Information, fidelity, and reversibility in general quantum measurements

We present the amount of information, fidelity, and reversibility obtained by arbitrary quantum measurements on completely unknown states. These quantities are expressed as functions of the singular values of a measurement operator corresponding to the obtained outcome. As an example, we consider a class of quantum measurements with highly degenerate singular values to discuss trade-offs among information, fidelity, and reversibility. The trade-offs are at the level of a single outcome, in the sense that the quantities pertain to each single outcome rather than the average over all possible outcomes.

Figure 1

Information $I\left(m\right)$ as a function of singular value $\lambda$ in $d=4$ for $(k,l)=(1,1),(1,2),(1,3),(2,1),(2,2)$, and $(3,1)$. The symbols $\left\{{\mathrm{P}}_r\right\}(r=1,2,3,4)$ denote projective measurements of rank $r$.

Figure 2

Fidelity $F\left(m\right)$ as a function of singular value $\lambda$ in $d=4$ for $(k,l)=(1,1),(1,2),(1,3),(2,1),(2,2)$, and $(3,1)$. The symbols $\left\{{\mathrm{P}}_r\right\}(r=1,2,3,4)$ denote projective measurements of rank $r$.

Figure 3

Reversibility $R\left(m\right)$ as a function of singular value $\lambda$ in $d=4$ for $(k,l)=(1,1),(1,2),(1,3),(2,1),(2,2)$, and $(3,1)$. The symbols $\left\{{\mathrm{P}}_r\right\}(r=1,2,3,4)$ denote projective measurements of rank $r$.

Figure 4

Efficiency with respect to fidelity, $E_F\left(m\right)$, as a function of singular value $\lambda$ in $d=4$ for $(k,l)=(1,1),(1,2),(1,3),(2,1),(2,2)$, and $(3,1)$. The symbols $\left\{{\mathrm{P}}_r\right\}(r=1,2,3,4)$ denote projective measurements of rank $r$.

Figure 5

Efficiency with respect to reversibility, $E_R\left(m\right)$, as a function of singular value $\lambda$ in $d=4$ for $(k,l)=(1,1),(1,2),(1,3),(2,1),(2,2)$, and $(3,1)$. The symbols $\left\{{\mathrm{P}}_r\right\}(r=1,2,3,4)$ denote projective measurements of rank $r$.