Efficiently Computable Strategies and Limits for Bosonic Channel Discrimination

Abstract

Discriminating between noisy quantum processes is a central primitive for quantum communication, metrology, and computing. While discrimination limits for finite-dimensional channels are well understood, the continuous-variable setting, particularly under experimentally relevant energy constraints, remains significantly less developed. In this work, we establish an energy-constrained chain rule for the Belavkin-Staszewski channel divergence, which yields a fundamental upper bound on the error exponents achievable by fully adaptive, energy-constrained quantum channel discrimination protocols. We then derive efficiently computable bounds on asymmetric error exponents for energy-constrained discrimination of bosonic dephasing and loss-dephasing channels. Specifically, we show that three operationally relevant quantities -- the measured relative entropy, the Umegaki relative entropy, and the geometric Renyi divergence -- admit semidefinite program (SDP) formulations when the input energy is bounded and the Hilbert space is suitably truncated. Applying these tools, we demonstrate that optimal probes for these channels under energy constraints are Fock-diagonal, and we also enable numerically precise evaluation of bounds on achievable error exponents across discrimination strategies ranging from separable to fully adaptive. The resulting SDPs provide practical benchmarks for quantum-limited sensing in low-energy bosonic platforms.

Figures (6)

• Figure 1: Tiered quantum channel discrimination strategies. The channel $\mathcal{E}$ is applied, chosen from $(\mathcal{N},\mathcal{M} )$: (a) the measured relative entropy, requiring only product measurements of the form depicted; (b) the quantum relative entropy, in general requiring a collective measurement; (c) a general, adaptive protocol for channel discrimination, when channel is called three times. The initial input state is $\tau$, the adaptive operations are $\mathcal{A}_1$ and $\mathcal{A}_2$, and the final measurement is $\mathcal{Q}$. The final states are denoted by $\rho_0^{(n)}$ and $\rho_1^{(n)}$, and $n=3$ in this case.
• Figure 2: Channel divergences for a pure dephasing channel as a function of the energy constraint $E$; the parameters are $\gamma_1 = 0.1$ and $\gamma_2 = 0.4$. The dimension of the reduced Hilbert space of the probe satisfies $\dim(R)=9$.
• Figure 3: Relative entropies for a loss-dephasing channel as a function of the energy constraint $E$. In this plot, the transmissivity parameters are $\eta_1 = 0.95$ and $\eta_2 = 0.85$, with a small dephasing parameter $\gamma_1=\gamma_2 = 0.01$. The dimension of the reduced Hilbert space of the probe satisfies $\dim(R)=9$.
• Figure 4: Relative entropies for the bosonic dephasing channel as a function of $\gamma_2$. Here we fix $\gamma_1 = 1$ and the energy constraint $E=0.5$. The environmental state upper bounds the ultimate quantum limit without energy constraints (see Eq. (31) of Ref. huang2024exact.
• Figure 5: Channel divergences for two dephasing channels as a function of the Hilbert space truncation, for $E=1$; the dephasing parameters are $\gamma_1 = 0.1$, and $\gamma_2 = 0.4$.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2