Information–disturbance tradeoff in continuous-variable Gaussian systems

We address the information–disturbance tradeoff for state measurements on continuous variable Gaussian systems and suggest minimal schemes for implementations. In our schemes, the symbols from a given alphabet are encoded in a set of Gaussian signals which are coupled to a probe excited in a known state. After the interaction the probe is measured, in order to infer the transmitted state, while the conditional state of the signal is left for the subsequent user. The schemes are minimal, i.e., involve a single additional probe, and allow for the nondemolitive transmission of a continuous real alphabet over a quantum channel. The tradeoff between information gain and state disturbance is quantified by fidelities and, after optimization with respect to the measurement, analyzed in terms of the energy carried by the signal and the probe. We found that transmission fidelity only depends on the energy of the signal and the probe, whereas estimation fidelity also depends on the alphabet size and the measurement gain. Increasing the probe energy does not necessarily lead to a better tradeoff, the most relevant parameter being the ratio between the alphabet size and the signal width, which in turn determine the allocation of the signal energy.

Figure 1

(Color online) Schematic diagram of an indirect measurement scheme for continuous variable systems. The symbols from the input alphabet $\mathbb{A}$ are encoded in a set of Gaussian signals $\mid {\psi }_{a\tau }{⟩}_s$ with real amplitude $a$ and fixed width $\tau$. The a priori probability of the transmitted symbols is a Gaussian with zero mean and width $\Delta$. The signal is coupled, by a $C_{\mathrm{sum}}$ gate, to a probe excited in the known state $\mid {\varphi }_{\theta \sigma }{⟩}_p$. After the interaction the probe is measured in order to infer the transmitted state, while the signal is left for the subsequent user. The measurement is described by the operator-valued measure $\Pi \left(b\right)$, which is optimized in order to maximize the estimation fidelity. The output signals $\mid {\phi }_b{⟩}_s$ are the conditional states of the signal after having observed the outcome $b$.

Figure 2

(Color online) Left: Transmission fidelity $F(\sigma ,\theta )$ for $\kappa =1$ and for different values of $\tau$; from top to bottom $\tau =0.4,1∕\sqrt{2},2$. Right: Estimation fidelity $G(\sigma ,\theta )$ for $\kappa =1$, $\Delta =1∕\sqrt{2}$ and for different values of $\tau$; from top to bottom $\tau =0.4,1\sqrt{2},2$.

Figure 3

Information–disturbance tradeoff $F_A(G,y)$ in the high energy limit for the probe, for different values of the width ratio $y={\Delta }^2∕{\tau }^2$. From darker to lighter gray we plot the tradeoff for $y=0.5,3,7,\mathrm{10\hspace{0.17em}000}$. The tradeoff gets worse for increasing values of $y$. For $y\gg 1$ the function $F_A(G,y)$ intercepts the axis $F=0$ for $G=\sqrt{2∕3}$. One moves along the curve, from right to left, by increasing $\theta$.

Figure 4

Information–disturbance tradeoff $F_B(G,y)$ for the probe in the undisplaced state $\mid {\psi }_{0\sigma }{⟩}_p$ and for different values of the ratio $y={\Delta }^2∕{\tau }^2$. From darker to lighter gray: the tradeoff for $y=0.1,1,3,7,\mathrm{10\hspace{0.17em}000}$. The tradeoff gets worse for increasing values of $y$. For $y\gg 1$ the function $F_B(G,y)$ intercepts the $G$ axis at $G=\sqrt{2∕3}$. One moves along the curves from right to left by increasing $\sigma$.

Figure 5

Information–disturbance tradeoff $F_C(G,\Delta )$ for the probe with minimum energy and for different values of the alphabet width $\Delta$. From darker to lighter gray: the tradeoff for $\Delta =0.1,0.2,2,5,\infty$. The tradeoff gets worse for increasing values of $\Delta$. For $\Delta \to \infty$ the function $F_C(G,\Delta )$ intercepts the $G$ axis at $G=\sqrt{2∕3}$. One moves along the curves from left to right by increasing $\tau$.

Figure 6

Information–disturbance tradeoffs $F_B(G,2{\Delta }^2)$ and $F_C(G,\Delta )$ (dashed lines) for different values of $\Delta$. From darker to lighter gray: the tradeoffs for $\Delta =0.2,1,2,5,100$. For $\Delta \gtrsim 3$ the two configurations lead to similar results. For $\Delta <3$ configuration B favors the information fidelity whereas configuration C privileges the estimation fidelity.