Upper bound for the success probability of unambiguous discrimination among quantum states Shengyu Zhang,* Yuan Feng, Xiaoming Sun, and Mingsheng Ying ² State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China ~Received 21 June 2001; published 8 November 2001! One strategy to the discrimination problem is to identify the state with certainty, leaving a possibility of undecidability. This paper gives an upper bound for the maximal success probability of unambiguous discrimi- nation among n states. This bound coincides with the known IDP limit when two states are considered. DOI: 10.1103/PhysRevA.64.062103 PACS number~s!: 03.65.Ta, 03.65.Ca Quantum state discrimination is a classically interesting and important problem @1#. A quantum system is prepared in a number of known, finite set of states, and we hope to de- termine what quantum state the system was actually in with the minimum probability of error. Ivanovic @2#, Dieks @3#, and Peres @4# consider the problem under the special require- ment that one must identify the state with certainty, leaving a possibility of undecidability. They find a higher probability of discrimination than merely using von Neumann measure- ment on a single-qubit state by adding an auxiliary quantum system, and the best result along their approach in the case of discrimination between two quantum states is 1 2u ^ p u q & u , where p and q are the two states to be distinguished. Jaeger and Shimony extend the problem in Ref. @5# to the case of unequal priori probabilities, and get the result as 1 22 A rs u ^ p u q & u , where r and s are the priori probabilities of the two states. This result is also discussed by Ban in Ref. @6# in the context of quantum communications. However, all these papers are along the Ivanovic’s approach and demon- strate only as far as their approach is considered that the probability of correct classification they get is the optimal one. This paper extends their work by showing that the bound they obtain is also the best one in a more general context: if one wants to unambiguously distinguish two quantum states only by arbitrary generalized measurements ~POVMs!, then the maximal probability of successful classi- fication is 1 22 A rs u ^ p u q & u . Another naturally intriguing extension is to prepare more than two states to be discriminated. Peres and Terno @7# give a solution to the problem of optimal distinction of three states having arbitrary priori probabilities and arbitrary de- tection values. More generally, for the distinction of n quan- tum states, an important fact shown by Chefles in Ref. @8# is that only linear independent states can be unambiguously discriminated. Another result, which solves a kind of special case known as equally probable symmetrical states, was given in Ref. @9#. But the optimal unambiguous distinction of arbitrary n quantum states is still unknown. This note gives an upper bound on this optimal value, and shows that the upper bound coincides with the known result 1 22 A rs u ^ p u q & u in the two-state case. In what follows, we assume a quantum system is prepared in one of the n states u c 1 & ,..., u c n & in a k-dimension Hilbert space with probabilities p 1 ,..., p n , respectively, where k is an arbitrary positive integer. We hope to identify the state of the system by one or more measurements. A measurement is described by a set of linear operators $ M m % such that ( m M m ² M m 5I . If the state of the quantum system is u c & before the measurement then the probability that result m occurs is ^ c u M m ² M m u c & , and the post-measurement state is M m u c & A ^ c u M m ² M m u c & . A measurement can also be described by a POVM mea- surement, which is a set of positive operators $ E m % such that ( m E m 5I . Similarly, If the state of the quantum system is u c & before the measurement then the probability that result m occurs is ^ c u E m u c & , and the post-measurement state is A E m u c & A ^ c u E m u c & . To present our results formally, we need an auxiliary defi- nition. The definition gives the probability of unambiguous identification among u c 1 & , ..., u c n & by measurement $ M m % . Definition 1. Suppose a quantum system is prepared in one of the n states u c 1 & ,..., u c n & in a k-dimension Hilbert space with probabilities of p 1 ,..., p n , respectively. The probability of unambiguous identification by measurement $ M m % is defined as follows: D~ p 1 ,..., p n , u c 1 & ,..., u c n & , $ M m % ) 5 ( i 51 n ( M m u c i & 0 M m u c j & 50,; j i p i ^ c i u M m ² M m u c i & . Intuitively, the i th summand of the right-hand side of the defining equation is the probability with which one can assert with certainty that the system is prepared in state u c i & . Thus the total summation of the right-hand side is the success probability of unambiguous discrimination among states u c 1 & ,..., u c n & . *Email address: sy@s1000e.cs.tsinghua.edu.cn ² Corresponding author; email address: yingmsh@tsinghua.edu.cn PHYSICAL REVIEW A, VOLUME 64, 062103 1050-2947/2001/64~6!/062103~3!/$20.00 ©2001 The American Physical Society 64 062103-1