Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1311.7571 · Journal of Operator Theory

On the convergence of output sets of quantum channels

Benoı̂t Collins, Motohisa Fukuda, Ion Nechita

Published May 1, 2015• 8 citations
arXivHTMLPDF

Abstract

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 $Φ_n$ such that along with any other quantum channel $Ξ$, we have convergence of the output set of $Φ_n$ and $Φ_n\otimes Ξ$ simultaneously; the case when $Ξ$ is entanglement breaking is investigated in details.

DOI
10.7900/jot.2013dec04.2008

Related papers

On the Minimum Output Entropy of Random Orthogonal Quantum Channels

Motohisa Fukuda, Ion Nechita

We consider the sequences of random quantum channels defined by 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 the 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.

arXiv:1703.08979•IEEE Transactions on Information Theory•Nov 23, 2017•4 citations

LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS

Benoı̂t Collins, Motohisa Fukuda, Ion Nechita

In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].

arXiv:1208.1449•Random Matrices Theory and Application•Jan 1, 2013•9 citations

Towards a state minimizing the output entropy of a tensor product of random quantum channels

Benoı̂t Collins, Motohisa Fukuda, Ion Nechita

We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.

arXiv:1111.6269•Journal of Mathematical Physics•Mar 1, 2012•15 citations

Estimation of Capacity for Channels Connected with Elementary Quantum Random Walks

S. Grishin

Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length $$\log(\frac{e}{2})$$ .

Lobachevskii Journal of Mathematics•Oct 1, 2024•1 citation

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

Motohisa Fukuda, Gilad Gour

We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.

arXiv:1502.06411•IEEE Transactions on Information Theory•Dec 19, 2016•5 citations

Regularized maximum pure-state input-output fidelity of a quantum channel

Moritz Ernst, Rochus Klesse

As a toy model for the capacity problem in quantum information theory we investigate finite and asymptotic regularizations of the maximum pure-state input-output fidelity $F(\mathcal{N})$ of a general quantum channel $\mathcal{N}$. We show that the asymptotic regularization $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}(\mathcal{N})$ is lower bounded by the maximum output $\ensuremath{\infty}$-norm ${\ensuremath{\nu}}_{\ensuremath{\infty}}(\mathcal{N})$ of the channel. For $\mathcal{N}$ being a Pauli channel, we find that both quantities are equal.

arXiv:1708.06660•Physical Review A•Dec 19, 2017•6 citations
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 ΦnΦ_nΦn​ such that along with any other quantum channel ΞΞΞ, we have convergence of the output set of ΦnΦ_nΦn​ and Φn⊗ΞΦ_n\otimes ΞΦn​⊗Ξ simultaneously; the case when ΞΞΞ is entanglement breaking is investigated in details.
We consider the sequences of random quantum channels defined by 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 the 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.
In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].
We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.
Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length log⁡(e2)\log(\frac{e}{2})log(2e​) .
We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.
As a toy model for the capacity problem in quantum information theory we investigate finite and asymptotic regularizations of the maximum pure-state input-output fidelity F(N)F(\mathcal{N})F(N) of a general quantum channel N\mathcal{N}N. We show that the asymptotic regularization $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}(\mathcal{N})$ is lower bounded by the maximum output $\ensuremath{\infty}$-norm ${\ensuremath{\nu}}_{\ensuremath{\infty}}(\mathcal{N})$ of the channel. For N\mathcal{N}N being a Pauli channel, we find that both quantities are equal.