Communication capacity of mixed quantum $t$-designs

We operationally introduce mixed quantum $t$-designs as the most general arbitrary-rank extension of projective quantum $t$-designs which preserves indistinguishability from the uniform distribution for $t$ copies. First, we derive upper bounds on the classical communication capacity of any mixed $t$-design measurement for $t\in [1,5]$. Second, we explicitly compute the classical communication capacity of several mixed $t$-design measurements, including the depolarized version of any qubit and qutrit symmetric, informationally complete (SIC) measurement and complete mutually unbiased bases, the qubit icosahedral measurement, the Hoggar SIC measurement, any anti-SIC (where each element is proportional to the projector on the subspace orthogonal to one of the elements of the original SIC), and the uniform distribution over pure effects.

Figure 1

Classical capacities of qubit mixed $t$-design POVMs as a function of the depolarizing parameter $\lambda$. Dashed lines, from top to bottom, represent the upper bounds on the capacity of the depolarized version of any projective 2-design (blue) and 3-design (red), as given by Theorem 1. Solid lines, from top to bottom, represent the capacities of the depolarized version of any SIC (blue), complete MUB (red), icosahedron (purple), and uniform distribution (green), as given by Theorem 2.

Figure 2

Classical capacities of qutrit mixed $t$-design POVMs as a function of the depolarizing parameter $\lambda$. Dashed lines, from top to bottom, represent the upper bounds on the capacity of the depolarized version of any projective 2-design (blue) and any antiprojective 2-design (red), as given by Theorem 1. Solid lines, from top to bottom, represent the capacities of the depolarized version of any SIC and MUB (blue), uniform distribution (green), and anti-SIC (red), as given by Theorem 2.

Figure 3

Classical capacities of eight-dimensional mixed $t$-design POVMs as a function of the depolarizing parameter $\lambda$. Dashed lines, from top to bottom, represent the upper bounds on the capacity of the depolarized version of any projective 2-design (blue) and any antiprojective 2-design (red), as given by Theorem 1. Solid lines, from top to bottom, represent the capacities of the depolarized version of any Hoggar SIC (blue), uniform distribution (green), and anti-SIC (red), as given by Theorem 2.