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arXiv:2007.03901 · Communications in Mathematical Physics

Quantum Channels with Quantum Group Symmetry

Hun Hee Lee, Sang-Gyun Youn

Published Jan 6, 2022• 7 citations
arXivHTMLPDF
DOI
10.1007/s00220-021-04283-9

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A unital completely positive map governing the time evolution of a quantum system is usually called a quantum channel, and it can be represented by a tuple of operators which are then referred to as the Kraus operators of the channel. We look at states of the system whose correlations with respect to the channel have a certain symmetry. Then we show that detailed balance holds if the Kraus operators satisfy a very interesting algebraic relation which plays an important role in the representation theory of any compact quantum group.
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.
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.