Deducing entanglement from signal fluctuations

This is a Perspective on "Single-shot energetic-based estimator for entanglement in a half-parity measurement setup" by Cyril Elouard, Alexia Auffèves, and Géraldine Haack, published in Quantum 3, 166 (2019).

By Alain Sarlette (Quantic lab, INRIA Paris).

The paper by Elouard et al, 2019 [1], considers the heralded entanglement between two qubits, from a continuous-time measurement that creates this entanglement. While the process is well-known, the originality of this work lies in the way the presence of entanglement is deduced.

To update the two qubits’ state (estimate) while they are measured, one usually runs a so-called quantum filter, essentially applying Bayes’ rule according to the principles of quantum measurement. In this filter, the measurement back-action drives the system depending on (continuous-time) measurement results, in a probabilistic way. A remarkable situation occurs when the continuous-time measurement performs the quantum non-demolition (QND) measurement of an observable, i.e. when it is the continuous-time unravelling of the usual projective quantum measurement. In this case, the value of the quantum filter at any given time t, only depends on its initial state and on the integral of the measurement signal up to t. Thus, in the four-dimensional complex Hilbert space associated to two qubits, the single real value corresponding to the integral of the measurement record, in association with a known initial state as is assumed here, fully determines the values of all other variables; in particular, it determines how much the system is entangled. This mathematical fact can be observed on Eq.(5) for instance. This direct relation between integrated measurement record and all other variables remains valid when the QND measurement efficiency is below 100%.
From a mathematical viewpoint, it is thus not too surprising that entanglement — here measured by concurrence — can be deduced from a single variable directly related to the measurement record — here the so-called quantum heat $Q(t)$, via the perfect mathematical equivalence between $(t,J(t))$ and $(t,Q(t))$ established in Eq.(11). Concretely, the concurrence could be deduced from $(t,Q(t))$, i.e. (23) could be rewritten — possibly quite nonlinearly — fully in terms of $(t,Q(t))$. Such nonlinear relation however could be physically hard to interpret.

A remarkable observation by Elouard et al, 2019 is that quite simple bounds on concurrence generation can be obtained in terms of the signal $Q(t)$ and of its $\textit{fluctuations}$ $\sigma(t)$. One can even push this non-equilibrium thermodynamics approach further and consider to deduce concurrence from the fluctuations $\sigma(t)$ only, discarding the actual signal values of $Q(t)$. Referring to Figure 3, we could ask: at a given time (horizontal cut), if we only know the color (intensity of fluctuations), can we deduce whether we are more likely to converge towards the plateau at $C=0$, or towards the one at $C=1$ ? Figure 1.b seems to suggest that this is indeed possible: fluctuations appear to persist much more when the system converges towards an entangled state ($C=1$).

Now let us take a look at the maths again. Equation (18) expresses the expected fluctuations $\sigma(t)$ as a function of time and of integrated measurement record. Of course we can use Equation (5) to deduce the population $p_0$ on the 0-eigenvalue subspace of the measurement operator, i.e. the one corresponding to entanglement $C=1$, still as a function of time and of integrated measurement record. Combining the two, we get an expression for the measurement fluctuations $\tilde{\sigma}(t)$, as a function of population $p_0$ on the entangled subspace $C=1$ and of time:
$$ \tilde{\sigma}(t) \propto p_0(t) (1 – (1-e^{-4t})p_0(t)) \; .$$
The latter expression shows that, at least for $t < ln(2) / 4$, higher population on the entangled subspace always implies more fluctuations. After this time, the system should begin to settle and the equation indeed says that highest fluctuations are now expected if the system did not ($p_0(t)\simeq 1/2$); lower values of $\tilde{\sigma}(t)$ are predicted for $p_0(t) \simeq 1$ (i.e. dominant population on $C=1$) and for $p_0(t) \simeq 0$ (dominant population on $C=0$), but the entangled situation is still predicted to yield more fluctuations than the other one. Altogether, this appears to confirm that carefully observing the transient profile of a thermodynamic quantity— here just fluctuation intensities in the measurement signal or in the quantum heat — does carry meaningful information on its own, about a variable of interest for quantum information, like entanglement concurrence. Of course this is a first step, and how far this can be extended still remains to be seen. For one thing, the detection of entanglement is conclusive only conditioned on having started from a particular initial state and with particular, known dynamics. This makes this continuous-time information not yet as indicative as a full independent tomography. The authors also state that it remains to be seen in which experimental setting we would have direct access to fluctuations but not to the values of the fully informative $J(t)$, thus making the focus on fluctuations practically justified. Also the system considered is particularly simple, so this type of information might be washed out or diluted in larger systems. However, the idea of checking how much can be deduced in terms of quantum information, purely from the profile of non-equilibrium fluctuation intensities, seems definitely appealing to pursue.

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[1] Cyril Elouard, Alexia Auffèves, and Géraldine Haack. Single-shot energetic-based estimator for entanglement in a half-parity measurement setup. Quantum 3, 166 (2019). 10.22331/​q-2019-07-15-166.

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