Optimal single-shot discrimination of optical modes

Retrieving classical information encoded in optical modes is at the heart of many quantum information processing tasks, especially in the field of quantum communication and sensing. Yet, despite its importance, the fundamental limits of optical mode discrimination have been studied only in a few specific examples. Here we present a toolbox to find the optimal discrimination of any set of optical modes. The toolbox uses linear and semidefinite programming techniques, which provide rigorous (not heuristic) bounds, and which can be efficiently solved on standard computers. We study both probabilistic and unambiguous single-shot discrimination in two scenarios: the channel-discrimination scenario, typical of metrology, in which the verifier holds the light source and can set up a reference frame for the phase; and the source-discrimination scenario, more frequent in cryptography, in which the verifier only sees states that are diagonal in the photon-number basis. Our techniques are illustrated with several examples. Among the results, we find that, for many sets of modes, the optimal state for mode discrimination is a superposition or mixture of at most two number states; but this is not general, and we also exhibit counterexamples.

Figure 1

The two scenarios considered in this paper for mode discrimination. (a) Channel-discrimination scenario, studied in Sec. 2. The mode $a_j$ to be discriminated is encoded by a unitary transformation from an input mode $a_0$. The verifier has control of the source (star) and can add a reference beam, with respect to which phase of the signal beam is well defined. If the reference is classical, the state in the signal beam can be taken as pure. (b) Source-discrimination scenario, studied in Sec. 3. The source itself is in the black box, whence the signal exits encoded in the mode to be discriminated. For the verifier, the phase of the signal beam is global, and therefore the state appears as a photon-number mixture.

Figure 2

Geometrical solution of the dual LP (28). The first constraint is associated to the line $L_0:x={\mathsf{a}}_0$, and is satisfied in the half-space $x\ge {\mathsf{a}}_0$. For $n>0$, the constraint is associated to the line $L_n:y=-\frac{1}{n}(x-{\mathsf{a}}_n)$ and the region satisfying that constraint is the upper half-space above $L_n$. The feasible region is the region where all constraints are satisfied (shaded area, drawn for $n\le 3$). The objective function is a line $L_{\text{obj}}$ with gradient $-1/\overline{n}$ (dashed; for the plot, $\overline{n}=0.5$). The dual problem can hence be understood as finding the minimum $y$-intercept by translating the line $L_{\text{obj}}$ vertically while ensuring that it still touches the feasible region. The ${\mathsf{a}}_n$ satisfy (29) and thus all the $L_n$ contribute nontrivially to the boundary of the feasible region. Had we chosen ${\mathsf{a}}_1^{\prime }$ instead of ${\mathsf{a}}_1$, the line $L_1^{\prime }$ (dotted) would not contribute to that boundary; that is, the constraint for $n=1$ would always be satisfied within the feasible region.

Figure 3

Probabilistic channel discrimination between two modes: dependence on the mode overlap $k\in \mathbb{C}$. Upper bound $P_{\text{corr}}^{\text{SDP}}$ on the guessing probability, solution of the SDP (11) for $n_{\text{max}}=300$, in a polar plot of $k$, for (a) $\overline{n}=0.5$ and (b) $\overline{n}=1$.

Figure 4

Unambiguous channel discrimination between two modes: dependence on the mode overlap $k\in \mathbb{C}$. Upper bound $P_{\text{UD}}^{\text{SDP}}$ on the guessing probability, solution of the SDP (15) for $n_{\text{max}}=300$, in a polar plot of $k$, for (a) $\overline{n}=0.5$ and (b) $\overline{n}=1$.

Figure 5

Phase discrimination: Dependence on $\overline{n}$. (a) Upper bound $P_{\text{corr}}^{\text{SDP}}$ on the guessing probability, solution of the SDP (11). (b) Upper bound $P_{\text{UD}}^{\text{SDP}}$ on the success probability, solution of the SDP (15). Both SDPs solved for $n_{\text{max}}=50$. In (a), the circles denote the analytical bounds from Theorem 3 of Ref. [4].

Figure 6

Computational and Fourier transform basis modes: Dependence on $\overline{n}$. (a) Upper bound on the guessing probability for probabilistic discrimination. (b) Upper bound on the success probability for unambiguous discrimination. The solid lines are the bounds in the source-discrimination scenario, obtained by solving the LP (22) or (26), respectively, whereas the dashed curves are the bounds in the channel-discrimination scenario, obtained by solving the SDP (11) or (15). For this family of modes, it is sufficient to set $n_{\text{max}}=50$.

Figure 7

Computational and Fourier transform basis modes: optimal state versus Fock states (for probabilistic discrimination in the source-discrimination scenario). The solid lines are the bound $P_{\text{corr}}^{\text{LP}}$, the solution to (22) with $n_{\text{max}}=50$. On the other hand, the dots represent the bound $P_{\text{corr}}^{\left(n\right)}$, the solution to the SDP (19).

Figure 8

Discriminating the differential-phase-shift modes: Dependence on the energy per pulse $\mu$. (a) Upper bound on the guessing probability for probabilistic discrimination. (b) Upper bound on the success probability for unambiguous discrimination. In both cases, the solid lines are the bounds for source-discrimination scenario: Solutions to the LPs (22) and (26), respectively. The dashed curves in (a) and the circles in (b) are the bounds for channel-discrimination scenario obtained by solving the SDPs (11) and (15), respectively. Here, we set $n_{\text{max}}=50$.

Figure 9

The two scenarios studied in this paper, with additional mode-independent losses before the measurement device ($t$ refers to the transmittivity).

Figure 10

Probabilistic discrimination of two modes as a function of losses (channel-discrimination scenario). Plot of $P_{\text{corr}}^{\text{opt}}(k,\overline{n},t)$ as a function of $t^2$ for $\overline{n}=1$ and $k=0,0.4,0.8$ (from top to bottom). The thick solid lines are the result of the heuristic optimization (for which the photon number was truncated at 5); on the line for $k=0.4$ are indicated the points reported in Table 1. The dotted lines are the values $P_{\text{corr}}^{\text{coh}}$ for coherent state input; the dashed lines are the values $P_{\text{corr}}^{\text{Fock}}$ for Fock state input.