Success probabilities for universal unambiguous discriminators between unknown pure states

Abstract

A universal programmable discriminator can perform the discrimination between two unknown states, and the optimal solution can be approached via the discrimination between the two averages over the uniformly distributed unknown input pure states, which has been widely discussed in previous works. In this paper, we consider the success probabilities of the optimal universal programmable unambiguous discriminators when applied to the pure input states. More precisely, the analytic results of the success probabilities are derived with the expressions of the optimal measurement operators for the universal discriminators and we find that the success probabilities have nothing to do with the dimension d while the amounts of the copies in the two program registers are equal. The success probability of programmable unambiguous discriminator can asymptotically approach to that of usual unambiguous discrimination (state comparison) as the number of copies in program registers (data register) goes to infinity.PACS numbers: 03.67. Hk, 03.65.Ta, 03.65.Fd The discrimination between quantum states is a basic tool in quantum information processing, and this is a nontrivial problem since an unknown state cannot be perfectly cloned [1]. Usually, there are two basic strategies to achieve the state discrimination: Minimum-error discrimination (MD) [2-4] with a minimal probability for the error, and unambiguous discrimination (UD) [5] with a minimum probability of inconclusive results. In those works, a quantum state is chosen from a set of known states but we do not know which and want to determine the actual states.The discrimination problems above are dependent on the set of states to be distinguished, and the device for the discrimination is not universal but specifically designed for the states. As in the spirit of programmable quantum devices [6], it is interesting to design a discrimination device that does not need to change as the input states change. Such a universal device that can unambiguously discriminate between two unknown qubit states has first been constructed by Bergou and Hillery [7]. 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. Furthermore, the case that each copy in the registers is the mixed state is also treated [11]. In most of these literatures, either the unambiguous discrimination or the minimum-error scheme is us...

Optimal programmable unambiguous discriminator between two unknown latitudinal states

“…This discriminator can distinguish any pair of states in this device with some probability, unless the two states |ψ 1 and |ψ 2 are identical. The generalizations and the experimental realizations of this programmable discriminator above were also introduced and widely discussed soon [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. In these works, either the minimum-error strategy or the unambiguous discrimination strategy was considered, and the two strategies were unified by introducing an error margin [57].…”

“…In these works, either the minimum-error strategy or the unambiguous discrimination strategy was considered, and the two strategies were unified by introducing an error margin [57]. The cases for the multiple copies of input states in both program and data registers are discussed, and the optimal solutions are obtained for n A = n C = n, n B = 1 [44,56], for n A = n C = 1, n B = n [41], for n A = n C = n, n B = m [48], and for arbitrary copies of states in both program and data registers [45,55,58], where n A , n B and n C are the copies of states in the registers A, B and C, respectively. The cases for high dimensional (qudit) states in the registers were also considered, and the unambiguous discrimination has been discussed for n A = n B = n C = 1 [42].…”

“…where |ψ 1 and |ψ 2 are two unknown pure states in ddimensional (d 2) Hilbert space. For this case, the optimal solutions for both the minimum-error discrimination and unambiguous discrimination were obtained in a group-theoretic approach [43] and the success probabilities are systematically derived for the unambiguous discrimination [58], while only the minimum-error discrimination was discussed in Ref. [55].…”

Optimal quantum state identification with qudit-encoded unknown states

“…In quantum state identification [1][2][3][4][5][6][7][8][9][10][11][12][13][14] the task is to identify the state of a quantum system, prepared with certain prior probability in a definite state out of a finite set of possible states, where some or all of the states are unknown. The unknown states are encoded into different reference copies, which together have been introduced as the program register in a programmable machine [1] for identifying the state of the probe, carrying the data.…”

“…The problem of quantum state identification has been first introduced for the optimal unambiguous identification of two unknown pure qubit states [1]. The in-vestigations have been soon extended to minimum-error identification [2], and generalizations have been also performed to take into account that more than one copy may be available for the probe system and the two reference systems [2][3][4][5][6][7][8][9], including the case that the two unknown states are mixed [7]. For two unknown pure qudit states with Hilbert space dimension d ≥ 2, minimum-error identification has been studied for arbitrary prior probabilities of the states [3], while the maximum overall success probability for unambiguous identification has been obtained in the case when the states are equally probable [3,4].…”