Probability and fidelity of teleportation in a two-mode continuous variable cluster state via an insufficiently selective measurement

TL;DR

This work addresses how finite squeezing and continuous-variable measurements affect teleportation fidelity and success probability in a two-mode CV cluster.It introduces insufficiently selective measurements as a mechanism to analyze and potentially optimize these metrics.The measurement outcome distribution is shown to correspond to the generalized Weierstrass transform, tying to the heat equation.Fidelity is expressed as a ratio of the non-homogeneous heat-equation solution to the standard heat equation and extended to sequential clusters with intermediate corrections.Using a squeezed-coherent state as the teleported input, the paper illustrates the practical trade-offs and suggests a pathway for CV one-way quantum computing with finite squeezing.

Abstract

Continuous-variable projective measurements can not select individual measurement results as in the discrete case; instead, the possible outcomes are bounded by the selectivity interval of the measurement; then, it is say that continuous-variable measurement devices are insufficiently selective. By utilizing this concept we show that the probability and fidelity of teleportation in a two-mode cluster state can be handled by the localization of the selectivity interval of the measurement apparatus. Besides, we provide a mathematical expression describing the probability distribution of the measurement outcomes in the two-mode cluster, which is a fundamental solution of the heat equation. In addition, we show that the fidelity of teleportation in the two mode cluster is given by the quotient between the squared solution of a non-homogeneous heat equation and the solution of the conventional heat equation. Furthermore, we extend our approach to a configuration involving successive clusters with intermediate corrections between each teleportation step. To exemplify our proposal, we consider the specific case of a squeezed-coherent state as the quantum state under teleportation.

Figures (10)

• Figure 1: Let be $\left|\psi(X)\right|^2=\left|\left\langle X\right.\left|\psi\right\rangle\right|^2$ the probability distribution (black curve with three peaks) of, for example, a continuous-variable observable $\hat{X}$. We consider a couple of detectors measuring $\hat{X}$ (such that $\hat{X}\left|X\right\rangle=X\left|X\right\rangle$) on the quantum system $\left|\psi\right\rangle$. These devices are centered on the point $X_{0}'$ on the $X$-axis, and can measure with resolutions $\Delta X_{1}$ and $\Delta X_{2}$; then, they can only detect points inside these intervals, giving indistinguishably the central value $X_{0}'$ as the measurement result. Besides, we have $\left|\Delta X_{1}\right| > \left|\Delta X_{2}\right|$. Then, while both detectors will give the outcome $X_{0}'$ when detecting points inside $\Delta X_{(1,2)}$, is the second which will provide a more accurate determination of the measured observable since all the set of points contained in $\Delta X_{2}$ are closer to $X_{0}'$ than the whole set contained in $\Delta X_{1}$. Notably, the probability of detecting $X_{0}'$ (represented by the blue and yellow areas below the curve and inside $\Delta X_{(1,2)}$) decreases as the measurement device becomes more accurate.
• Figure 2: Quantum circuit for noisy Gaussian teleportation of an arbitrary state $\left|\psi\right\rangle$ using a two-mode CV cluster state. The input states become entangled (dotted lines) through a $\hat{C}_Z$ gate. Subsequently, we perform an insufficiently selective projective measurement on the first mode; then, for all possible outputs inside the selectivity region of the measurement, we get the state $\mathcal{\hat{M}}\hat{X} (p_{1}) \hat{F} \left|\psi\right\rangle$ in the second mode of the circuit.
• Figure 3: We assume that the probability distribution of the measurement results within the cluster, $P(p_{1})$, is encompassed by a set of finite-resolution detectors, each having associated a measurement result, which is determined by the central coordinate of the corresponding selectivity interval. We identify the $i$-esim detector $D_{i}$ with an associated measurement result $p_{1}'$ and a selectivity interval $\Delta p_{1}$. From a phase-space perspective, each finite-resolution detector selects a region of the momentum marginal distribution $P(p_{1})$ (see Fig. \ref{['fig:3']} for a specific instance) associated with the quantum system under meaurement.
• Figure 4: Probability of teleportation of the squeezed-coherent state versus (a) the width of the selectivity interval $\Delta p_{1}$ ($V_{s}^2 = 1, \theta =\pi$) for different squeezing parameters $r_{1}$, and (b) the squeezing parameter of the squeezed vacuum state ($\delta_{q}^2 (r_{1}, \theta) = 0.25, V_{s}^2 =e^{2r_{2}}$) for various widths of the selectivity interval. The black curves represent the case where $\Delta p_{1}$ is centered on the central coordinate of the position of the squeezed-coherent state ($p_{1}'=-q_{0}$), while red curves represent the case where $\Delta p_{1}$ is not centered on the momentum coordinate allowing maximum teleportation; in particular for (a) we have $(p_{1}'+q_{0})=0.75$ and (b) $(p_{1}'+q_{0})=0.3$. Figures (a) and (b) show that the probability of teleportation increases when the selectivity interval of the measurement and the squeezing in position of both states building the cluster increases; besides, as we move away from the central value $p_{1}'=-q_{0}$, the probability of teleportation decreases.
• Figure 5: Behavior of the fidelity of teleportation $\mathcal{F}$ of a squeezed-coherent state inside the three-dimensional region $\mathcal{R}=\left\lbrace (r_{1}, r_{2}, \theta) | -1\leq r_{1}\leq 1, -1\leq r_{2}\leq 1, 0\leq\theta \leq 2\pi\right\rbrace$ for a two-mode cluster state with finite squeezing. The effective displacements are (a) $X_{0}=0$, (b) $X_{0}=\pm 1$, (c) $X_{0}=\pm 1.75$, (d) $X_{0}=\pm 3.5$. The fidelity inside $\mathcal{R}$ diminishes as the effective displacement increases.