Minimum Output Entropy Research Articles

Local additivity revisited

We make a number of simplifications in Gour and Friedland’s proof of local additivity of the minimum output entropy of a quantum channel. We follow them in reframing the question as one about the entanglement entropy of bipartite states associated with a dB × dE matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state, which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information.

Concentration estimates for random subspaces of a tensor product and application to quantum information theory

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.

Entanglement entropy and vacuum states in Schwarzschild geometry

Recently, it was proposed that there must be either large violation of the additivity conjecture or a set of disentangled states of the black hole in the AdS/CFT correspondence. In this paper, we study the additivity conjecture for quantum states of fields around the Schwarzschild black hole. In the eternal Schwarzschild spacetime, the entanglement entropy of the Hawking radiation is calculated assuming that the vacuum state is the Hartle-Hawking vacuum. In the additivity conjecture, we need to consider the state which gives minimal output entropy of a quantum channel. The Hartle-Hawking vacuum state does not give the minimal output entropy which is consistent with the additivity conjecture. We study the entanglement entropy in other static vacua and show that it is consistent with the additivity conjecture.

Additivity violation of the regularized minimum output entropy

The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by Matthew B. Hastings [“Superadditivity of communication capacity using entangled inputs”, Nature Physics 5, 255–257 (2009; )] in the one-shot case by exhibiting a pair of random quantum channels. However, the initial motivation was arguably to understand regularized quantities, and there was so far no way to solve additivity questions in the regularized case. The purpose of this paper is to give a solution to this problem. Specifically, we exhibit a pair of quantum channels that unearths additivity violation of the regularized minimum output entropy. Unlike previously known results in the one-shot case, our construction is non-random, infinite-dimensional, and in the commuting-operator setup. The commuting-operator setup is equivalent to the tensor-product setup in the finite-dimensional case for this problem, but their difference in the infinite-dimensional setting has attracted substantial attention and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's and Connes embedding problem [ Z. Ji et al., “ \mathsf{MIP}^*= \mathsf{RE} ”, Preprint, ], Likewise, it is not clear that our approach works in the finite-dimensional setup. Our strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.

Additivity violation of quantum channels via strong convergence to semi-circular and circular elements

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.

On classical capacity of Weyl channels

The additivity of minimal output entropy is proved for the Weyl channel obtained by the deformation of a q-c Weyl channel. The classical capacity of channel is calculated.

Algebraic Methods of the Study of Quantum Information Transfer Channels

Kraus representation of quantum information transfer channels is widely used in practice. We present examples of Kraus decompositions for channels that possess the covariance property with respect to the maximal commutative group of unitary operators. We show that in some problems (for example, the problem on the estimate of the minimal output entropy of the channel), the choice of a Kraus representation with nonminimal number of Kraus operators is relevant. We also present certain algebraic properties of noncommutative operator graphs generated by Kraus operators for the case of quantum channels that demonstrate the superactivation phenomenon.

Universal upper bounds for Gaussian information capacity

The most natural way to describe an information-carrying system containing a specific noise is an additive white Gaussian-noise (AWGN) channel. In bosonic quantum systems (especially the Gaussian case), although the classical information capacity for a phase-insensitive and thermal-noise channel is additive based on a proof of the minimum output entropy conjecture, several open questions remain. By generalizing the Gaussian noise model from thermal noise to general Gaussian noise, we rigorously revisit and calculate these strong upper bounds on the information capacity for single-mode with general Gaussian-noise channels. In this study, we use the quantum entropy power inequality (QEPI) approach. This framework gives a new formula for finding upper bounds on the information capacity of bosonic Gaussian channels.

Quantum informational properties of the Landau–Streater channel

We study the Landau–Streater quantum channel Φ:B(Hd)↦B(Hd), whose Kraus operators are proportional to the irreducible unitary representation of SU(2) generators of dimension d. We establish SU(2) covariance for all d and U(3) covariance for d = 3. Using the theory of angular momentum, we explicitly find the spectrum and the minimal output entropy of Φ. Negative eigenvalues in the spectrum of Φ indicate that the channel cannot be obtained as a result of Hermitian Markovian quantum dynamics. Degradability and antidegradability of the Landau–Streater channel is fully analyzed. We calculate classical and entanglement-assisted capacities of Φ. Quantum capacity of Φ vanishes if d = 2, 3 and is strictly positive if d ⩾ 4. We show that the channel Φ ⊗ Φ does not annihilate entanglement and preserves entanglement of some states with Schmidt rank 2 if d ⩾ 3.

Energy-Constrained Private and Quantum Capacities of Quantum Channels

This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.

An algorithm to explore entanglement in small systems.

A quantum state’s entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.

On the Minimum Output Entropy of Random Orthogonal Quantum Channels

We consider sequences of random quantum channels defined using the Stinespring formula with Haar-distributed random orthogonal matrices. For any fixed sequence of input states, we study the asymptotic eigenvalue distribution of the outputs through tensor powers of random channels. We show that the input states achieving minimum output entropy are tensor products of maximally entangled states (Bell states) when the tensor power is even. This phenomenon is completely different from the one for random quantum channels constructed from Haar-distributed random unitary matrices, which leads us to formulate some conjectures about the regularized minimum output entropy.

Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups

In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels. An application to the construction of $d$-positive maps on matrix algebras is also presented.

Additive Bounds of Minimum Output Entropies for Unital Channels and an Exact Qubit Formula

We investigate minimum output (R\'enyi) entropy of qubit channels and unital quantum channels. We obtain an exact formula for the minimum output entropy of qubit channels, and bounds for unital quantum channels. Interestingly, our bounds depend only on the operator norm of the matrix representation of the channels on the space of trace-less Hermitian operators. Moreover, since these bounds respect tensor products, we get bounds for the capacity of unital quantum channels, which is saturated by the Werner-Holevo channel. Furthermore, we construct an orthonormal basis, besides the Gell-Mann basis, for the space of trace-less Hermitian operators by using discrete Weyl operators. We apply our bounds to discrete Weyl covariant channels with this basis, and find new examples in which the minimum output R\'enyi $2$-entropy is additive.

Minimum output entropy of a non-Gaussian quantum channel

We introduce a model of non-Gaussian quantum channel that stems from the combination of two physically relevant processes occurring in open quantum systems, namely amplitude damping and dephasing. For it we find input states approaching zero output entropy, while respecting the input energy constraint. These states fully exploit the infinite dimensionality of the Hilbert space. Upon truncation of the latter, the minimum output entropy remains finite and optimal input states for such a case are conjectured thanks to numerical evidences.

Entropy power inequalities for qudits

Shannon’s entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: f(a X+1−a Y)≥af(X)+(1−a)f(Y)∀ a∈[0,1]. Here, X and Y are continuous random variables and the function f is either the differential entropy or the entropy power. König and Smith [IEEE Trans. Inf. Theory 60(3), 1536–1548 (2014)] and De Palma, Mari, and Giovannetti [Nat. Photonics 8(12), 958–964 (2014)] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where X and Y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields. In this paper, we similarly establish a class of EPI analogues for d-level quantum systems (i.e., qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a ∈ [0, 1] and acts like a finite-dimensional analogue of a beam splitter with transmissivity a, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as a is changed from zero to one. We obtain analogues of Shannon’s EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but also for a much larger class of functions. This class includes the Rényi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity.

Almost One Bit Violation for the Additivity of the Minimum Output Entropy

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} 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 l^p norms on the set K_{k,t} 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 eps > 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 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

Random matrix techniques in quantum information theory

The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.

On the constrained classical capacity of infinite-dimensional covariant quantum channels

The additivity of the minimal output entropy and that of the χ-capacity are known to be equivalent for finite-dimensional irreducibly covariant quantum channels. In this paper, we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper [Giovannetti et al., Commun. Math. Phys. 334(3), 1553-1571 (2015)] to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the “threshold condition,” obtained for single mode in Schäfer et al. [Phys. Rev. Lett. 111, 030503 (2013)], and the proof of the fact that under this condition the classical “Gaussian capacity” resulting from optimization over only Gaussian inputs is equal to the full classical capacity. Complex structures correspond to different squeezings, each with its own normal modes, vacuum and coherent states, and the gauge. Thus our results apply, e.g., to multimode channels with a squeezed Gaussian noise under the standard input energy constraint, provided the squeezing is not too large as to violate the generalized threshold condition. We also investigate the restrictiveness of the gauge-covariance condition for single- and multimode bosonic Gaussian channels.

On the convergence of output sets of quantum channels

We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in [BCN13]. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences $\Phi_n$ such that along with any other quantum channel $\Xi$, we have convergence of the output set of $\Phi_n$ and $\Phi_n\otimes \Xi$ simultaneously; the case when $\Xi$ is entanglement breaking is investigated in details.