Mixed State Discrimination Using Optimal Control

We present theory and experiment for the task of discriminating two nonorthogonal states, given multiple copies. We implement several local measurement schemes, on both pure states and states mixed by depolarizing noise. We find that schemes which are optimal (or have optimal scaling) without noise perform worse with noise than simply repeating the optimal single-copy measurement. Applying optimal control theory, we derive the globally optimal local measurement strategy, which outperforms all other local schemes, and experimentally implement it for various levels of noise.

Figure 1

Layout of the experiment. A polarizing beam splitter (PBS) acts as a filter to ensure high fidelity horizontally polarized photons. A half wave plate (HWP) in a motorized rotation stage determines the measurement basis. A polarizing beam displacer (PBD) and single-photon counting modules (SPCMs) discriminate between horizontally and vertically polarized photons with high contrast. The result of the measurement is fed to a processor which, depending on the protocol being tested, adjusts the operation of the HWP controller. Single photon inputs are obtained through type-I spontaneous parametric down-conversion—a 410 nm diode laser pumps a BiBO (bismuth borate) crystal, producing pairs of 820 nm single photons in the state $|HH⟩$, with photons in separate spatial modes. One of the photon pair is guided to the input of the experiment through a single-mode optical fiber. The other photon is guided directly to a single-photon counting module. Detection in coincidence ensures high fidelity single photons are measured in the experiment.

Figure 2

Probability of error $C_N$ in $N$-copy state discrimination using various schemes in the absence of noise. Lines represent theoretical predictions, points represent experimental data, each 2000 measurements. Error bars represent 1 standard deviation of the mean of a binomial distribution. The locally optimal local measurement scheme performs best for all $N$; where $N=1$ it is equivalent to unbiased measurements, both schemes using one single-copy optimal measurement.

Figure 3

Error probability $C_N$ of discrimination schemes under $\nu =10\%$ depolarizing noise. Points represent 1000 experimental discriminations. The addition of noise detrimentally impacts the locally optimal local measurement scheme more than the unbiased scheme. Indeed, theory predicts that the latter outperforms the former for $N=5$, $N=7$, and $N\ge 9$. The globally optimal local measurement scheme performs better than all other local measurement schemes in the presence of noise. The theoretical optimal collective measurement cost is plotted for comparison.

Figure 4

Error probability $C_N$ of discrimination schemes under various levels of noise $\nu$ for $N=10$ measured copies. Points each represent 2000 ($\nu =0$) or 1000 ($\nu >0$) experimental discriminations. Here, unbiased measurements outperform the locally optimal local measurement scheme for noise $\nu \gtrsim 10\%$.