Exact maximum purity of absolutely PPT qutrit–qudit states
Establish the exact maximum purity of absolutely positive partial transpose (PPT) states ρ ∈ M_3 ⊗ M_n for n ≥ 2 by proving that the maximum equals 4/(9n) when n ≡ 0 (mod 4), equals (36n+8)/( (9n+1)^2 ) when n ≡ 1 (mod 4), equals (36n+16)/( (9n+2)^2 ) when n ≡ 2 (mod 4), and equals (36n−8)/( (9n−1)^2 ) when n ≡ 3 (mod 4), and demonstrate that the maximizers are precisely those states whose spectra match the corresponding piecewise patterns specified in Conjecture 2.
References
The numerical finding in Table 2 and Conjecture \ref{conj1} lead us to the following conjecture for the qutrit-qudit case. Conjecture \ref{conj2} The maximum purity of absolutely PPT states \rho \in M_3\otimes M_n, n\ge 2, is given by \frac{4}{9n} if n \equiv 0 \mod 4, \frac{36n+8}{(9n+1)2} if n \equiv 1 \mod 4, \frac{36n+16}{(9n+2)2} if n \equiv 2 \mod 4, and \frac{36n-8}{(9n-1)2} if n \equiv 3 \mod 4. Moreover, the maximum value is attained when the corresponding eigenvalue multiplicities and values specified in the conjecture hold.