Certified answers for ordered quantum discrimination problems

We investigate the quantum state discrimination task for sets of linear independent pure states with an intrinsic ordering. These structured discrimination problems allow for a scheme that provides a certified level of error; that is, answers that never deviate from the true value by more than a specified distance and hence control the desired quality of the results. We obtain an efficient semidefinite program and also find a general lower bound valid for any error distance that only requires the knowledge of an optimal minimum-error scheme. We apply our results to the cases of quantum change point and quantum state anomaly detection.

Figure 1

Structure of the source states. The blue dot labels the true state. The adjacent positions (in pale red) label the possible errors of the discrimination scheme outcomes.

Figure 2

Example of the structure of $F_r^{\mathrm{\Delta }}=R^{†}{E}_r^{\mathrm{\Delta }}R$. The matrices $F_r^{\mathrm{\Delta }}$ have dimensions $n\times n$ and their nonvanishing elements are depicted as colored boxes. For $\mathrm{\Delta }=0$ (no errors), the central blue box is the only nonvanishing element. For $\mathrm{\Delta }=1$ (one error) the nonvanishing elements are contained in the red $3\times 3$ block, and in the $5\times 5$ green block for $\mathrm{\Delta }=2$, etc. The remaining entries of $F_r^{\mathrm{\Delta }}$ are all zero.

Figure 3

Structure of the matrix variable $Z$ for $\mathrm{\Delta }=1$. The blue boxes correspond to the free matrix elements and the blank ones are fixed to be zero. The highlighted boxes are the elements that appear in the objective function $(1/n)Tr\left[ZA\right]$ of Eq. (5).

Figure 4

The correspondent map takes the nonzero parts of the variable $Z$ and accommodates it in $n\times n$ matrices with zeros in the remaining places. Observe that matrix sum is defined only for matrices of the same dimensions.

Figure 5

We depict a specific block $r$ of $Z^{\mathrm{ME}}-Z^{\mathrm{\Delta }}$. The light blue block corresponds to $Z_r^{\mathrm{ME}}$ and the darker one to $Z_r^{\mathrm{\Delta }}$. The small black boxes show the elements of the minor of interest to obtain the bound (9).

Figure 6

A machine produces a signal state and suddenly it produces another signal. Our task is to determine by measurements the exact moment when this change happens.

Figure 7

Probability of success versus the allowed error distance $\mathrm{\Delta }$ for QCP with $n=25$ and $c=0.6$. The blue solid curve is the exact numerical SDP values (5). The dashed green curve is the analytical lower bound (21). We also show as a reference the red straight line with the value of the minimum-error scheme.

Figure 8

Outcome probability profile of the minimum-error scheme of the QCP. The parameter $\delta =\widehat{k}-k_0$ is the distance of the output guess $\widehat{k}$ with respect to the position $k_0$ of the true change point. Here $n=25$, $c=0.6$ and we take the change point to occur at the central position $k_0=13$.

Figure 9

Probability of success against the error distance $\mathrm{\Delta }$ for QSAD with $n=25$ and $c=0.6$. The blue solid line shows the numerical results from SDP (5) and the dashed green line shows the lower bound [Eqs. (28) and (30)]. We also show as a reference a black straight line with the value of the minimum-error scheme.

Figure 10

Path of the contour integral of $\mu (z,c)$ in the complex plane.