Standard forms of noisy quantum operations via depolarization

Abstract

We consider noisy, non-local unitary operations or interactions, i.e. the corresponding evolutions are described by completely positive maps or master equations of Lindblad form. We show that by random local operations the completely positive maps can be depolarized to a standard form with a reduced number of parameters describing the noise process in such a way that the noiseless (unitary) part of the evolution is not altered. A further reduction of the parameters, in many cases even to a single one (i.e. global white noise), is possible by tailoring the decoherence process and increasing the amount of noise. We generalize these results to the dynamical case where the ideal unitary operation is given by some interaction Hamiltonian. The resulting standard forms may be used to compute lower bounds on channel capacities, to simplify quantum process tomography or to derive error thresholds for entanglement purification and quantum computation.

Exact and approximate unitary 2-designs and their application to fidelity estimation

“…A related protocol, twirling a quantum channel with Haar-random unitary operators, has been studied recently for the purpose of reducing a generic gate to a standard form, which is among the results of Ref. [10], where it was also shown that the Clifford group is also an adequate substitute for that task. Moreover, the task of generating typical generic entanglement has been studied in Ref.…”

“…with the symplectic inner product S P on the index space (see [19] for further details; these techniques are discussed also in [10]). Therefore, setting r k = α k β k leads to the above form.…”

Entanglement purification and quantum error correction

“…[22]) in such a way that the diagonal elements are not altered, p k = p k,k . This often allows one to restrict considerations to Pauli-diagonal channels, and makes such channels particularly important.…”

“…Consider a quantum channel E and the associated bipartite mixed statê E = 1l ⊗ E|Φ Φ| (22) that is obtained by sending part of a maximally entangled state |Φ through the channel E. Consider also the channelÊ which is generated when using a state E for teleportation. Notice that for Pauli-diagonal channels E P (see Eq.…”

Multipartite secure state distribution

“…In standard purification protocols this is enforced by depolarizing the transmitted state. Instead, here we depolarize the channels E j to a Pauli-diagonal form [16]…”