Abstract
A quantum channel is said to be amixed-unitarychannel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channelΦ, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or notΦis a mixed-unitary channel. This hardness result holds even under the assumption thatΦis not within an inverse-polynomial distance (in the dimension of the space upon whichΦacts) of the boundary of the mixed-unitary channels.
Highlights
In the theory of quantum information, quantum channels represent discrete-time changes in systems that can, in an idealized sense, be realized by physical processes
A unitary channel is a channel of the form Φ : L(Cn) → L(Cn) that is given by Φ(X) = U XU ∗ for every X ∈ L(Cn), for some fixed choice of a unitary operator U ∈ L(Cn)
The main purpose of this section is to clarify some of the notation and conventions we use throughout the paper, and to define two decision problems: one is the mixed-unitary detection problem, whose hardness is the primary focus of this paper, and the second is the unitary quadratic minimization problem, which serves as an intermediate problem through which an NP-complete problem is reduced to the mixed-unitary detection problem
Summary
In the theory of quantum information, quantum channels represent discrete-time changes in systems that can, in an idealized sense, be realized by physical processes. En}, the operator Ei,j is represented by the matrix having a 1 in entry (i, j) and all other entries 0.) We prove that it is NP-hard, with respect to polynomial-time Turing reductions, to determine whether or not a given quantum channel is mixed-unitary. We note that our main result can, be closely linked with the problem of separability testing, in the sense that testing if a channel is mixed-unitary may alternatively be formulated as a problem concerning the expression of a bipartite density operator in a certain way.
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