space-like separated finite region. Recently, in algebraic quantum field theory, Rédei captured the idea of the locality principle by the notion of operational separability. The operation in operational separability is performed in some finite space-time region, and leaves unchanged the state in its space-like separated finite space-time region. This operation is defined with a completely positive map. In the present paper, we justify using a completely posit…
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment, or if the applied transformation is correlated with the state of the system. We provide a physically motivated definition of accessible non-CP maps and derive two necessary conditions for a map to be accessible. We also show that entanglement between the system and the environment is not necessary to generate a non-CP dynamics. We describe two simple approxi…
We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3x3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles t…
We consider open quantum systems with factorized initial states, providing the structure of the reduced system dynamics, in terms of environment cumulants. We show that such completely positive (CP) and trace preserving (TP) maps can be unraveled by linear stochastic Schrödinger equations (SSEs) characterized by sets of colored stochastic processes (with n-th order cumulants). We obtain both the conditions such that the SSEs provide CPTP dynamics, and those for unraveling an open system dynamics. We then focus on Gaussian non-Markovian unravellings, whose known structure displays a functional …
The problem of conditions on the initial correlations between the system and the environment that lead to completely positive (CP) or not-completely positive (NCP) maps has been studied by various authors. Two lines of study may be discerned: one concerned with families of initial correlations that induce CP dynamics under the application of an arbitrary joint unitary on the system and environment; the other concerned with specific initial states that may be highly entangled. Here we study the latter problem, and highlight the interplay between the initial correlations and the unitary applied.…
This article introduces PnCP, a MATLAB toolbox for constructing positive maps which are not completely positive. We survey optimization and sum of squares relaxation techniques to find the most numerically efficient methods for this construction. We also show how this package can be applied to the problem of classifying entanglement in quantum states.
It is a well-known result due to E. Størmer that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. Fillipov and K. Magadov. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show …
Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.
Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.
King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained.
In this work we show that, given a linear map from a general operator space into the dual of a C$^*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(1,cb)$-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.
One of the interesting problems on optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable witness which neither has the spanning property nor is associated with extremal positive linear map. Here, we answer this question negatively by examining the extremality of the positive linear maps constructed by Qi and Hou [J. Phys. A {\bf 44}, 215305 (2100)].
We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous one-parameter semigroups of roots. We present structural and general existence and non-existence results, some special examples in settings where we understand the situation better, and several challenging open problems. Our study is closely related to Elfving's embedding problem in classical probability and the divisibility problem of quantum channels.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one parameter family of indecomposabl…
We study the convex set of all bipartite quantum states with fixed marginal states. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182, [1]]. Here we present an alternative necessary and sufficient condition for a state with given marginals to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2 x 2 we give a simple new…
King and Ruskai asked whether the $p\to q$ norm of a completely positive map $Φ$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}$, is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.
For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in d…
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results we use a symmetrization technique from the theory of entanglement distillation, and analyze when certain symmetric …
In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce…
