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We analyse the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance. [ABSTRACT FROM AUTHOR]

Quantum cloning is a fundamental protocol of quantum information theory. Perfect universal quantum cloning is prohibited by the laws of quantum mechanics, only imperfect copies being reachable. Symmetric quantum cloning is concerned with case when the quality of the clones is identical. In this work, we study the general case of 1 → N asymmetric cloning, where one asks for arbitrary qualities of the clones. We characterize, for all Hilbert space dimensions and number of clones, the set of all possible clone qualities. This set is realized as the nonnegative part of the unit ball of a newly introduced norm, which we call the Q -norm. We also provide a closed-form expression for the quantum cloner achieving a given clone quality vector. Our analysis relies on the Schur–Weyl duality and on the study of the spectral properties of partially transposed permutation operators. [ABSTRACT FROM AUTHOR]

We introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition; hence, they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit counting and interlace graph polynomials for directed graphs. [ABSTRACT FROM AUTHOR]

We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class S 1 to the Euclidean space ℓ 2 , which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and Gühne by deriving systematic relations between the performance of these two criteria. [ABSTRACT FROM AUTHOR]

We prove that the PPT 2 conjecture holds for linear maps between matrix algebras which are covariant under the action of the diagonal unitary group. Many salient examples, like the Choi-type maps, depolarizing maps, dephasing maps, amplitude damping maps, and mixtures thereof, lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two. [ABSTRACT FROM AUTHOR]

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the achievable region of the universal 1 → 2 asymmetric quantum cloning problem. This concerns the quantum cloning of any quantum state to two approximate clones of different qualities. Measuring the cloning performance with the probabilities of the mixed clone states as the figure of merit, we show that the physical region is a union of ellipses in the plane. The study of these regions has consequences for the eavesdropping on quantum cryptography, and a wide variety of tasks. Equivalently, we characterize the region of quantum-state compatibility of two possibly different isotropic states, considering, for the first time, negative figures of merit. [ABSTRACT FROM AUTHOR]

In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K. We also showed that the set K is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of $${\ell^p}$$ norms on the set K and prove that the maximum is attained on a vector of shape ( a, b, . . . , b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any $${\varepsilon > 0}$$ , it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as $${\log 2 -\varepsilon}$$ . Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less. [ABSTRACT FROM AUTHOR]

It is now well-known that, with high probability, the additivity of minimum output entropy does not hold for pairs of the random quantum channel and its complex conjugate. We investigate asymptotic behavior of output states of r-tensor powers of such pairs, as the dimension of inputs grows. We compute the limit output states for any sequence of well-behaved inputs, which consist of a large class of input states having a nice set of parameters. Then, we show that among these input states tensor products of Bell states give asymptotically the least output entropy, giving positive mathematical evidence towards additivity of above pairs of channels. [ABSTRACT FROM AUTHOR]

We study the partial transposition W=(id⊗t) W∈ M(ℂ) of a Wishart matrix W∈ M(ℂ) of parameters ( dn, dm). Our main result is that, with d→∞, the law of mW is a free difference of free Poisson laws of parameters m( n±1)/2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half-line. [ABSTRACT FROM AUTHOR]

Given two positive integers n and k and a parameter t∈(0,1), we choose at random a vector subspace V⊂ℂ⊗ℂ of dimension N∼ tnk. We show that the set of k-tuples of singular values of all unit vectors in V fills asymptotically (as n tends to infinity) a deterministic convex set K that we describe using a new norm in ℝ. Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy. [ABSTRACT FROM AUTHOR]

We consider a generalized model of repeated quantum interactions, where a system $${\mathcal{H}}$$ is interacting in a random way with a sequence of independent quantum systems $${\mathcal{K}_n, n \geq 1}$$. Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between $${\mathcal{H}}$$ and $${\mathcal{K}_n}$$. The other involves random quantum states describing each copy $${\mathcal{K}_n}$$. In the limit of a large number of interactions, we present convergence results for the asymptotic state of $${\mathcal{H}}$$. This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble. [ABSTRACT FROM AUTHOR]

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel. As an application, we study variations of random matrix models introduced by Hayden [7], and show that their eigenvalues converge almost surely. In particular we obtain, for some models, sharp improvements on the value of the largest eigenvalue, and this is shown in further work to have new applications to minimal output entropy inequalities. [ABSTRACT FROM AUTHOR]

An important problem in quantum information theory is the mathematical characterization of the phenomenon of quantum catalysis: when can the surrounding entanglement be used to perform transformations of a jointly held quantum state under LOCC (local operations and classical communication)? Mathematically, the question amounts to describe, for a fixed vector y, the set T( y) of vectors x such that we have $$x \otimes z \prec y \otimes z$$ for some z, where $$\prec$$ denotes the standard majorization relation. Our main result is that the closure of $$T(y)$$ in the $$\ell_1$$ norm can be fully described by inequalities on the $$\ell_p$$ norms: $$||x||_p \leq ||y||_p$$ for all p ≥ 1. This is a first step towards a complete description of T( y) itself. It can also be seen as a $$\ell_p$$ -norm analogue of the Ky Fan dominance theorem about unitarily invariant norms. The proof exploits links with another quantum phenomenon: the possibiliy of multiple-copy transformations ( $$x^{\otimes n} \prec y^{\otimes n}$$ for given n). The main new tool is a variant of Cramér’s theorem on large deviations for sums of i.i.d. random variables. [ABSTRACT FROM AUTHOR]