Programmable quantum-state discriminators with simple programs

Abstract

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to n copies. In comparison to other works we find that providing n > 1 copies of the data state yields higher success probabilities than providing n > 1 copies of the program states.

Programmable discrimination with an error margin

“…On the unambiguous side, R = 0, the only possible choice is r α = 0 for all values of α, and the success probability is given by (17). At the other end point, if R ≥ R c = n+1 α=1 p α r c,α , where r c,α is the critical margin in the subspace H α , given by (7) with c = c α , we immediately recover the minimum-error result (18).…”

“…(19) below]. Bases obeying an orthogonality relation of the form (16) exist for any two subspaces and are known as Jordan bases [17]. Since a state of the first basis has nonzero overlap with only one element of the second basis, the problem of discriminating σ 1 from σ 2 can be cast as pure state discrimination in each Jordan subspace, which we label by j (note that the overlaps c j do not depend on the magnetic number m).…”

Success probabilities for universal unambiguous discriminators between unknown pure states

“…Moreover, if we have the PSP, take the average of it and we can obtain the ASP. Unfortunately, except the results for few special simple cases [7,8], PSPs for the general case with multi-copy and high-dimensional states are left behind and still not obtained.The main purpose of this paper is to evaluate the PSPs of the optimal universal unambiguous discriminators between two unknown pure states for the general case. Following the optimal measurement operators for the universal unambiguous discriminator in Ref.…”

“…Moreover, if we have the PSP, take the average of it and we can obtain the ASP. Unfortunately, except the results for few special simple cases [7,8], PSPs for the general case with multi-copy and high-dimensional states are left behind and still not obtained.…”

“…In this programmable quantum device, two possible states enter two program registers as "programs" respectively, and the data register is prepared with a third state (guaranteed to be one of the two possible states) which one wishes to identify. One amazing feature of this discriminator is that the states in the device can be unknown, which means no classical knowledge on the states is provided, and it is capable to distinguish any pair of states in this device.Later, the generalizations and the experimental realizations of programmable discriminators are introduced and widely discussed [8][9][10][11][12][13]. The problems with multiple copies in program and data registers or with high-dimensional states in the registers are considered.…”