Complete optimal convex approximations of qubit states under $$B_2$$ B 2 distance

Abstract

We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the B 2 -distance of an arbitrary target state. Both the analytical formulae of the B 2 -distance and the corresponding complete optimal decompositions are obtained. The tradeoff relations for both the sum and the squared sum of the B 2 -distances have been analytically and numerically investigated.

The best approximation of a given qubit state with the limited pure-state set

“…Case 2.1.2: M 32 = 0. Substituting equation ( 26) into equation ( 22), and using a similar process from equations ( 27) to (35), we can get the same conclusion as equations (36)…”

“…Recently, the generalized problem, optimally approximating a desired and unavailable quantum channel Φ (quantum state ρ) by the convex mixing of a given set of other channels {Ψ i } (quantum states {χ i }) was addressed in references [34][35][36][37]. The convex approximation problem is to seek for the least distinguishable channel (quantum state) from Φ(ρ) among the convex set i p i Ψ i ( i p i χ i ).…”

“…Namely, the disposable quantum states come from six eigenstates of Pauli matrices and the distance metric is taken as the trace norm. In addition, the B 2 -distance between any target state and the eigenstates of two of three Pauli matrices was considered in references [36,37].…”

The Best Approximation of an Objective State With a Given Set of Quantum States

“…The optimal approximation of a quantum state with limited states has been addressed in various cases. [47][48][49][50][51][52] In refs. [47,48], optimally approximating an unavailable quantum state 𝜌 (quantum channel Φ) by the convex mixing of states (channels) drawn from a set of available states {v i } (channels {Ψ i }) was considered.…”

“…Then the approximation by the eigenstates of any two Pauli matrices was investigated in ref. [50] and some interesting tradeoff relations were also proposed. Later, the disposable quantum state set was extended from the eigenstates of the Pauli matrix to the eigenstates of any quantum logic gate, [51] and then to arbitrary quantum states without any restriction.…”