Projective measurements can probe non-classical work extraction and time-correlations

We demonstrate an experimental technique to characterize genuinely nonclassical multi-time correlations using projective measurements with no ancillae. We implement the scheme in a nitrogen-vacancy center in diamond undergoing a unitary quantum work protocol. We reconstruct quantum-mechanical time correlations encoded in the Margenau-Hills quasiprobabilities. We observe work extraction peaks five times those of sequential projective energy measurement schemes and in violation of newly-derived stochastic bounds. We interpret the phenomenon via anomalous energy exchanges due to the underlying negativity of the quasiprobability distribution.

There is no unique way of defining a joint probability for the multi-time statistics of quantum mechanical quantities since even a single quantum observable does not always commute with itself at different times.Nevertheless, across the quantum sciences, multi-time fluctuations of quantum-mechanical quantities, especially energy, play a crucial role.Correlation functions between events at different times resemble joint probability distributions for the eigenvalues of the observables, but they are in general neither real nor positive.In fact, they are represented by quasiprobabilities, akin to the well-known Wigner function in quantum optics [1], but associated with a process rather than a state.The standard definition of a temporal correlation function between two events described by projectors Π i (0) and Π f (t), coincides with a Kirkwood-Dirac quasiprobability (KDQ) [2][3][4], and the same goes for multi-time extensions.
As we survey in [5], the centrality of the KDQ and its real part, the Margenau-Hill quasiprobability (MHQ), has only recently come to be fully appreciated.These quasiprobabilities underpin analysis from perturbation theory [5] to information scrambling [6,7].Weak values [5,8,9] are conditional KDQ [5,7] and generalized anomalous weak values are in one-to-one correspondence with non-classical (negative or complex) KDQ quasiprobabilities [10,11].Negative values of the MHQ are linked to quantum metrological advantages in both local and postselected setups [12][13][14] and to power output advantages in quantum thermodynamics [14].
In such a context, the aim of our work is two-fold: First, pave the way to the experimental study of the role of non-commutativity in temporal correlations via the MHQ.We do so by providing the first experimental demonstration of a weak two-point measurement (wTPM) scheme [5,28], a protocol that -in contrast to weak measurement schemes -requires neither ancillae nor fine-tuned system-ancilla couplings.In fact, we reconstruct the back-reaction-free limit encapsulated by quasiprobabilities not by a weak measurement, but by linearly combining different projective measurement schemes in such a way that different back-reactions cancel.Conceptually our idea can be seen as a twist on probabilistic error cancellation techniques in quantum computing, where several noisy circuits are sampled from to reconstruct an ideal error-free limit [29].This technique can be deployed in quantum thermodynamics beyond the scope of our study.
Second, we lay down the theoretical and experimental ground for the study of non-classical energetic processes via the MHQ.We measure work extraction peaks in a driven three-level system up to five times those of the TPM scheme [30] and in violation of a newly introduced stochastic bound.We explain this phenomenon by interpreting negative probabilities as non-classical pathways of a stochastic process.Remarkably, the data required to witness genuinely non-classical effects via violations of the stochastic work bound can be obtained from measurements routinely performed in TPM experiments.
We put this forward as a theoretical and experimental framework to interpret the recently observed energetics of superconducting qubits experiments [31], and more generally in quantum thermodynamics experiments showcasing genuinely non-classical features.
Non-classicality.-The KDQ encodes temporal correlations between quantum observables and so does its real part, the MHQ.Here we focus on the latter.Given two observables A(0 in terms of their eigenvalues a i , b f and their eigenprojectors Π i (0), Ξ f (t), and a quantum channel E describing the system dynamics in the time interval [0, t], the MHQ reads as where E † denotes the adjoint of E and ρ is the quantum state at t = 0.The MHQ is a quasiprobability, as it satisfies if q if = 1 and q if ∈ R. The marginals over i (f ) reproduce the quantum outcome statistics of a measurement of B(t) carried out at time t (A(0), carried out at time 0).The MHQ, being a two-time correlator [32], encodes information about the process, including its linear response and quantum currents [5,33].
In our experiments A(0) and B(t) are the Hamiltonian at times 0 and t and the channel is a unitary work protocol U , i.e., E(•) ≡ U(•) ≡ U (•)U † .The 'unperturbed' work w t := Tr [H(t)ρ(t)] − Tr [H(0)ρ(0)] can be obtained as the average with . Clearly, the same framework applies to the study of temporal correlations beyond work processes.
The quantum process has a stochastic (classical) interpretation when q if ≥ 0 for all i, f .In fact, for fixed i, f , commutativity implies positivity: if The converse does not hold, i.e., negativity is a stronger property than non-commutativity [4].In cases (a-b), q if reduces to the TPM probability of observing outcomes i followed by f in a sequential projective measurement of the observables A(0) and B(t), with the intermediate evolution E.
Negative values of q if indicate non-classicality in the temporal correlations.These are associated with proofs of contextuality [11,34] and correspond to elementary non-classical processes.For work protocols, crucially an anomalous excitation process E f > E i (classically associated to work done, not extracted!)occurring with 'negative probability' q if < 0 is equivalent to a classical de-excitation process occurring with probability |q if |, and hence contributes to the extracted work.
The non-classicality of the MHQ is defined via the negativity functional [4,5,35] [36] For work extraction from pure states, we prove in the supplemental material [37] an upper bound on the extracted work W ext that holds whenever a stochastic interpretation is possible, i.e., ℵ = 0: where are the joint probabilities from the TPM scheme, and is the END-time energy measurement probability [38].Standard TPM experiments satisfy this inequality.Hence, its violations indicate work extraction peaks above TPM that can only occur because negativity is at play.We will look for these peaks in the experimental data by optimizing the negativity of anomalous excitation processes within the experimentally achievable parameters.
Measurement scheme.-Wepresent an experimental implementation of the wTPM measurement scheme [28], a non-selective (NS) 2-outcome projective measurement that checks whether the initial energy is E i or NOT E i , followed by unitary evolution and a projective measurement of the final Hamiltonian.The wTPM joint probabilities read where The state ρ NS,i can be obtained by performing non-selective projective measurements with projectors Π i and I − Π i or, equivalently, by the preparation of the states ρ i and ρ i with the corresponding probabilities (as in our experiments).This joint probability is related to the MHQ [5,28] by i.e., the MHQ is given by three distinct contributions [39] that stem from applying in three separate sets of runs the wTPM protocol, the TPM and END schemes [40].
Experimental setting.-Weuse as a quantum system the electronic spin of an NV center in bulk diamond at room temperature.NV centers are defects in a diamond lattice with an orbital ground state that is a spin triplet S = 1.The degeneracy in the spin quantum number m S is lifted due to the zero field splitting and to the presence of an external bias field aligned with the spin quantization axis.The NV spin qutrit can be optically initialized into m S = 0 (|0 ) by illuminating the defect with a green laser [41].Moreover, the spin state can be read out by detecting the photoluminescence (PL) after a laser illumination, as the PL intensity depends on the spin projection m S [42,43].In addition, on-resonance microwave fields are used to coherently drive the spin, with coherence times up to milliseconds (at room temperature) [44,45].By virtue of these properties, NV centers are broadly used for quantum technologies, such as quantum sensing [46][47][48], quantum information [49,50] and, recently, for quantum thermodynamics [51][52][53].
A time-varying Hamiltonian is implemented by coherently driving the NV spin with a microwave field with phase changing in time.More specifically, the spin qutrit is driven by a bi-chromatic microwave field on-resonance with both the transitions |0 ↔ |−1 and |0 ↔ |+1 .In the microwave rotating frame, the Hamiltonian of the system (after the rotating wave approximation) is where = 1, S α are the spin operators defined in terms of the Gell-Mann matrices, and the Hamiltonian amplitude Ω 1 (Ω 2 ) and rate of phase increase φ 1 (φ 2 ) correspond respectively to the Rabi frequency and the phase of the driving field for the transition |0 ↔ |+1 (|0 ↔ |−1 ), as detailed in [37].See also [37] for more details on the energy level structure and the driving fields.To simplify the measurements in the time-varying energy eigenbasis, we remove the time dependency on one of the Hamiltonian eigenstates by setting Ω 1 = Ω 2 = Ω and The eigenstates of the Hamiltonian (8) are: 0), H(t)] = 0 for tφ/π ∈ Z.Hence, the system energy changes during the unitary evolution under H(t), meaning that work is exchanged between the NV spin and the microwave field.
Work quasiprobabilities.-TheMHQ plays a role in thermodynamics, where the characterization of work, heat, and internal energy fluctuations in quantum processes calls for novel tools to account for exquisitely quantum effects.The seminal TPM scheme [30,54,55] is unable to capture non-commutativity [56].This motivated the use of the MHQ to characterize non-classical work fluctuations [57] and anomalous heat exchanges due to quantum correlations [58].However, experimental realizations remained limited, due to the challenge of adapting TPM experiments to access work quasiprobabil-  8) with Ω = (2π)2.219MHz and φ = 1.09Ω.Note that the data for f=0 are always constant.This is a consequence of setting φ1 = φ2 = φ.In such case, the interaction between the NV center and the two microwave fields corresponds to a Stimulated Raman Adiabatic Passage (STIRAP) in the two-photon resonance condition [59] (see also [37]).The data outside the interval [0, 1] is originated by photon shot noise during the PL read-out, hence affecting the PL normalization (see text).
ities.Here, we experimentally reconstruct the MH work quasiprobability on the NV center using only projective energy measurements and pure state preparations.This paves the way for a range of other TPM experiments to adopt the same strategy.
We take the initial state to be pure ρ = |ξ ξ| [60].One can therefore reconstruct p END f , p TPM if , and p wTPM if by measuring a set of conditional probabilities of the form where |ψ depends on the scheme that we want to implement.We directly measure p END f = p(f |ξ) for the END scheme.Instead, for the TPM and wTPM schemes we measure the conditional probabilities p(f |i) and p(f |i) by initializing the quantum states ρ i and ρ i , respectively, and we combine them as p TPM if = p i p(f |i) and The results for the measurements of p END f , p(f |i) and p(f |i) are shown in Fig. 1.The protocol to measure these conditional probabilities is based on our previous works [51][52][53]; however, the full description of the protocol is also included in [37].The main idea is the following: First, the qutrit is prepared into the pure state |ψ = |ξ ,  4)] as a function of time (solid green line represents the simulated data).For almost all the interaction time t the negativity is larger than zero, hence overcoming the classical limit (dotted line) and entering into the non-classical region (blue area).This region is bounded from above by √ d − 1 (dashed line), where d = 3 is the dimension of the system's Hilbert space [5].
|i or |i , depending on the measurement scheme (END, TPM, wTPM).Then, it evolves under the time-varying Hamiltonian H(t) in the time interval [0, t].At the end of the protocol, we optically read out the probability that the energy of the system is E f (t), i.e., p(f |ψ).As mentioned before, the PL intensity (averaged over ∼ 10 6 repetitions of the experiment) encodes information about the spin state.Hence, by normalizing the average PL with respect to reference PL levels we obtain p(f |ψ) [37].
Note that the optical read-out is destructive, hence, for each given initial state, we perform independent experiments for each value of t and for each of the three Hamiltonian projectors Ξ f (t).
We can now obtain the MHQ work distributions at each t by combining the results of all the previous measurements as dictated by Eq. ( 7).The results are shown in Fig. 2a-c.From Eq. ( 4) the non-classicality is quantified by the negativity of the measured work distribution.Its experimental values are plotted in Fig. 2d.
Work extraction.-Letus focus now on the thermodynamics of the driven qutrit.In Fig. 3 we compare the experimental data for the average extracted work in the TPM scheme, − w TPM t , with the unperturbed extracted work W ext = − w t .In our experiment, the TPM (projective measurements) reduces the efficiency of the work extraction process.Comparing Fig. 3 and Fig. 2 we observe that peaks in the average work coincide with peaks in negativity (non-classical process).What is more, Fig. 3 shows that the stochastic bound of Eq. ( 5) for work extraction is violated, showing that these peaks are high enough that they can only occur when q if turns negative.The bound in Eq. ( 5) is a powerful tool for witnessing non-classicality, as it relies only on the combination of the TPM statistics and END-time energy measurements.
A physical interpretation of the non-classical work can  5) and [37]].
be given by noting that, in our experiments, negativity is concentrated in the anomalous excitation processesnegativity of the MHQ distribution is associated with the largest exciting transition, w −+ = 2Ω [Fig.2a-c].Classically this transition contributes to work done but quantumly it enhances work extraction.This negativity is destroyed in the TPM scheme, resulting in decreased work extraction.Theoretical considerations often focus on the total negativity ℵ, but from a thermodynamic point of view, our experiments indicate that it is the distribution of negativity among the outcomes that plays a crucial role.In fact, numerical simulations [37] show that our experimental conditions are both close to minimizing q −+ as well as maximizing the extracted work.
Conclusions.-Wepresented the first experimental implementation of a wTPM scheme, reconstructing the Margenau-Hills work quasi-distribution for a spin qutrit.Our platform of choice has been an NV center in diamond driven by a microwave field acting as a work reservoir.
Our experiment demonstrates how to reconstruct genuinely non-classical effects in a work process using projective measurements only, without the need for ancillae and fine-tuned couplings, with similar resource requirements as the TPM.
Furthermore, we found that peaks in the work extraction are associated with peaks in non-classicality in the form of negativity.In fact, the height of the observed work peaks was such that they can only be explained by the presence of negativity.We proved this by introducing a new stochastic bound for work processes [Eq.(5)] that allows inferring negativity without even implementing the wTPM measurement scheme.Thus, we witness non-classicality with minimal adjustments to TPM experiments already in place.Since the TPM scheme has been implemented in a variety of platforms [17,18,[51][52][53], we expect our demonstration will herald further experimental studies on such set-ups.
We gave a general interpretation of the nonclassical work phenomenon as an anomalous energy process, where negativity flips the conventional directionality of selected stochastic transitions, transforming contributions to work done into contributions to work extracted -leading to the observed peaks.We highlighted the thermodynamic relevance of the 'negativity distribution', beyond usual considerations of total negativity.The enhanced work extraction resembles recently reported anomalous energy exchanges in superconducting qubit systems [31,61].We believe our work will provide a suitable interpretative and experimental framework for that setup, but we leave this study for future work.
Finally, our experiment can provide access to the multi-time correlations of a driven unitary dynamics.This suggests an alternative path to witnessing relevant process properties, such as scrambling [35], by combining independent experiments that involve projective measurements only and no ancillae.This is particularly relevant in view of the results in [62] linking, in the case of diagonal initial states, a TPM characteristic function to the out-of-order-correlations (OTOC) witnessing information scrambling.These results can be generalized beyond the diagonal case when access to the MHQ is granted.The potential use of wTPM schemes to extract information from the dynamics of many-body systems is another future line of research suggested by our work.
Acknowledgments.We gratefully thank F. Poggiali for critical reading of the manuscript.S.H.G. acknowledges the financial support from CNR-FOE-LENS-2020.S.G. acknowledges The Blanceflor Foundation for financial support through the project "The theRmodynamics be-hInd thE meaSuremenT postulate of quantum mEchanics (TRIESTE)".As described in the main text, the three level system realized for our experiments is based on the spin triplet S = 1 of the orbital ground state of an NV center, with Hamiltonian where ∆ = 2.87GHz is the zero-field-splitting, γ e denotes the electron gyromagnetic ratio, and B is a bias magnetic field aligned with the NV quantization axis z (determined by the orientation of the NV defect in the diamond lattice).
The spin triplet is driven by two continuous on-resonance microwave (MW) fields addressing the transitions |0 ↔ |−1 and |0 ↔ |+1 .Hence, overall the spin dynamics can be described by the Hamiltonian where Ω 1 and Ω 2 are the Rabi frequencies for the transitions |0 ↔ |+1 and |0 ↔ |−1 , respectively.In addition, ω ±1 = ∆ ± γ e B denote the frequencies, and ϕ 1 (t) and ϕ 2 (t) are the time-varying phases of the MW fields.The energy level structure of the qutrit and its interaction with the MW fields is depicted in Fig. S1(a).
In the microwave rotating frame (defined by the unitary transformation , with j 2 = −1) and after applying the rotating wave approximation, the Hamiltonian H(t) reads as The Hamiltonian (S3) is defined in terms of the spin operators , where λ i are the Gell-Mann matrices: In our experiments, we select the time-varying phases so that they change linearly in time, i.e., ϕ 1 (t) = φ 1 t and ϕ 2 (t) = −φ 2 t.Therefore, Eq. ( 8) in the main text is recovered from Eq. (S3).The energy level structure in the MW rotating frame is sketched in Fig. S1(b).Finally, let us note that we can describe the system dynamics in a different rotating frame for which the Hamiltonian is time independent.In fact, instead of the MW rotating frame, we can express Eq.(S2) by transforming it in the rotating frame determined by the unitary transformation V = exp [jt ((ω +1 + φ 1 ) |+1 +1| + (ω −1 + φ 2 ) |−1 −1|)].In this new frame and after the rotating wave approximation, the time-independent Hamiltonian is:

FIG. 1 .
FIG.1.Experimental results from the END scheme [panel (a)], and from the measurements of the conditional probabilities obtained by initializing the qutrit in the states ρi [panels (b)] and ρ i [panels (c)].The solid lines denote the simulations using Eq.(8) with Ω = (2π)2.219MHz and φ = 1.09Ω.Note that the data for f=0 are always constant.This is a consequence of setting φ1 = φ2 = φ.In such case, the interaction between the NV center and the two microwave fields corresponds to a Stimulated Raman Adiabatic Passage (STIRAP) in the two-photon resonance condition[59] (see also[37]).The data outside the interval [0, 1] is originated by photon shot noise during the PL read-out, hence affecting the PL normalization (see text).

FIG. 2 .
FIG. 2. (a-c) MeasuredMHQs q if as a function of time.The blue circles correspond to f = −, the orange squares to f = 0, and red triangles to f = +.The solid line represents the simulated data, while the dashed black line corresponds to f |q if |.Note that only q−+ [panel (c)] exhibits negative values.(d) Experimental measurements (green circles) of the negativity [Eq.(4)] as a function of time (solid green line represents the simulated data).For almost all the interaction time t the negativity is larger than zero, hence overcoming the classical limit (dotted line) and entering into the non-classical region (blue area).This region is bounded from above by √ d − 1 (dashed line), where d = 3 is the dimension of the system's Hilbert space[5].

FIG. 3 .
FIG. 3.Average unperturbed extracted work Wext = − w t = i piEi(0) − f p END f E f (t), and average extracted work in the TPM scheme, − w TPM t

FIG. S1 .
FIG. S1.Energy diagram for the NV center spin triplet.(a) In the laboratory frame according to the Hamiltonian H(t) in Eq. (S2).(b) In the MW rotating frame, as described by the Hamiltonian H(t) in Eq (S3) (or Eq. (8) in the main text).(c) In a rotating frame where the Hamiltonian is effectively time-independent, as in Eq. (S5).It is worth noting that in frame (c), the time-varying phase of each MW field can be reinterpreted as detuning.

FIG. S4 .
FIG. S4. Results of the numerical simulations.In both panels (a)-(b), each empty blue circle represents the result for a set of random parameters of the system Hamiltonian and a random initial pure state.For each empty blue circle there are two orange crosses corresponding to the cases φ1 = φ2 = φ R 1 and φ1 = φ2 = φ R 2 .Instead, the red circle with the error bars represents the experimentally measured values, as detailed in the main text.Finally, in panel (b), the horizontal line denotes the upper bound of the non-classicality measure ℵ.Such a bound is equal to √ d − 1, where d = 3 is the dimension of the Hilbert space of the system [5].
A.B. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number BR 5221/4-1.A.L. acknowledges support from the Israel Science Founda-tion (Grant No. 1364/21).The work was also supported by the European Commission under GA n. 101070546-MUQUABIS.