Port-based teleportation in arbitrary dimension

Abstract

Port-based teleportation (PBT), introduced in 2008, is a type of quantum teleportation protocol which transmits the state to the receiver without requiring any corrections on the receiver’s side. Evaluating the performance of PBT was computationally intractable and previous attempts succeeded only with small systems. We study PBT protocols and fully characterize their performance for arbitrary dimensions and number of ports. We develop new mathematical tools to study the symmetries of the measurement operators that arise in these protocols and belong to the algebra of partially transposed permutation operators. First, we develop the representation theory of the mentioned algebra which provides an elegant way of understanding the properties of subsystems of a large system with general symmetries. In particular, we introduce the theory of the partially reduced irreducible representations which we use to obtain a simpler representation of the algebra of partially transposed permutation operators and thus explicitly determine the properties of any port-based teleportation scheme for fixed dimension in polynomial time.

Optimal port-based teleportation

“…We can also reproduce expression for the fidelity of the teleported state in the case of the maximally entangled state using certain choice of the coefficients a m ( ) p in the most general form of the POVM given by(61). N and plugging them in(69) we reproduce fidelity for the maximally entangled state as a resource state (see theorem 12 of [12] or expression(120) above).…”

“…, where t C is a partial transposition with respect to subsystem C performed on permutation operator V(a, C) acting between subsystems a and C. The operator ρ in(1) is called the PBT operator, and can be expressed as (see [12]):…”

“…which project onto irreps of S(N) contained in M α labelled by Young diagrams μ and induced from the irreps of S(N−1) labelled by α [12]. Denoting by P α a Young projector onto irrep labelled by  a -N 1 we get the following representation of η from equation (4):…”

“…The support of every η(α) is the space S(M α ) which is invariant with respect to the action of S(n−1), so we see that F μ (α) are eigenprojectors of η(α). From [12] we know that projectors F μ (α) can be written as:…”

“…In our previous work [12] we give an explicit expression for the fidelity F given in equation (1) in terms of N d , , the dimensions d μ , and multiplicities m μ of irreps of the permutation group S(N) when the resource state is given by as a N-fold tensor product of y ñ + |…”