Fidelity of Fock-state-encoded qubits subjected to continuous-variable Gaussian processes

When a harmonic oscillator is under the influence of a Gaussian process such as linear damping, parametric gain, and linear coupling to a thermal environment, its coherent states are transformed into states with Gaussian Wigner function. Qubit states can be encoded in the $|0\rangle$ and $|1\rangle$ Fock states of a quantum harmonic oscillator, and it is relevant to know the fidelity of the output qubit state after a Gaussian process on the oscillator. In this paper we present a general expression for the average qubit fidelity in terms of the first and second moments of the output from input coherent states subjected to Gaussian processes.

Figure 1

(a) A single mode of a harmonic oscillator is subjected to a Gaussian process, which maps the input observables ${\widehat{X}}_{\mathrm{in}}$ and ${\widehat{P}}_{\mathrm{in}}$ into the output variables ${\widehat{X}}_{\mathrm{out}}$ and ${\widehat{P}}_{\mathrm{out}}$, of the same or a different quantum system. The wavy arrows represent, e.g., absorption losses or addition of thermal noise associated with the possible coupling to environment degrees of freedom. (b) Schematic view of the transformation of a coherent input state with $\mathrm{var}\left({\widehat{X}}_{\mathrm{in}}\right)=\mathrm{var}\left({\widehat{P}}_{\mathrm{in}}\right)=\frac{1}{2}$. The solid arrows indicate the mean values, and the circle and the ellipse show the standard deviation of the continuous quadrature variables in the $xp$ plane. For the output state, $\theta$ denotes the angle between the $x$ axis and the major axis of the uncertainty ellipse with ${\sigma }_1\ge {\sigma }_2$.

Figure 2

A contour plot of the qubit fidelity for symmetric gain and variances according to Eq. (22) as a function of the gain imperfection, $1-g$, and the excess variance relative to the vacuum noise limit, $2{\sigma }^2-1$. A few values of $F_q$ are marked on the graph with the dashed curve enclosing the nonclassical limit $F_q>\frac{2}{3}$.

Figure 3

(a) The decay of qubit fidelity when the harmonic oscillator hosting the qubit is coupled to a heat bath by a decay rate $\gamma$. Each curve corresponds to a specific bath temperature with $\overline{N}$ denoting the mean excitation level of the oscillator in equilibrium. From top to bottom: $\overline{N}=0$ (red), $\overline{N}=0.3$ (green), $\overline{N}=1$ (blue), $\overline{N}=3$ (cyan), and $\overline{N}=10$ (magenta). The horizontal dashed line denotes the nonclassical limit $F_q>\frac{2}{3}$ and the vertical dashed lines mark the time $T_{F_q=2/3}$ at which this limit is reached. This characteristic time is also shown in (b) on the vertical axis (in units of ${\gamma }^{-1}$) as a function of the equilibrium excitation level $\overline{N}$.

Figure 4

The qubit fidelity from Eq. (25) as a function of $\epsilon$, which is conveniently used to parametrize the asymmetry in gain and variance by $g_x=g_0\epsilon$, $g_p=g_0/\epsilon$, ${\sigma }_x^2={\sigma }_0^2{\epsilon }^2$, and ${\sigma }_p^2={\sigma }_0^2/{\epsilon }^2$. From top to bottom: $g_0=1$ and ${\sigma }_0^2=\frac{1}{2}$ (black), $g_0=1$ and ${\sigma }_0^2=\frac{1.05}{2}$ (red), $g_0=0.9$ and ${\sigma }_0^2=\frac{1}{2}$ (green), and $g_0=0.9$ and ${\sigma }_0^2=\frac{1.05}{2}$ (blue).