Harvesting stabilizer entropy and non-locality from a quantum field

Abstract

The harvesting of quantum resources from the vacuum state of a quantum field is a central topic in relativistic quantum information. While several proposals for the harvesting of entanglement from the quantum vacuum exist, less attention has been paid to other quantum resources, such as non-stabilizerness, commonly dubbed {\em magic} and quantified by the Stabilizer Rényi Entropy (SRE). In this work, we show how to harvest SRE from the vacuum state of a massless field using accelerated Unruh-DeWitt detectors in Minkowski spacetime. In particular, one can harvest a particular non-local form of SRE that cannot be erased by local unitary operations. This non-local SRE is a fundamental quantity to study the interplay between entanglement and non-stabilizer resources. We conclude our work with an analysis of the CHSH inequalities: when restricting to stabilizer measurements, i.e. Pauli measurements, one cannot extract a violation from the quantum field.

Figures (7)

• Figure 1: Trajectories of accelerated UDW detectors: (a) detectors with parallel acceleration $a_{\parallel}$; (b) detectors with antiparallel acceleration $a_{\nparallel}$; (c) detectors with perpendicular acceleration $a_{\perp}$.
• Figure 2: Plot of the SRE as a function of $\Omega\sigma$ and of the concurrence for different values of $\sigma/L$. The initial state of the detectors is $\rho_{AB}=\dyad{00}$. For small values of the product $\Omega\sigma$ one can see that the final state of the detectors has significant amount of non-stabilizerness. While this is in contrast with the vacuum state of the field being a stabilizer state (i.e. SRE-free state), the presence of SRE is explained by noticing that the interaction between the detectors and the field is non-Clifford, such that some non-stabilizerness is introduced in the system for small values of $\sigma\Omega$. On the other hand, for $\Omega\sigma\gtrsim1$, one can get rid of this spurious contribution and genuinely harvest SRE from the field.
• Figure 3: Variations of SRE $\Delta\tilde{M}_2(\rho_{AB})$ (solid line) and concurrence $\Delta\mathcal{C}(\rho_{AB})$ (dashed line), as a function of the (parallel) acceleration $a_\parallel$, for various choices of parameters, with the detectors initialized in the state $\rho_{AB}=\dyad{00}$. In Fig. \ref{['fig:parameters_omega']} we study the two resources for different values of $\Omega$, setting $\sigma=1$ and $L=1$. Notice that for $\Omega=1$, for $a=0$ one has a non zero SRE, in agreement with the observation made in the inertial setting. One can also notice that the harvested SRE becomes larger for lower values of $\Omega$. In Fig. \ref{['fig:parameters_L']} we plot the resources for different values of the initial separation $L$ between the detectors, setting $\Omega=2$ and $\sigma=1$. Also in this case, one can observe the same behavior obtained in the inertial setting, as the SRE does not depend on $L$, while the concurrence does.
• Figure 4: Variations of SRE $\Delta\tilde{M}_2(\rho_{AB})$ (solid line), concurrence $\Delta\mathcal{C}(\rho_{AB})$ (dashed line), non-local SRE $\Delta M^{NL}_2 (\rho_{AB})$ (dashed-dotted line) and value of Bell operator $\Delta B_0$ (dotted line) as a function of the (parallel) acceleration $a_\parallel$, together with the corresponding purity of the state (see upper plots). Panels (a) Initial state $\ket{00}$, (b) initial state $\ket{\Phi^+}$, (c) initial state $\ket{0}_A\otimes\ket{T}_B$. In panel (a) and (b) the curves for $\Delta\tilde{M}_2(\rho_{AB})$ and $\Delta M^{NL}_2 (\rho_{AB})$ overlap perfectly as all the harvested SRE is non-local. Notice that the state of the detectors, for the initial state $\ket{\psi_0}_{AB}=\ket{00}$, is always pure with very good approximation.
• Figure 5: Plot of the variation of the non-stabilizerness, concurrence, expectation value of the Bell operator and non-local non-stabilizerness as a function of the acceleration when the detectors are initialized in the maximally entangled state $\rho_{AB}(0)=\dyad{\Phi^+}$ for different values of the detectors frequency. In the plot we have set $\sigma=1$ and $L=1$. Different line shapes represent the various quantities, while different colours represent different values of $\Omega$, as in the legend. From a qualitative point of view, one can once again observe that all the harvested non-stabilizerness is non-local. This feature, as explained in the main text, is the one responsible for the non violation of the CHSH inequality. The only difference with respect to the scenario where the detectors are initialized in the state $\dyad{00}$ is that the SRE also depends on the initial separation $L$. However, there are no choice of parameters for which the CHSH inequality is violated.