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arXiv:2012.05761 · Quantum

Entanglement-symmetries of covariant channels

Dominic Verdon

Published Feb 29, 2024• 3 citations
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Abstract

Let G and G′ be monoidally equivalent compact quantum groups, and let H be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan( G ) → Chan( G′ ), where Chan( G ) is the category whose objects are finite-dimensional C∗ -algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object H has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.

DOI
10.22331/q-2024-02-29-1272

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Let G and G′ be monoidally equivalent compact quantum groups, and let H be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan( G ) → Chan( G′ ), where Chan( G ) is the category whose objects are finite-dimensional C∗ -algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object H has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.
We define two channels N and N' to be entanglement-equivalent if there exists an entanglement-assisted channel coding scheme for N from N' and vice versa. For example, the noiseless channel on a d^2-level classical system and the noiseless channel on a d-dimensional quantum system are entanglement-equivalent by teleportation and dense coding. We provide a construction of entanglement-equivalences of channels based on the principle of covariance. Let G be a compact quantum group of Kac type, let F': Rep(G) -> Hilb be a fibre functor accessible from the canonical fibre functor F: Rep(G) -> Hilb by a unitary pseudonatural transformation, and let G' be the compact quantum group of Kac type obtained from the fibre functor F' by Tannaka duality. Then for any G-covariant channel N we construct a G'-covariant channel N' and entanglement-assisted channel coding schemes witnessing an entanglement-equivalence of N and N'. This construction induces an equivalence of categories E: Chan(G) -> Chan(G'), where Chan(G) is the category whose objects are G-C*-algebras and whose morphisms are covariant channels. We show that all tight teleportation and dense coding schemes are a special case of our construction, using covariance for the action of the quantum permutation group of the algebra M_d(\mathbb{C}). Where G is a finite abelian group, we explicitly compute the effect of the equivalence E on algebras and channels. We thereby produce concrete new examples of entanglement-equivalences of channels in the case where G=Z_2 \times Z_2 and the noncanonical fibre functor is specified by the Pauli matrices.
Abstract We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional C∗C^*C∗ C ∗ -algebras) carry an action of a compact quantum group G , and all channels (completely positive maps preserving a certain G -invariant functional) are covariant with respect to the G -actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: (1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. (2) We show that every quantum confusability graph with a G -action (which we call a quantum G -graph) arises as the confusability graph of a covariant channel. (3) We show that a covariant channel is reversible precisely when its confusability G -graph is discrete. (4) When G is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability G -graphs.
We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.
Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects.
A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem ), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation , which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.