Stinespring Representation Research Articles

Diagonal unitary and orthogonal symmetries in quantum theory

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

The block Schur product is a Hadamard product

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product.
 We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product.
 We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

Operator -norms and their use

We consider a family of equivalent norms (called operator -norms) on the algebra of all bounded operators on a separable Hilbert space induced by a positive densely defined operator on . By choosing different generating operators we can obtain the operator -norms producing different topologies, in particular, the strong operator topology on bounded subsets of . We obtain a generalised version of the Kretschmann-Schlingemann- Werner theorem, which shows that the Stinespring representation of completely positive linear maps is continuous with respect to the energy-constrained norm of complete boundedness on the set of completely positive linear maps and the operator -norm on the set of Stinespring operators. The operator -norms induced by a positive operator are well defined for linear operators relatively bounded with respect to the operator , and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between operator -norms and the standard characteristics of -bounded operators. Operator -norms allow us to obtain simple upper bounds and continuity bounds for some functions depending on -bounded operators used in applications. Bibliography: 29 titles.

Strong convergence of quantum channels: Continuity of the Stinespring dilation and discontinuity of the unitary dilation

It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics on the sets of quantum channels and Stinespring isometries is given. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. The discontinuity of the unitary dilation is shown by constructing a strongly converging sequence of quantum channels that cannot be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows us to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

Factorization of an adjontable Markov operator

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

О дилатации одного класса вполне положительных отображений

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.

Quantum Query Algorithms Are Completely Bounded Forms

We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials. Revision note: A mistake was found in the proof of the second result on degree-4 polynomials far from 2-query quantum algorithms. An explanation of the issue, a corrected proof and stronger examples are presented in work of Escudero Guti\'errez and the second author.

Modular Sesquilinear Forms and Generalized Stinespring Representation

We consider completely positive maps defined on locally C*-algebra and taking values in the space of sesquilinear forms on Hilbert C*-module M. We construct the Stinespring type representation for this type of maps and show that any two minimal Stinespring representations are unitarily equivalent.

On the complete boundedness of the Schur block product

We give a Stinespring representation of the Schur block product on pairs of square matrices with entries in a C$^*$-algebra as a completely bounded bilinear operator of the form \begin{equation*} A:=(a_{ij}), B:= (b_{ij}): A \square B := (a_{ij}b_{ij}) = V^*\lambda (A)F \lambda (B) V, \end{equation*} such that $V$ is an isometry, $\lambda$ is a *-representation, and $F$ is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new inequalities on the diagonals of matrices: \begin{align*} \|A \square B\| &\!\leq \! \|A\|_r \|B\|_c \text { operator, row, and column norm;} \\ - \mathrm {diag}(A^*A) &\!\leq \! A^*\square A \leq \mathrm {diag}(A^*A), \\ \forall \Xi , \Gamma \!\in \! \mathbb {C}^n\!\otimes \! H: |\langle (A \square B) \Xi , \Gamma \rangle | & \!\leq \! \|\big (\mathrm {diag}(B^*B)\big )^{1/2}\Xi \| \|\big (\mathrm {diag}(AA^*)\big )^{1/2}\Gamma \|. \end{align*}

Extremal Unital Completely Positive Maps and Their Symmetries

We consider the set $$P_1({\mathcal A},{\mathcal M})$$ (respectively $$CP_1({\mathcal A},{\mathcal M})$$ of unital positive (completely) maps from a $$C^*$$ algebra $${\mathcal A}$$ to a von-Neumann sub-algebra $${\mathcal M}$$ of $${\mathcal B}({\mathcal H})$$ , the algebra of bounded linear operators on a Hilbert space $${\mathcal H}$$ . We study the extreme points of the convex set $$P_1({\mathcal A},{\mathcal M})$$ ( $$CP_1({\mathcal A},{\mathcal M})$$ ) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from $$\hat{{\mathcal A}}$$ to $${\mathcal M}$$ , where $${\mathcal A}^{**}$$ is the universal enveloping von-Neumann algebra over $${\mathcal A}$$ . If $${\mathcal A}={\mathcal M}$$ then a (completely) positive map $$\tau $$ admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if $${\mathcal M}$$ is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on $${\mathcal M}$$ to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.

Matrix KSGNS construction and a Radon–Nikodym type theorem

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C∗-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon–Nikodym theorem for this type of completely positive n×n matrices.

Data processing for the sandwiched R\xe9nyi divergence: a condition for equality

The alpha -sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for alpha ge 1/2. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the alpha -sandwiched Rényi divergence for all alpha ge 1/2. For the range alpha in [1/2,1), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

Classical and Quantum Parts of the Quantum Dynamics: The Discrete-Time Case

In the study of open quantum systems modeled by a unitary evolution of a bipartite Hilbert space, we address the question of which parts of the environment can be said to have a “classical action” on the system, in the sense of acting as a classical stochastic process. Our method relies on the definition of the Environment Algebra, a relevant von Neumann algebra of the environment. With this algebra we define the classical parts of the environment and prove a decomposition between a maximal classical part and a quantum part. Then we investigate what other information can be obtained via this algebra, which leads us to define a more pertinent algebra: the Environment Action Algebra. This second algebra is linked to the minimal Stinespring representations induced by the unitary evolution on the system. Finally, in finite dimension we give a characterization of both algebras in terms of the spectrum of a certain completely positive map acting on the states of the environment.

Dequantization Via Quantum Channels

For a unital completely positive map $\Phi$ ("quantum channel") governing the time propagation of a quantum system, the Stinespring representation gives an enlarged system evolving unitarily. We argue that the Stinespring representations of each power $\Phi^m$ of the single map together encode the structure of the original quantum channel and provides an interaction-dependent model for the bath. The same bath model gives a "classical limit" at infinite time $m\to\infty$ in the form of a noncommutative "manifold" determined by the channel. In this way a simplified analysis of the system can be performed by making the large-$m$ approximation. These constructions are based on a noncommutative generalization of Berezin quantization. The latter is shown to involve very fundamental aspects of quantum-information theory, which are thereby put in a completely new light.

Local operator multipliers and positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and characterise them extending both the characterisation of operator multipliers from [16] and that of local Schur multipliers from [27]. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.

Weak Multiplicativity for Random Quantum Channels

It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p>1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p>1, given a random quantum channel N (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of n copies of N decays exponentially with n. The proof is based on relaxing the maximum output infinity-norm of N to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.

Relations for Certain Symmetric Norms and Anti-norms Before and After Partial Trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.

Optimal unitary dilation for bosonic Gaussian channels

A generic quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. Here, the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multi-mode bosonic Gaussian channel is analyzed both for mixed and pure environment corresponding to the Stinespring representation. In particular, for the case of pure environment we compute this quantity and present an explicit unitary dilation for arbitrary bosonic Gaussian channel. These results considerably simplify the characterization of these continuous-variable maps and can be applied to address some open issues concerning the transmission of information encoded in bosonic systems.

The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation

Stinespring's dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinespring's dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimension-independent bounds. This result generalizes Uhlmann's theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. We briefly discuss further implications for quantum cryptography, thermalization processes, and the black hole information loss puzzle.