Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: Applications to quantum reading and target detection

We consider the problem of distinguishing, with minimum probability of error, two optical beam-splitter channels with unequal complex-valued reflectivities using general quantum probe states entangled over $M$ signal and $M^{\prime }$ idler mode pairs of which the signal modes are bounced off the beam splitter while the idler modes are retained losslessly. We obtain a lower bound on the output state fidelity valid for any pure input state. We define number-diagonal signal (NDS) states to be input states whose density operator in the signal modes is diagonal in the multimode number basis. For such input states, we derive series formulas for the optimal error probability, the output state fidelity, and the Chernoff-type upper bounds on the error probability. For the special cases of quantum reading of a classical digital memory and target detection (for which the reflectivities are real valued), we show that for a given input signal photon probability distribution, the fidelity is minimized by the NDS states with that distribution and that for a given average total signal energy $N_s$, the fidelity is minimized by any multimode Fock state with $N_s$ total signal photons. For reading of an ideal memory, it is shown that Fock state inputs minimize the Chernoff bound. For target detection under high-loss conditions, a no-go result showing the lack of appreciable quantum advantage over coherent state transmitters is derived. A comparison of the error probability performance for quantum reading of number state and two-mode squeezed vacuum state (or EPR state) transmitters relative to coherent state transmitters is presented for various values of the reflectances. While the nonclassical states in general perform better than the coherent state, the quantitative performance gains differ depending on the values of the reflectances. The experimental outlook for realizing nonclassical gains from number state transmitters with current technology at moderate to high values of the reflectances is argued to be good.

Figure 1

Schematic of setup for determining which of two beam-splitter channels (indexed by $b$) of the form of Eq. (1) is present within a black box B. A quantum state source $S$ produces a pure state of $M$ signal modes ${\left\{{\widehat{a}}_{\mathrm{in}}^m\right\}}_{m=1}^M$ and $M^{\prime }$ idler modes ${\left\{{\widehat{b}}_{\mathrm{in}}^{m^{\prime }}\right\}}_{m^{\prime }=1}^{M^{\prime }}$ of which one signal-idler pair is shown. Each signal mode reflects off the beam splitter while the idler mode is unaffected. The environment modes ${\left\{{\widehat{e}}_{\mathrm{in}}^m\right\}}_{m=1}^M$ coupling to the beam splitter are all in the vacuum state. The detection module $D$ performs the minimum error probability (Helstrom) measurement for distinguishing the two possible $(M+M^{\prime })$-mode quantum states corresponding to $b=0$ and $b=1$.

Figure 2

Number state vs. coherent state Chernoff exponent gain G of Eq. (69) for values of $R_0$ and $R_1$ above $0.4$. Appreciable gain is seen for larger values of the reflectances.

Figure 3

Error probability bounds versus $N_s$ for target detection with $R_0=0$ and $R_1=0.3$. The number of modes $M=50$ for the EPR state curves.

Figure 4

Error probability bounds versus $N_s$ for quantum reading with $R_0=0.3$ and $R_1=0.6$. The number of modes $M=50$ for the EPR state curves.

Figure 5

Error probability bounds versus $N_s$ for quantum reading with $R_0=0.5$ and $R_1=0.75$. The number of modes $M=50$ for the EPR state curves.

Figure 6

Error probability bounds versus $N_s$ for quantum reading with $R_0=0.2$ and $R_1=0.8$. The number of modes $M=50$ for the EPR state curves.

Figure 7

Error probability bounds versus $N_s$ for reading of an ideal memory with $R_0=0.5$ and $R_1=1$. The number of modes $M=50$ for the EPR state curves.