Practical Framework for Conditional Non-Gaussian Quantum State Preparation

We develop a general formalism, based on the Wigner function representation of continuous-variable quantum states, to describe the action of an arbitrary conditional operation on a multimode Gaussian state. We apply this formalism to several examples, thus showing its potential as an elegant analytical tool for simulating quantum optics experiments. Furthermore, we also use it to prove that Einstein-Podolsky-Rosen steering is a necessary requirement to remotely prepare a Wigner-negative state.

Popular Summary

Phase space is a useful tool to represent physical systems, such as the position and momentum of a particle, or the electric field of propagating light. It is common to describe quantum states of such systems by their Wigner function: the quantum generalization of the joint probability distribution. Of particular interest are the states for which the Wigner function reaches negative values. This “Wigner negativity” is a hallmark of quantum phenomena, and is known to be necessary for quantum computing. Here, we theoretically study a method to remotely generate Wigner negativity. First, we divide our system into two subsystems. Then, we show that upon appropriate measurements on one subsystem one can induce Wigner negativity in the other subsystem, provided there are strong quantum correlations between the two. We precisely quantify which level of correlations and measurements are necessary and sufficient for success.

More specifically, we start from a Gaussian state and provide a general expression for the Wigner function of one part of this system (kept with Alice), when it is conditioned upon a measurement performed on another part of the system (kept with Bob). We prove that Alice can only acquire Wigner negativity when they initially share a state with the following property: upon Alice’s measurements of position and momentum, Bob obtains conditional measurement statistics that violate Heisenberg’s inequality. Alice can then perform Einstein-Podolsky-Rosen steering on Bob’s subsystem.

Our work thus fundamentally connects Wigner negativity and Einstein-Podolsky-Rosen steering, having crucial consequences for quantum state engineering.

Figure 1

Sketch of the conditional state-preparation scenario: a multimode quantum state with density matrix $\hat{\rho }$ is separated over two subsets of modes, $\mathbf{f}$ and $\mathbf{g}$. A measurement is performed on the modes in $\mathbf{g}$, yielding a result associated with a POVM element $\hat{A}$. Conditioning on this measurement outcome “projects” the subset of mode $\mathbf{f}$ into a state ${\hat{\rho }}_{\mathbf{f}\mid \hat{A}}$. The directional EPR steering, discussed in Sec. 4, is highlighted.

Figure 2

Photon heralding with a particular Gaussian input state, generated by mixing two equal squeezed thermal states (b) on a balanced beam splitter (a). On one of the outputs of the beam splitter, a projective measurement is performed on the Fock state $∣n⟩$, which heralds a non-Gaussian state in the other mode. The Wigner functions of this non-Gaussian state are shown for the case where $n=5$, with varying degrees of EPR steering $\mu$, controlled by varying the thermal noise $\delta$ for a fixed squeezing $s=5\mathrm{dB}$. The Wigner negativity, measured by $\mathcal{N}$ given in Eq. (32), is shown in (c) for varying degrees of EPR steering and a varying number of measured photons $n$.

Figure 3

Conditional state Wigner function $W_{f\mid \hat{A}}({\overrightarrow{x}}_f)$, obtained by subtracting a photon in two of the three modes in a three-mode entangled state. This entangled state is generated by mixing three squeezed vacuum states in a sequence of beam splitters with transmittances of $75\mathrm{\%}$ (left) and $25\mathrm{\%}$ (right). Two of the squeezed vacuum states are squeezed by 5 dB in the $x$ quadrature (left, right) and one is squeezed by 5 dB along the $p$ quadrature (middle). The photon subtraction is represented by highly transmitting beam splitters that send a small fraction of light to a photon detector, which effectively implements the operators ${\hat{a}}_{g_1}$ and ${\hat{a}}_{g_2}$ on the modes $g_1$ and $g_2$, respectively.