Probabilistic phase-insensitive optical squeezer in compliance with causality

A non-trace-preserving map describing a probabilistic but heralded noiseless linear amplifier has recently been proposed and experimentally demonstrated. Here, we exhibit another remarkable feature of this peculiar transformation, namely its ability to serve as a universal single-mode squeezer regardless of the quadrature that is initially squeezed. Hence, it acts as a heralded phase-insensitive optical squeezer, conserving the signal-to-noise ratio just as a phase-sensitive optical amplifier but for all quadratures at the same time, which may offer interesting perspectives in quantum optical communications. Although this ability to squeeze all quadratures seemingly opens a way to instantaneous signaling by circumventing the quantum no-cloning theorem, we explain the subtle mechanism by which such a causality violation is forbidden, even on an heralded basis.

Figure 1

Fidelity $F$ of the approximate phase-insensitive squeezer (with respect to the ideal one) as a function of the truncation size $N$ for $g=1.1$. We consider a vacuum-squeezed state at the input with several values of the squeezing $r$ (from 2 dB to 8 dB). The success probability $P_{\mathrm{succ}}$ is also shown. For $N=2$ and a squeezing of 4 dB, we have $F=0.9694$ and $P_{\mathrm{suc}}=0.7099$.

Figure 2

Simple illustration of the peaceful coexistence of the HNLA with special relativity. Alice and Bob share an entangled state $|{\mathrm{EPR}}_s\rangle$ and we compare two situations: (i) Bob amplifies his mode with the HNLA of gain $g$, which creates a stronger entangled state $|{\mathrm{EPR}}_{s^{\prime }}\rangle$, and then Alice measures her mode, thereby preparing some mixture at Bob's side; (ii) Alice measures her mode first, thereby preparing some mixture at Bob's side, and Bob later amplifies each component pure state of this mixture. It is verified that these two situations yield the same average state ${\rho }_{s^{\prime }}^{\mathrm{th}}$ at Bob's side as a consequence of Bayes rule (both photon number or heterodyne measurement are considered).

Figure 3

Continuous-variable analog to Dieks' scheme exploiting the heralded phase-insensitive squeezer, which seemingly opens the possibility for instantaneous signaling on a heralded basis. Alice and Bob share an entangled state $|{\mathrm{EPR}}_s\rangle$, and Alice measures the $x$ or $p$ quadrature in order to send the bit 0 or 1, respectively. This results into two possible (indistinguishable) mixtures of squeezed states at Bob's side. The perfect cloning (or amplification) of each component squeezed state of these two mixtures realized with the heralded phase-insensitive squeezer of gain $g$ seems to provide Bob with a way to discriminate the phase-encoded information ($x$ or $p$), which would lead to heralded signaling. In fact, the state dependence of this heralded transformation conspires with Bayes rule and forbids this to happen. Bob's mixtures coincide with the same state ${\rho }_{s^{\prime }}^{\mathrm{th}}$ regardless of Alice's bit.