and then X ′ commutes with S . Finally, and the proof is complete. ■ Therefore, the search of the unknown optimization op- erator X can be restricted to the subclass of P composed by the PSD operators that commute with the symmetry operator S . The optimal Π$_{0}$ can be found by the relations ( X$_{opt}$ − 1 $_{M}$ρ$_{0}$ )Π$_{0}$ = Π$_{0}$( X$_{opt}$ − 1 $_{M}$ρ$_{0}$ ) = 0 subject to Π$_{0}$ ∈ P and ∑ M − 1 i =0 S $^{i}$Π$_{0}$ S − i = I . In the next section, we develop a formulation, where the commutation condition XS = SX is replaced by an alternative constraint. Since the symmetry operator S is known a-priori, it is possible to exploit its spectral characterization to rewrite the dual problem DP2 as follows. Other form for the dual problem for symmetric quantum states ( DP3 ). Let λ$_{0}$,λ$_{1}$,...,λ ¯ N − 1 be the ¯ N ≤ N distinct eigenvalues of S , and N$_{i}$ be the multi- plicity of λ$_{i}$ . Let U be a basis of eigenvectors of S , with the first N$_{0}$ eigenvectors corresponding to λ$_{0}$ , the next N$_{1}$ eigenvectors corresponding to λ$_{1}$ , and so on. Then, minimize the trace of the optimization operator ˜ X over the subclass of P consisting of block-diagonal operators with blocks of dimension N$_{i}$ , under the constraint ˜ X ≥ 1 $_{M}$U $^{†}$ρ$_{0}$ U . Once found a minimum trace operator ˜ X$_{opt}$ , its trace gives the maximum probability of correct detection, P$_{c}$ = Tr ( ˜ X$_{opt}$ ) . Proof: By DP2 the optimization operator X can be chosen to commute with S . Therefore, given an eigenbasis U for S , we can write X = U ˜ XU † where, by well-known results on simultaneous diagonalization of commuting self-adjoint operators [10], ˜ X turns out to be block diagonal with the size of the blocks given by the multiplicity of the eigenvalues of S . Moreover, we find that Tr ( X ) = Tr ( U ˜ XU $^{†}$) = Tr ( ˜ X ) and the constraint X ≥ 1 $_{M}$ρ$_{0}$ becomes ˜ X ≥ 1 $_{M}$U $^{†}$ρ$_{0}$ U . ■ The new formulation of the dual problem is given in a form that is particularly suitable for SDP computational tools [11] and, moreover, it is analytically more tractable that the previous version. We now quantify the complex- ity of the different approaches. The commutative requirement XS = SX is then replaced by fixing the block-diagonal structure on ˜ X . Given an optimal ˜ X$_{opt}$ , the reference optimal operator Π$_{0}$ is solution of ( U ˜ X$_{opt}$U † − 1 $_{M}$ρ$_{0}$ )Π$_{0}$ = Π$_{0}$( U ˜ X$_{opt}$U † − 1 $_{M}$ρ$_{0}$ ) = 0 , subject to Π$_{0}$ ∈ P and ∑ M − 1 i =0 S $^{i}$Π$_{0}$ S − i = I . The new formulation of the dual problem is given in a The complexity of a linear program, to get the solution of a convex optimization problem, is hard to evaluate in terms of arithmetic operations (see [8] for details). Nev- ertheless, since we are seeking a global optimum, the di- mension of the feasible region for the objective function gives an order of the complexity of the problem [8]. The set of self-adjoint operators on H forms an N $^{2}$– dimensional real vector space. Therefore, for a given op- timization problem we can find the dimension d of the correspondent real space on which the considered objec- tive function is defined. In other terms, we find the num- ber of real variables d in a given objective function. The results are summarized in Table I where C$_{e}$ and C$_{i}$ repre- sent, respectively, the number of equality and inequality constraints for a given problem. Note that the PSD con- dition on self-adjoint operators is counted as an inequal- ity, e.g., the relation Π$_{0}$ ∈ P is counted as the inequality Π$_{0}$ ≥ 0 . The dual problem DP3 presents the smaller number of variables and constraints among the considered optimiza- tion problems and, in particular, d = ∑ ¯ N − 1 i =0 N 2 i is smaller than N $^{2}$, depending on the spectrum of the symmetry op- erator S . If S has all distinct eigenvalues, then ¯ N = N , N$_{i}$ = 1 for each i , and the optimization operator ˜ X in DP3 becomes diagonal, giving d = N < N $^{2}$. In this section, the previous results are applied to the discrimination of an ensemble of M symmetric mixed quantum states on a 2-dimensional ( N = 2 ) complex Hilbert space. The symmetry operator is and it represents a linear transformation given by a coun- terclockwise rotation through angle π/M . We assume that the reference density operator has the general form ρ$_{0}$ = [ α β β 1 − α ] , (5) S = [ cos ( π M ) − sin ( π M ) sin π M cos π M ] , (4) Tr ( X $^{′}$) = 1 M M − 1 ∑ i =0 Tr ( S − $^{i}$XS$^{i}$) = M − 1 ∑ i =0 Tr ( XS $^{i}$S − $^{i}$) = 1 M M − 1 ∑ i =0 Tr ( X ) = Tr ( X ) , Table I: Number of real decision variables d , equality constraints C e and inequality constraints C i for the opti- mization problems. d C e C i PP1 MN 2 1 M PP2 N 2 1 1 DP1 N 2 0 M DP2 N 2 0 2 DP3 ∑ ¯ N i $_{=0}$N 2 i 0 1 3 D. Problem dimension and number of constraints C. Alternative Formulation of the Dual Problem IV. EXAMPLE OF APPLICATION