A quasiprobability distribution for heat fluctuations in the quantum regime

The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system. Here we overcome this difficulty by showing that the heat exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by the Margenau-Hill quasiprobability distribution. Its negativities, being associated to proofs of contextuality, witness non-classicality. We discuss the thermodynamic interpretation of the negative probabilities, and provide inequalities for the heat flows that can only be violated in their presence. We test these results on data collected in a recent experiment studying the heat transfer between two qubits.

The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system. Here we overcome this difficulty by showing that the heat exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by the Margenau-Hill quasiprobability distribution. Its negativities, being associated to proofs of contextuality, witness non-classicality. We discuss the thermodynamic interpretation of the negative probabilities, and provide inequalities for the heat flows that can only be violated in their presence. We test these results on data collected in a recent experiment studying the heat transfer between two qubits.
Consider two systems, C for 'cold' and H 'hot', in thermal states at temperatures T C < T H . If the overall system is isolated and energy is conserved, and there are no initial correlations between C and H, heat will flow on average from the hot to the cold body: as mandated by the second law of thermodynamics. Eq. (1) can be derived for both classical and quantum systems. The fluctuations of Q are governed by Jarzinsky's exchange fluctuation theorem (XFT) [1].
The situation is less straightforward if the initial state is locally thermal but correlated. Initial correlations allow for violations of the bound of Eq. (1) [2,3]. Moreover, the specific scheme used to probe the heat flow becomes crucial, since quantum correlations between C and H can be destroyed in the initial measurement phase. To observe this, consider what is perhaps the leading scheme to measure heat and work in quantum systems; the 'two-projective-measurement' or TPM scheme [4,5]. This scheme consists of projective energy measurements carried out on C and H at the start and at the end of the protocol. When the Hamiltonians H C and H H are nondegenerate, the measurement disturbance transforms the initial state ρ CH into a state classically correlated in the energy basis, i.e. all quantum correlations are destroyed at the very beginning of the protocol. This, in turn, deeply modifies the subsequent heat flows.
In the standard setting [1], CH undergoes a unitary dynamics U that conserves the overall energy. Here and in the rest of this work we denote by ρ CH an arbitrary bipartite quantum state with thermal marginals at temperatures T C and T H , respectively. The largest violation of Eq. (1) ('backflow') can be shown to be proportional * The authors contributed equally to this work; lostaglio@gmail.com to I CH , the initial quantum mutual information between C and H: where ∆β = β C − β H , β X = 1/(kT X ), with k Boltzmann's constant. If the local Hilbert spaces of C and H both have dimension d, I CH ≤ log d for non entangled states, whereas for entangled states I CH can be at most 2 log d. This matches the intuition that quantum systems can exhibit stronger than classical correlations which, if not destroyed in the measurement process, can give stronger than classical backflows [3].
In this work we compare the heat flows measured within the TPM scheme, denoted by Q TPM , with those of Eq. (2). We analyse both the backflow from C to H as well as the direct flow from H to C. We show in generality that, when Q goes beyond certain threshold values, an underlying quasiprobability for heat fluctuations must turn negative; these negativities arise naturally due to noncommutativity, when we extend the notion of heat fluctuations beyond the uncorrelated scenario of the original XFT of Ref. [1] and beyond its extension to classically correlated systems of Ref. [6]. These negative 'probabilities' have a thermodynamic interpretation as contributions to the heat flows of Eq. (2) and, as we will discuss, cannot be explained within the framework of stochastics thermodynamics. As a bonus, we derive an XFT for the Margenau-Hill quasiprobability that incorporates quantum correlations in the initial state. This turns out to be considerably more complicated than the original XFT, while it recovers known results in the relevant classical limits.
A quasiprobability for heat fluctuations. Classically one can measure the heat flows between C and H and their flucuations by measuring H C at the start and at the end of protocol. However, if we do that in the quantum regime we do not measure the fluctuations of the process in Eq. (2). Rather, we measure the fluctuations of a process from which quantum correlations and local coherence in the initial state are removed by the initial projective measurement. This is arguably unsatisfactory since, as mentioned before, strong backflows have been related to the presence of quantum correlations [6].
This impasse can be traced back to the informationdisturbance tradeoff of quantum mechanics; the Heinsenberg picture shows that the construction of a probability distribution for the heat fluctuations of the process in Eq. (2) would require accessing two generally incompatible observables. These are the energy at the initial and final times, i.e. H C ⊗ H H and U † (H C ⊗ H H )U . Since quantum mechanics does not allow for the construction of such joint probabilities, quasiprobabilities are a natural tool [7]. Another way to see the obstacle at hand is the no-go result of Ref. [8]. Adapted to our context, it implies that we cannot (1) reproduce the average value Q of Eq. (2) and (2) reproduce the TPM predictions for input states diagonal in the energy basis, if fluctuations are associated with a probability distribution convex linear in the input. If we wish to witness strong heat backflows encoded in Q, these desiderata essentially force us to consider quasiprobability distributions for the heat fluctuations.
While relatively uncommon in the thermodynamic setting (but see, e.g., [9][10][11][12][13]), the use of quasiprobabilities to probe quantum effects is widespread in other fields of the quantum sciences. Most aptly for our considerations, they find application in the study of quantum transport [14][15][16]. In quantum optics, the use of the Wigner function as a quasiprobability distribution over phase space is commonplace [17,18] [19]. The basic property of the Wigner function is that its marginals correctly reproduce the statistics of a position and a momentum measurement, while the joint probability can turn negative. The 'marginals' in the scenario under consideration are the energy statistics, at the initial and final time, of the process in Eq. (2). Hence we look for a quasiprobability distribution p W iCiH )fCfH for the transition from the initial energies (i C , i H ) to the final energies ( where Y iC,iH,fC,fH W := iC,iH,fC,fH This ensures that, if ∆E iCfC = E iC − E fC , the average heat flows of the process in Eq. (2) are reproduced: As discussed, this entails defining a quasiprobability distribution for the observables H C ⊗ H H and While presumably alternative choices would be relevant in different settings, here we argue that the Margenau-Hill (MH) quasiprobability is a natural candidate (in fact, it was put forward in Ref. [9] in the context of work fluctuations [20]). To understand this choice, recall the definition of fluctuations in the standard TPM scheme: It is enlightening to look at the characteristic function for the heat where D X is the dephasing in the basis of the Hamiltonian H X , D X (·) = E X ∈ spec(HX) Π E X (·)Π E X , with Π E X the projector on the eigenspace labelled by E X . Eq. (6) tells us, as is well-known [4], that the TPM characteristic function is associated with a two-time correlation function. However, note that this correlation function is not computed on the initial state ρ CH ; rather, it is computed on a locally dephased state in the energy basis [21]. This does not make any difference if ρ CH is in thermal equilibrium, as in the standard XFT of Ref. [1], or even if it is a state classically correlated in the energy basis, as in [6]. However, it implies that the contribution of the initial coherence and entanglement in the energetic degrees of freedom are removed from the correlation function. A natural generalization that avoids this issue is i.e. the same two-point correlation function, but computed on the actual (rather than dephased) initial state. If we move back to the heat distribution and focus our attention on the real part we find [22] This distribution is normalized and satisfies the marginal properties of Eq. (4) (and, hence, Eq. (5)). It also coincides with p TPM iCiH )fCfH whenever ρ CH is only classically correlated in the energy basis ([Π iCiH , ρ CH ] = 0), which ensures a meaningful classical limit. Hence, as implied by the mentioned no-go theorems, Eq. (8) defines a quasiprobability rather than a probability for heat fluctuations. In fact, Eq. (8) coincides with the definition of a quasiprobability distribution for any two noncommuting observables A = a µ a Π a , B = b µ b Π b given by Margenau and Hill in 1961 [23] and known since the late 1930s [24]: Concerning the negativities, from [9] (Eq. 41) or [25] (Theor. 7.5) one has p W iCiH )fCfH ∈ [−1/8, 1]. In fact, whenever [U † (Π fCfH ), Π iCiH ] = 0 (non-commutativity), there exist states ρ CH for which p W iCiH )fCfH < 0 [9]. The next question concerning Eq. (8) is the experimental accessibility of these heat fluctuations (see Appendix Sec. 1 for more details). There are different schemes to estimate p W , which replace the first projective measurement of the TPM scheme with a weak measurement (using either a continuous variable pointer or a qubit probe). Appealingly, p W iCiH )fCfH can hence be reconstructed in a 'minimally disturbing' version of the TPM scheme. Furthermore, p W iCiH )fCfH is proportional to a quantity inferred from weak measurements and known as the (generalized) weak value [26], first introduced by Aharonov, Albert and Vaidman [27,28] and later generalized to mixed states [29,30].
Another appealing property, which follows from Pusey's theorem [31] and the abovementioned relation between p W and weak values, is that the negativities of p W are witnesses of contextuality [13] in the generalized form defined in [32]. Specifically, the statistics collected by the weak measurement scheme probing p W (with either the one dimensional pointer or the qubit pointer) cannot be explained by any noncontextual hidden variable model, a claim valid even in the presence of noise [33]. An equivalent statement follows from Ref. [34]: in the presence of negativities in p W , there is no positive quasiprobability representation of the quantum protocol probing p W that is able to reproduce the collected data. Loosely speaking, no underlying classical stochastic process can explain the relevant operational statistics when p W < 0.
In conclusion, due to its natural definition, its favourable properties and the strong notion of nonclassicality that follows from its negativity, we propose to use the MH quasiprobability to investigate heat fluctuations between two quantum systems in the fully quantum regime.
Thermodynamic role of negativities in the heat fluctuations. Q can be decomposed as follows. Let us split the quasiprobability in positive and negative com- fCfH is positive (negative), and zero otherwise. Then, with This equation gives a very suggestive interpretation of the role of negative probabilities in heat flows. Recall that, with our sign convention, Q > 0 means backflow.
The total flow Q splits into the 'backflow' from C to H, denoted by Q back , minus the 'direct flow' from H to C, denoted by Q direct . Consider first Q back . Since E iC > E fC , Q back has two positive contributions: (1) One from the transition i C → f C , which removes energy from C (as expected). (2) Another from the transition f C → i C when p − fCfH )iCiH < 0. The transition f C → i C would add energy to C, yet it contributes to the back flow from C to H when the correspondent quasiprobability turns negative. Symmetrically, a transition taking energy away from C can contribute to the direct flow, when p − iCiH→fCfH < 0, see illustration in Fig. 1. Note that if one measures C and H projectively in the energy basis before applying the unitary U , then ρ CH will commute with Π iCiH in Eq. (8); this forces p W iCiH )fCfH ≥ 0, i.e. all negativities disappear.
As discussed in the introduction, it is known that certain strong backflows Q cannot be reproduced within the TPM scheme. The above considerations suggest that we can understand strong flows as the result of negative probabilities, which are destroyed by the measurement process; in fact, Eq. (11) suggests similar considerations may apply to the direct flows as well. It's important to bear in mind that negativity will not always result in strong heat flows. There are two kinds of negativity: the back flow negativity in Eq. (10), and the direct flow negativities of Eq. (11). Since only Q = Q back − Q direct is observed, these two negativites must not cancel each other if we are going to observe an overall effect in the heat flow. In the next section, we will put these intuitions to a firmer ground, showing that 'strong' heat flows can only happen in the presence of negativities.
Heat flows witnessing negativities. We start with the archetypal (and fully solvable) two-qubit scenario, which is also the minimal model in which heat flows can be observed. Since U conserves energy, nontrivial energy exchanges can only happen if H C = H H . Then Inequality 1. Let ρ CH be a two qubit system with thermal marginals (β C = β H ) and U an energy-preserving unitary, [U, by renormalizing the temperature. If p W is nonnegative,
See Appendix Sec. 2 for the proof. Witnessing a violation of the previous inequality ensures that negativity is at play in p W . Note that to violate the bound of Eq. (12), we necessarily need to witness |Q| > |Q TPM |, i.e. a direct or inverse flow bigger than the corresponding flow observed in the TPM scheme. While violations of Eq. (1) can in general occur in a purely classical setup, due to initial correlations in the energetic degrees of freedom, violations of inequality 1 imply negativity, which is a proxy for genuinely quantum effects.
Let us analyze in more detail when the above inequality is violated. Microscopic energy conservation requires [U, H C +H H ] = 0. In the Appendix Sec. 2 we show that a simple reparametrization allows us to restrict U and ρ CH to take the form where z C(H) = 1 + e −β C(H) and η ∈ R. Hence, the parameter space for fixed β C , β H reduces to (θ, η, P 00 ). In Fig. 2 we compare the parameter region in which p W is negative (larger yellow region) with the regions where our inequalities (12) are violated. The blue (red) region corresponds to the detection of negativity in the direct (back) flow probabilities [35]. In a recent NMR experimental setup, the heat backflow between two local-thermal qubits with initial correlations was measured [36]. While other general approaches (e.g., that of Ref. [3]) fail to witness quantum effects for this setup, here we report that, for the parameters considered in the experiment, inequality 1 is violated (see dashed line in Fig. 2). Negativity can hence be detected in relevant accessible experiments (see Appendix Sec. 3 for more details). Note that the experimental state preparation [36], despite being nonentangled, generates heat fluctuations strong enough to prove negavity.
We mention in passing a nontrivial result that holds specifically for two qubit systems: the existence of backflow for an appropriate energy-preserving dynamics is necessary and sufficient for the presence of quantum correlations, i.e. to conclude that ρ CH is not a classically correlated state in the energy basis. These points are detailed in the Appendix Sec. 2.
A quantum exchange fluctuation theorem. To derive a negativity condition for an arbitrary finite-dimensional system, we present the quantum XFT. Traditionally, fluctuation theorems are derived by considering the ratio of the forward and backward process probabilities [1,6]. Hence, in the spirit of [9,37], we consider This can be derived by applying Jensen's inequality to Eq. (16). ∆I W in Eq. (18) is a classical mutual information-like term satisfying ∆I W = ∆I TPM if the initial state only has classical correlations in the energy basis, and ∆I W = 0 in the absence of initial quantum and classical correlations [38]. Violations of the inequality imply that some quasiprobabilities are necessarily negative. Note that the inequality becomes trivial if no initial coherence is present (χ = 0), as expected.
In Appendix Sec. 4 we investigate the heat flows between two qutrits and show that our inequality can witness negativity in these higher dimensional systems as well. We also generalize these considerations to arbitrary finite-dimensional systems with equal Hamiltonians and nondegenerate energy gaps. This allows us to prove an additional generic property of negativity: If all quasiprobabilities that contribute to direct or backflow are negative, then the flows are necessarily strong, i.e. |Q| > |Q TPM |. When only some of them are negative, the last statement becomes convoluted as cancellation effects between positive and negative probabilities (weighted according to the corresponding energy gap) appear.
Outlook. Recent no-go results [8,13] strongly suggest that, if we are to probe the fully quantum thermodynamic regime, due to noncommutativity we need to renounce to a straightforward statistical interpretation of microscopic fluctuation processes. Moving in this direction, here we introduced a natural quasiprobability for heat fluctuations whose negativity captures nonclassical effects in the strong form of contextuality, which formally implies the impossibility of describing the probing scheme through any underlying classical stochastic process. We showed that negativities in the heat fluctuations have a thermodynamic interpretation and that certain heat flows between two locally thermal states at different temperatures can only happen in their presence. We used these tools to witness nonclassicality by analysing data collected in a recent experiment. This is evidence that quasiprobabilities, already widely used in the fields of quantum optics and quantum information, are a useful tool for investigating quantum thermodynamic effect without classical counterpart. ativities in the underlying quasiprobability. In the qubit scenario one overcomes this issue by making the bound in Eq. (12)  The scheme for measuring p W iCiH )fCfH is through a weak measurement at the start of the protocol and a projective measurement at the end. Specifically, define a family of measurement schemes where at the start the system projectors Π iCiH are coupled for a unit time to the momentum P of a one-dimensional pointer device through the interaction Hamiltonian Π iCiH ⊗ P . The pointer is initially in a pure state (πs 2 ) −1/4 dxe −x 2 /2s 2 |x . Then the dynamics U takes place on CH, at the end of which a final projective energy measurement is performed on CH and outcome (f C , f H ) is observed with probability q fCfH . One can then verify that the expected position of the pointer given that some energies (f C , f H ) are observed in the final energy measurement, denoted by X |fCfH , can be directly related to p W iCiH )fCfH in the weak measurement limit s → ∞: The same expression gives p TPM iCiH→fCfH if s → 0 (see Appendix B of Ref. [13]). In this sense, the protocol probing p W iCiH )fCfH can be understood as a 'minimally invasive' version of the standard TPM scheme.

b. Qubit probe
An alternative scheme to estimate p W iCiH )fCfH uses only a qubit pointer [33,39] and, for completeness, will be briefly discussed here. Take a qubit ancilla in a state |ψ = cos |0 − sin |1 . Couple system and ancilla through the unitary V = Π iCiH⊥ ⊗ I + Π iCiH ⊗ σ z (generated by the Hamiltonian H int = Π iCiH ⊗ |1 1| by V = e −igHintt , setting t = π/g). The ancilla is then measured in the |± = (|0 ± |1 )/ √ 2 basis. The corre-sponding Kraus operators on the system are with the correspondent positive operator valued measurement (POVM) on the system given by Denote the joint probability of observing outcome + on the pointer and outcome (f C , f H ) on the system by q fCfH,+ ( ). Define, with obvious notation, also q fCfH,− ( ). Then, if ∆q fCfH ( ) := q fCfH,+ ( )−q fCfH,− ( ), where p fCfH = Tr U † (Π fCfH )ρ is the probability of observing energy outcomes (f C f H ) if no measurement scheme is performed (V = I). Clearly, This gives a way to reconstruct p W iCiH )fCfH from the joint statistics on pointer and system in the abovementioned measurement scheme, together with the probabilities p fCfH that can be inferred from a second experiment where no measurement scheme is applied. Note that, given the knowledge of from the initialization of the ancilla, one can reconstruct p W iCiH )fCfH without taking the limit → 0.
where spec(X) is the spectrum of X and Π E is the projector on the eigenspace of energy E. One can verify that, for all U with [U, H C + H H ] = 0, both p TPM and p W are not affected by the application of D on the initial state ρ CH . For the two qubit case this implies that, without loss of generality, we can take the state with thermal marginals to have the form where z C(H) = 1 + e −β C(H) and we set H C = H H = |1 1| by renormalizing the temperature. Since ρ CH has to be a valid density operator, P 00 must satisfy The MH and the TPM probabilities satisfy p W 01,10 = p TPM 01,10 + η cos θ sin θ cos ξ, p W 10,01 = p TPM 10,01 − η cos θ sin θ cos ξ, p TPM 01,10 = 1 − P 00 z C − P 00 sin 2 θ, The probabilities p W and p TPM are independent of κ, hence we set κ = 0. Furthermore, p TPM are independent of λ, ξ and φ, while p W only depends upon them by a factor cos(λ + ξ + φ). A simple reparametrization then allows one to set λ = φ = 0. One can also set ξ = 0 by a redefinition of η. We can always take η ≥ 0 by a reparametrization of θ (if η ≤ 0 map θ → −θ). In the main text, we allowed η ∈ R for an easier comparison with the experimental parameters of Ref. Note that Q does not depend on P 00 and that ρ CH ≥ 0 implies |η| ≤ (1/z C − P 00 )(1/z H − P 00 ). Since the bound is monotonically decreasing in the allowed parameter regime of P 00 , the largest set of accessible η is achieved for P 00 = zC+zH−zCzH zCzH .
Concerning Q TPM , we have Hence Q TPM ≤ 0, i.e. for the two-qubit case no backflow exists in the TPM scheme. Note that Q(η = 0) = Q TPM .
b. Backflow and quantum correlations in two-qubit systems Next we show that, for a state that is diagonal in the energy basis (even if it includes classical correlations), no heat backflow can be observed. This can be checked by simply replacing η = 0 in Eq. (B8), which gives Hence, η = 0 is necessary for backflow. Conversely, suppose η = 0. Expanding Q at first order in θ we find Recalling that we set wlog η ≥ 0, it follows that when η = 0 one has Q > 0 (backflow) for θ < 0 small enough.

Appendix C: Experimental realization
In a recent NMR experiment [36], the heat backflow Q between two nuclear spin 1/2, prepared in an initial locally thermal state, was measured. The initial Hamiltonian is A locally thermal state is prepared with the form A simple yet important observation is the following: the measurement of Q when α = 0 corresponds to the measurement of Q TPM for the case α = −0.19. This follows from the fact that the initial energy measurement on C and H (the first step of the TPM protocol) has the effect of setting α = 0. Thanks to this observation, the data collected in the experiment can be used to test inequality 1 in the main text.
In Fig. 3 we plot the heat transfer between C and H for various interaction times. The green line corresponds to the heat flow in the correlated scenario, whereas the orange line correspond to the uncorrelated scenario. Equivalently, they correspond to the measurements of Q (green) and Q TPM (orange). Q > 0 corresponds to backflow from C to H, whereas Q < 0 corresponds to direct flow from H to C. The shaded areas indicate the interaction times in which negativity in heat transfer quasiprobability is present (for the backflow p W 01,10 < 0 and for the direct flow p W 10,01 < 0). The dark shaded area indicates a violation of inequality 1 in the main text, where our bound detects negativity. This shows that the data collected in the experiment [36] allows us to detect negativity. While in the experiment only the backflow regime was explored, which is sufficient for detecting negativity, here we theoretically extend the interaction time to the regime where direct flow is observed. In this regime, we see that a measurement of the heat flows would allow us to detect the negativity of the MH quasiprobability also in the direct flow.
Appendix D: Extensions to arbitrary finite-dimensional systems 1. Alternative bound to inequality 2 As was discussed in the main text,χ in inequality 2 may diverge. Here we derive an alternative bound (inequality 3) for the detection of negativity for arbitrary finite-dimensional systems that avoids this problem. However, we note that, for all the case studies considered, we found inequality 2 to be superior to inequality 3.
Direct calculation of p i C i H )f C f H e ∆β∆EiCfC leads to the following Note that c(ρ CH ) = 0 if the populations of ρ CH coincide with those of ρ C ⊗ ρ H , even if ρ CH has coherence. This is why we use the notation c to indicate "classical correlations". On the other hand, q(ρ CH ) = 0 whenever there is no coherence, even in the presence of classical correlations, hence the notation q. The term 1 in Eq. (D1) was obtained noting that Inequality 3. Let ρ CH be an arbitrary finite-dimensional system with thermal marginals (β C = β H ) and U an energy-preserving unitary, [U, The result is an application of Jensen's inequality to Eq. (D1). Using norm inequalities it can be shown that so the r.h.s is finite.

Heat flow between two qudits
To study the heat flow between two qudits we first consider the case of two qutrits, which can then be generalized easily to d dimensional systems. Setting E 0 = 0, we will take H C = H H = 2 n=1 E n |n n| and assume no degeneracy of the energy gaps (the 'Bohr spectrum') throughout this section (see Fig.4). The general twoqutrit state takes the form, we are left with the free parameters η ij , ξ ij and four undetermined populations ρ i that must comply with the nonnegativity of ρ CH . The general energy preserving unitary takes the form Here we used the short-cut notation c nm := cos(θ nm ) and s nm := sin(θ nm t), where θ nm = θ mn , φ nm = φ mn and λ nm = λ mn are parameters characterizing the coupling between the n level of the first qutrit and the m level of second qutrit. Within the manifolds (01,02,12), κ are global phases that do not change the energy transfer transition probabilities. One can notice the similarity between the U for the qutrits with the one of the qubits. The heat flow probabilities can be split into contributions from independent manifolds (01,02,12) that possess the same structure seen in Eq. (13). The generalization to higher dimensions is now straightforward, as the structure of ρ CH and U in equations (D6) and (D8) is preserved.
Here we set E 0 = 0 and ordered the levels such that E 0 ≤ E 1 ≤ E 2 ≤ · · · and d the dimension of the qudit. Next, we show that if all quasiprobabilities that contribute to the direct or backflow are negative, then necessarily |Q| > |Q TPM |. The difference ∆Q = Q − Q TPM reads ∆Q = − En>Em η nm √ ρ nd+m ρ md+n sin(2θ nm ) cos(ξ nm + φ nm + λ nm )(E n − E m ).
For the direct flow (Q < 0), if ∀E n > E m we have p W nm )mn < 0, then from Eq. (D9) we immediately conclude that ∆Q < 0. For the backflow (Q > 0), if ∀E n < E m we have p W nm )mn < 0, then from Eq. (D9) we now have that ∆Q > 0, which is exactly what we wanted to prove. Note that Eq. (D11) also suggests a way of maximizing the difference ∆Q for a given initial state. Choosing a protocol such that ξ nm + λ nm = −φ nm and θ nm = ±π/4 for all n, m, we obtain |∆Q| max = En>Em η nm √ ρ nd+m ρ md+n (E n −E m ). (D12) In Fig. 5 we consider the two qutrits setup of Eq. (D6). We plot the negativity in the direct flow and compare it to the violations of Inequalities 2 and 3 for different protocols. The protocols are determined by varying θ 01 and θ 02 .