Quantum entanglement in phase space

While commonly used entanglement criteria for continuous variable systems are based on quadrature measurements, here we study entanglement detection from measurements of the Wigner function. These are routinely performed in platforms such as trapped ions and circuit QED, where homodyne measurements are difficult to be implemented. We provide complementary criteria which we show to be tight for a variety of experimentally relevant Gaussian and non-Gaussian states. Our results show novel approaches to detect entanglement in continuous variable systems and shed light on interesting connections between known criteria and the Wigner function.

💡 Research Summary

The paper addresses the problem of detecting entanglement in continuous‑variable (CV) quantum systems without relying on quadrature (homodyne) measurements, which are often impractical in many experimental platforms such as trapped ions, circuit QED, and quantum acoustics. Instead, the authors propose to use direct measurements of the joint Wigner function, which can be obtained via displaced‑parity operations at individual phase‑space points.

The central idea is to reduce the four‑dimensional joint Wigner function (W_{AB}(x_A,p_A,x_B,p_B)) to a two‑dimensional “slice” by imposing a linear relation between the coordinates of mode B and those of mode A: ((x_B,p_B) = (x’(x_A,p_A),p’(x_A,p_A))). This relation is described by a general linear transformation with matrix elements (a,b,c,d) and possible offsets, constrained to have determinant (\Delta = \pm 1). The case (\Delta = -1) corresponds to a mirror reflection (partial transposition in phase space), while (\Delta = +1) corresponds to a symplectic (canonical) transformation.

Three entanglement criteria are derived from basic properties of the Wigner function (normalization, purity bounds, and positivity of overlaps).

1. Criterion (I)
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