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Entanglement-equivalences of covariant channels

Dominic Verdon

Published Dec 10, 2020• 1 citation
OpenAlex

Abstract

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.

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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.
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.
In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].
We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert–Schmidt scalar product and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be `tight' in the sense that all Hilbert spaces involved have the same finite dimension d , and the classical channel involved distinguishes d 2 signals. A general construction procedure for orthonormal bases of unitaries, involving Latin squares and complex Hadamard matrices is also presented.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.
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.
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.