Quantum reading under a local energy constraint

Nonclassical states of light play a central role in many quantum information protocols. Very recently, their quantum features have been exploited to improve the readout of information from digital memories, modeled as arrays of microscopic beam splitters [Pirandola, Phys. Rev. Lett. 106, 090504 (2011)]. In this model of “quantum reading,” a nonclassical source of light with Einstein-Podolski-Rosen correlations has been proven to retrieve more information than any classical source. In particular, the quantum-classical comparison has been performed under a global energy constraint, i.e., by fixing the mean total number of photons irradiated over each memory cell. In this paper we provide an alternative analysis which is based on a local energy constraint, meaning that we fix the mean number of photons per signal mode irradiated over the memory cell. Under this assumption, we investigate the critical number of signal modes after which a nonclassical source of light is able to beat any classical source irradiating the same number of signals.

Figure 1

Inset (a): Quantum reading of Ref. [56] is formulated under a global energy constraint. This means that we fix the total average number of photons $N$ irradiated over the memory cell. Thus, if the number of input signals is $M$, each one has an average of $N/M$ photons, which goes to zero for $M\to \infty$. Inset (b): In this paper, we consider an alternative model of quantum reading under a local energy constraint (locally constrained quantum reading). In this case, we fix the average number of photons $N_S$ for each input signal. Since the total number of signals $M$ can be arbitrary, we have that the total energy irradiated over the cell $M{N}_S$ is generally unbounded.

Figure 2

Model of memory. Digital information is stored in a disk whose memory cells are beam-splitter mirrors with different reflectivities: $r=r_0$ encoding bit value $u=0$, and $r=r_1$ encoding bit value $u=1$. Readout. A reader is generally composed of a transmitter and a receiver. It retrieves a stored bit by probing a memory cell with a signal system $S$ (composed of $M$ bosonic modes) and detecting the reflected system $R$ together with an idler system $I$ (composed of $L$ bosonic modes). In general, the output system $R$ combines the signal system $S$ with a bath system $B$ ($M$ bosonic modes in thermal states). The transmitter is in a state $\rho$ which can be classical (a classical transmitter) or nonclassical (a quantum transmitter). In particular, we consider a quantum transmitter with EPR correlations between the signal and idler systems. In this paper, the quantum-classical comparison is performed under a local energy constraint, i.e., by fixing the average number of photons $N_S$ per signal mode. (The signal system $S$ has a total average number of photons $N=M{N}_S$, which is generally unbounded.)

Figure 3

Contour plots of the information gain $G^{*}$ as a function of the signal energy $N_S$ and the number of signal modes $M$ (in logarithmic scale). Reflectivities are $r_0=0.6$ and $r_1=0.95$. Thermal noise is $N_B={10}^{-3}$ (left plot) and $N_B={10}^{-2}$ (right plot). In the bottom black area, we have $G^{*}=0$. The maximum values of $G^{*}$ are taken in the intermediate white area where $G^{*}\gtrsim 0.3$ bits. For a large number of modes $M$, we have $G^{*}\to 0$.

Figure 4

Contour plots of the information gain $G^{*}$ as a function of the signal energy $N_S$ and the number of signal modes $M$ (in logarithmic scale). Reflectivities are $r_0=0.95$ and $r_1=0.98$. Thermal noise is $N_B={10}^{-3}$ (left plot) and $N_B={10}^{-2}$ (right plot). In the bottom black area, we have $G^{*}=0$. The maximum values of $G^{*}$ are taken in the intermediate white area where $G^{*}\gtrsim 0.7$ bits. For a large number of modes $M$, we have $G^{*}\to 0$.

Figure 5

Contour plots of the information gain $G^{*}$ as a function of the signal energy $N_S$ and the number of signal modes $M$ (in logarithmic scale). Thermal noise is equal to $N_B={10}^{-5}$. In the left plot, reflectivities are $r_0=0.95$ and $r_1=0.98$. In the right plot, reflectivities are $r_0=0.95$ and $r_1=0.999$. In the bottom black area, we have $G^{*}=0$. The maximum values of $G^{*}$ are taken in the intermediate white area where $G^{*}\gtrsim 0.7$ (left) and $G^{*}\gtrsim 0.8$ (right). For large number of modes $M$, we have $G^{*}\to 0$.

Figure 6

Number of signals $M$ (logarithmic scale) versus pit reflectivity $r_0$. The curves refer to $N_S=0.01$, $0.1$, and $0.5$ photons. For each value of the energy $N_S$, we plot the critical number $M^{\left(N_S\right)}\left(r_0\right)$ as a function of $r_0$. All the curves have an asymptote at $r_0=1$. For $N_S\gtrsim 2.5$ photons (curves not shown), we have another asymptote at $r_0=0$.

Figure 7

Minimum number of signals $M$ versus energy $N_S$. The solid curve represents $M^{\left(N_S\right)}\left(0\right)$ while the dashed curve represents $\tilde{M}$. Notice that the minimum number of signals is actually given by $\lceil M\rceil$, where $\lceil \cdots \rceil$ is the ceiling function.