Quantum reading of digital memory with non-Gaussian entangled light

It has been shown recently S. Pirandola [Phys. Rev. Lett. 106, 090504 (2011)] that entangled light with Einstein-Podolsky-Rosen correlations retrieves information from digital memory better than any classical light. In identifying this, a model of digital memory with each cell consisting of a reflecting medium with two reflectivities (each memory cell encoding the binary numbers 0 or 1) is employed. The readout of binary memory essentially corresponds to discrimination of two bosonic attenuator channels characterized by different reflectivities. The model requires an entire mathematical paraphernalia of a continuous variable Gaussian setting for its analysis when arbitrary values of reflectivities $0\le r_0,r_1\le 1$ are considered. Here we restrict ourselves to a basic quantum readout mechanism with two different families of non-Gaussian entangled states of light, in which the binary channels to be discriminated are (i) ideal memory characterized by reflectivity $r_1=1$ (identity channel) and (ii) a thermal noise channel—where the signal light illuminating the memory location gets completely lost ($r_0=0$) and only a white thermal noise hitting the upper side of the memory reaches the decoder. We compare the quantum reading efficiency of two families of non-Gaussian entangled light [($m,m^{\prime }$) family of path-entangled photon states and entangled state obtained by mixing a single photon with coherent light in a 50:50 beam splitter] with any classical source of light in this model. We identify that the classes of non-Gaussian entangled transmitters studied here offer significantly better reading performance than any classical transmitters of light in the regime of low signal intensity. We also demonstrate that the ($m,m^{\prime }$) family of entangled light exhibits better reading performance than NOON states.

Figure 1

Quantum Chernoff bound on error $P_{\mathrm{err},\mathrm{QCB}}^{(m,m^{\prime })}(M,N_S,N_B,m)$ of a ($m,m^{\prime }$) family of states $\left\{\right|m::2N_S-m\rangle ,N_S<m\le 2N_S\}$ and the classical error bound $\mathcal{C}(M,N_S,N_B)$ vs number of copies $M$ for the values (a) $N_S=1,m=2$, (b) $N_S=3,m=4,$ $5,6$, and thermal noise $N_B={10}^{-1}.$ It is clear from Fig. 1b that NOON states ($m=6$) fail to beat the general ($m,m^{\prime }$) class of states as well as classical light.

Figure 2

Maximum information retrieved $J_{\max ,\mathcal{C}}$ in the readout process from a classical transmitter and the minimum information decoded from quantum light of the ($m,m^{\prime }$) state for (a) $N_S=0.5$ and $N_B={10}^{-5}$, and (b) $N_S=2.5$ and $N_B=1$, with $m=3,4$ and $m=5$ (NOON state of the family) plotted as a function of number of copies $M$. It may be seen that quantum reading information retrieved via NOON states is smaller compared to that of other states of the ($m,m^{\prime }$) family and, also, classical light.

Figure 3

Error probability bounds $P_{\mathrm{err},\mathrm{QCB}}^{\left(\Psi \right)}(M,N_S,N_B)$ and $\mathcal{C}(M,N_S,N_B)$ of the entangled state $|{\Psi }_{SI}\rangle$ [see Eq. (16)] as a function of number of copies $M$ for different choices of mean signal intensity and thermal noise (a) $N_S=1$, $N_B=0.01$, (b) $N_S=5,N_B=1.5$. In Fig. 1a, we also plot the error probability of the single-photon Fock state, which is found to be much smaller than the Chernoff error bound of the entangled state $|{\Psi }_{SI}\rangle$.

Figure 4

Maximum information $J_{\max ,\mathcal{C}}$ retrieved from digital memory when classical light is used, and minimum information $J_{\min ,\mathcal{Q}}$ decoded by employing quantum entangled state (16) of light as a function of number of copies M for the entangled state for $N_S=0.75$ and $N_B={10}^{-5}.$