where α and β are real numbers. Since ρ$_{0}$ is PSD, the feasible values of α and β are constrained as 0 ≤ α ≤ 1 and | β | ≤ √ α (1 − α ) , respectively. Without loss of generality we assume α ≥ 1 / 2 . The operator S has two non-degenerate eigenvalues equal to λ$_{1}$ ( S ) = e i π/M and λ$_{2}$ ( S ) = e − $^{iπ/M}$. There- fore, the corresponding eigenvectors define the basis and ∑ i =0 S $^{i}$Π$_{0}$ S = I . It is useful to observe that the condition ∑ M − 1 i =0 S $^{i}$Π$_{0}$ S − i = I implicitly fixes the value of the trace of Π$_{0}$ . In fact, Tr ( ∑ M − 1 i =0 S $^{i}$Π$_{0}$ S − $^{i}$) = M − 1 i =0 Tr ( S $^{i}$Π$_{0}$ S − $^{i}$) = ∑ M − 1 i =0 Tr (Π$_{0}$ S − $^{i}$S$^{i}$) = M Tr (Π$_{0}$) and being Tr ( I ) = N it follows that Tr (Π$_{0}$) = N/M . By simple algebra it can be found a closed-form expression for Π$_{0}$ in the following two cases. 1. β = 0 . The optimal Π$_{0}$ is given by The optimal reference measure operator Π$_{0}$ can be found from the conditions given in Section III C ( U ˜ X$_{opt}$U † − 1 $_{M}$ρ$_{0}$ )Π$_{0}$ = Π$_{0}$( U ˜ X$_{opt}$U † − 1 $_{M}$ρ$_{0}$ ) = 0 , that in this example simplify as (˜ x$_{1}$ I − 1 $_{M}$ρ$_{0}$ )Π$_{0}$ = Π$_{0}$(˜ x$_{1}$ I − 1 $_{M}$ρ$_{0}$ ) = 0 , being ˜ X$_{opt}$ = ˜ x$_{1}$ I . Note that these condi- tions also imply that ρ$_{0}$ Π$_{0}$ = Π$_{0}$ ρ$_{0}$ . The optimal Π$_{0}$ can then be numerically found by solving a linear system of equations including the additional requirements Π$_{0}$ ∈ P M − 1 − i and after some simple algebra, 1 we find that it can be rewritten as where, both ˜ x$_{1}$ and ˜ x$_{2}$ are real and non-negative, being ˜ X PSD. The constraint ˜ X ≥ 1 $_{M}$U $^{†}$ρ$_{0}$ U reads Consequently, for the dual problem DP3 the optimization operator ˜ X has to be diagonal with ˜ x$_{1}$ ≥ 1 / (2 M ) and ˜ x$_{2}$ ≥ 1 / (2 M ) . Hence, the min- imum of Tr ( ˜ X ) = ˜ x$_{1}$ + ˜ x$_{2}$ is obtained for ˜ x$_{1}$ = ˜ x$_{2}$ = 1 / (2 M )(1 + √ (2 α − 1) 2 + (2 β ) $^{2}$. In conclusion, the min- imum error probability P$_{e}$ = 1 − P$_{c}$ = 1 − Tr ( ˜ X ) is We note that the first term on the right-hand side of (10) corresponds to a blind guessing on the equiproba- ble elements belonging to the quantum state ensemble, while the second term represents the gain due to the op- timal quantum discrimination. It is interesting to ob- serve that the minimum error probability is obtained for β = √ α (1 − α ) and it results 2 For instance, (11) holds for α = 1 and β = 0 which is the case of study considered by Helstrom [1] and Ban et al. [5], which models linearly dependent spin–1/2 quantum states, where the generating density operator has rank- one ρ$_{0}$ = | ψ$_{0}$ 〉〈 ψ$_{0}$ | with pure state | ψ$_{0}$ 〉 = [ 1 0 ] . When ρ$_{0}$ is diagonal, i.e., β = 0 , (10) simplifies as The constraint ∑ M − 1 i =0 S $^{i}$Π$_{0}$ S − i = I becomes 2 M ∑ M − 1 i =0 ρ$_{i}$ = I showing that the quantum state ensemble has a particular structure. Indeed, such a specific geometry has been considered by Yuen et al. in [3, (IV.4)] and the results therein reported are in agreement with (11) and (14) . Π$_{0}$ = 2 M ρ$_{0}$ . (14) Π$_{0}$ = 2 M [ 1 0 0 0 ] . (13) In particular, setting α = 1 / 3 the same numerical results obtained in [12] are found. 2. β = √ α (1 − α ) . The measure operator results 2 P$_{e}$ = M − 1 M − 1 M √ (2 α − 1) 2 + (2 β ) 2 . (10) (2 M ˜ x$_{1}$ − 1)(2 M ˜ x$_{2}$ − 1) − [(2 α − 1) $^{2}$+ (2 β ) $^{2}$] ≥ 0 , (9) ˜ X ≥ 1 2 M [ 1 (2 α − 1) + i 2 β (2 α − 1) − i 2 β 1 ] , (8) ˜ X = [ ˜ x$_{1}$ 0 0 ˜ x$_{2}$ ] , (7) U = 1 √$_{2}$ [ 1 1 − i i ] . (6) P$_{e}$ = 1 − M . (11) P$_{e}$ = 1 − α 2 M . (12) The proposed formulation of the dual problem has also proved useful to numerically solve systems of large dimen- sions, where the computational complexity sets a severe limit to the possibility of finding an optimal solution. This is the case of pulse position modulation (PPM), a modulation format candidate for deep space communica- tions [13]. In quantum PPM the states are defined in a Hilbert space given by the tensorial product of M sub- spaces, each of dimension n , and, therefore, the overall space dimension N grows exponentially with the PPM order M , being N = n M [14]. In [15], DP3 is applied to quantum PPM using the software CVX for SDP [11]. The solution of DP3 results considerably faster than DP1 . As a limit case, for values of N about 1000 the discrimina- tion problem was successfully solved with DP3 while, on the same processing unit, CVX fails with DP1 , because of computer memory limits. We have studied the dual problem for minimum error probability discrimination of symmetric quantum states. V. CONCLUSIONS 4 1 It is recalled that an Hermitian matrix is PSD if and only if its principal minors are all non-negative.