Verifying bound entanglement of dephased Werner states

The verification of quantum entanglement under the influence of realistic noise and decoherence is crucial for the development of quantum technologies. Unfortunately, a full entanglement characterization is generally not possible with most entanglement criteria such as entanglement witnesses or the partial transposition criterion. In particular, so-called bound entanglement cannot be certified via the partial transposition criterion. Here we present the full entanglement verification of dephased qubit and qutrit Werner states via entanglement quasiprobabilities. Remarkably, we are able to reveal bound entanglement for noisy mixed states in the qutrit case. This example demonstrates the strength of the entanglement quasiprobabilities for verifying the full entanglement of quantum states suffering from noise.

Figure 1

The entanglement quasiprobability distribution for the qubit Werner state for $\alpha =0.8$, without dephasing. Each bar represents the value of the entanglement quasiprobability distribution $P_{\mathrm{Ent}}(a,b)$ for the vector $|a,b\rangle$. The negativities indicate the entanglement of the given state.

Figure 2

Verifying entanglement of qutrit Werner states undergoing Gaussian dephasing. Each curve shows the minimal value ${\alpha }_{\min }$ in dependence of the dephasing strength $\delta$ so that the corresponding criterion certifies entanglement for $\alpha >{\alpha }_{\min }$. The red solid line refers to the entanglement quasiprobabilities and the blue dashed line refers to partial transposition. The gray colored area between the solid and dashed line marks the entangled dephased qutrit Werner states with positive partial transposition.