Schur multipliers and mixed unitary maps

2009

Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that n copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that n(n + 1)/2 copies of a symmetric unital quantum channel may be arbitrarily-well approximated by a mixture (convex combination) of unitarily implemented channels. In addition, we prove that any extremal properties of a channel are preserved over n (and thus n(n + 1)/2) copies. The result has the potential to be completely generalized to include non-symmetric channels. I. INTRODUCTION AND BACKGROUND There is a famous theorem attributed to Garrett Birkhoff that states that doubly stochastic matrices are convex combinations of permutation matrices. In the quantum context, doubly stochastic matrices become doubly stochastic channels, i.e. completely positive maps preserving both the trace and the identity. Quantum mechanically we understand the permutations to be the unitarily implemented channels. That is, we expect doubly stochastic quantum channels to be convex combinations of unitary channels. Unfortunately it is wellknown that some quantum channels cannot be written that way [1, 2]. Recent work has suggested that n copies of a single channel might approximate a mixture (convex combina-*

Reports on Mathematical Physics

We investigate the quantum privacy properties of an important class of quantum channels, by making use of a connection with Schur product matrix operations and associated correlation matrix structures. For channels implemented by mutually commuting unitaries, which cannot privatise qubits encoded directly into subspaces, we nevertheless identify private algebras and subsystems that can be privatised by the channels. We also obtain further results by combining our analysis with tools from the theory of quasiorthogonal operator algebras and graph theory.

Proceedings of the American Mathematical Society, 2008

We give a formula for Markov dilation in the sense of Anantharaman-Delaroche for real positive Schur multipliers on B(H).

2004

We address the question of the multiplicativity of the maximal p-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the reduction of unitary dynamics in larger Hilbert spaces. It is shown that the maximal output purity of tensor products of single-mode channels under Gaussian inputs is multiplicative for any p ∈ (1, ∞) for products of arbitrary identical channels as well as for a large class of products of different channels. In the case of p = 2 multiplicativity is shown to be true for arbitrary products of generic channels acting on any number of modes.

Banach Center Publications, 2010

Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made in [29]. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with the classical Schur multipliers defined by Schur and their characterisation by Grothendieck, and make our way through measurable multipliers studied by Peller and Spronk, operator multipliers defined by Kissin and Shulman and, finally, multidimensional Schur and operator multipliers developed by Juschenko and the authors. We point out connections of the area with Harmonic Analysis and the Theory of Operator Integrals. 1. Classical Schur multipliers For a Hilbert space H, let B(H) be the collection of all bounded linear operators acting on H equipped with its operator norm • op. We denote by ℓ 2 the Hilbert space of all square summable complex sequences. With an operator A ∈ B(ℓ 2), one can associate a matrix (a i,j) i,j∈N by letting a i,j = (Ae j , e i), where {e i } i∈N is the standard orthonormal basis of ℓ 2. The space M ∞ of all matrices obtained in this way is a subspace of the space M N of all complex matrices indexed by N × N. It is easy to see that the correspondence between B(ℓ 2) and M ∞ is one-to-one. Any function ϕ : N × N → C gives rise to a linear transformation S ϕ acting on M N and given by S ϕ ((a i,j) i,j) = (ϕ(i, j)a i,j) i,j. In other words, S ϕ ((a i,j) i,j) is the entry-wise product of the matrices (ϕ(i, j)) i,j and (a i,j) i,j , often called Schur product. The function ϕ is called a Schur multiplier if S ϕ leaves the subspace M ∞ invariant. We denote by S(N, N) the set of all Schur multipliers. Let ϕ be a Schur multiplier. Then the correspondence between B(ℓ 2) and M ∞ gives rise to a mapping (which we denote in the same way) on B(ℓ 2). We first note that S ϕ is necessarily bounded in the operator norm. This follows from the Closed Graph Theorem; indeed, suppose that A k → 0 and S ϕ (A k) → B in the operator norm, for some elements A k , B ∈ B(ℓ 2),

Theoretical and Mathematical Physics, 2015

In this paper we find, for a class of bipartite quantum states, a nontrivial lower bound on the entropy gain resulting from the action of a tensor product of identity channel with an arbitrary channel. By means of that we then estimate (from below) the output entropy of the tensor product of dephasing channel with an arbitrary channel. Finally, we provide a characterization of all phase-damping channels resulting as particular cases of dephasing channels.

arXiv: Quantum Physics, 2018

In this work, we prove a lower bound on the difference between the first and second singular values of quantum channels induced by random isometries, that is tight in the scaling of the number of Kraus operators. This allows us to give an upper bound on the difference between the first and second largest (in modulus) eigenvalues of random channels with same large input and output dimensions for finite number of Kraus operators $k\geq 169$. Moreover, we show that these random quantum channels are quantum expanders, answering a question posed by Hastings. As an application, we show that ground states of infinite 1D spin chains, which are well-approximated by matrix product states, fulfill a principle of maximum entropy.

2016

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M = 2, N become large or M become large and N = 2.

Taiwanese Journal of Mathematics, 2018

In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces B(H) which define bounded operators acting on ℓ 2 (H) and use it to get the description of the right Schur multipliers acting on ℓ 2 (H) in terms of certain operator-valued measures.

A multiplicativity conjecture for quantum communication channels is formulated, validity of which for the values of parameter p close to 1 is related to the solution of the fundamental problem of additivity of the channel capacity in quantum information theory. The proof of the conjecture is given for the case of natural numbers p.

2000

Journal of Physics A-mathematical and General, 2003

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on Bbb Cn otimes Bbb Cn whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on Bbb C3 otimes Bbb C3 with operator-Schmidt number S for every S in {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on Bbb C3 otimes Bbb C3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator Bbb CM1 otimes Bbb CM2 rightarrow Bbb CN1 otimes Bbb CN2, with M1M2 = N1N2. This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022-5).

Journal of Physics A: Mathematical and Theoretical, 2014

We develop a framework which unifies seemingly different extension (or 'joinability') problems for bipartite quantum states and channels. This includes known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Jamiołkowski isomorphism between bipartite states and linear transformations, which we term the homocorrelation map: in contrast to the better-known Choi isomorphism which emphasizes the preservation of the positivity constraint, use of the Jamiołkowski isomorphism allows one to characterize the preservation of the statistical correlations of bipartite states and quantum channels. The resulting homocorrelation map thus acquires a natural operational interpretation. We define and analyze state-joining, channel-joining, and local-positive-joining problems in three-party settings with collective ⊗ ⊗ U U U symmetry, obtaining exact analytical characterizations in low dimensions. We find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, whereas quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the strength of positive correlation across two subsystems at a single time, as well as on the strength of negative correlation between the state of a single system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party ⊗ ⊗ U U U-invariant setting. J. Phys. A: Math. Theor. 48 (2015) 035307 P D Johnson and L Viola 3 2 The circumflex, 'ˆ', over a set denotes the complement of that set. Hence, tr ℓˆi s a trace over all subsystems in … N {1, , } except for those in subset ℓ.

Bulletin of the London …, 1975

Let 2tf and j f be Hilbert spaces, and let M and N be von Neumann algebras of operators on Jf and Jf respectively . Let Jf ® Jf denote the Hilbert space tensor product of 3V with X, and let M (g) N denote the von Neumann algebra on #? (g) Jf generated by the operators m ® « for meM, neN [2], We will denote the cornmutant of a von Neumann algebra M by M'. Although there were a number of earlier proofs of special cases, it was not until 1967 that Tomita [8, 6] gave a proof in full generality for: THEOREM 1. Let M and N be von Neumann algebras on 2tf and J f respectively. Then Received 11 May, 1974. t The research for this paper was conducted while we were both visiting at the University of Pennsylvania. We would like to thank the members of the Mathematics Department there for their warm hospitality during our visits.

Journal of Functional Analysis, 2012

We study positive maps of B (K) into B (H) for finite-dimensional Hilbert spaces K and H. Our main emphasis is on how Choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps. Special attention will be given to entanglement properties and k-positive maps, in particular tensor products of 2-positive maps. The latter problem is directly related to the question of n-copy distillability of quantum states, for which we obtain a partial result.