Quantum process tomography with informational incomplete data of two $J$-coupled heterogeneous spins relaxation in a time window much greater than $T_1$

We reconstruct the time dependent quantum map corresponding to the relaxation process of a two-spin system in liquid-state NMR at room temperature. By means of quantum tomography techniques that handle informational incomplete data, we show how to properly post-process and normalize the measurements data for the simulation of quantum information processing, overcoming the unknown number of molecules prepared in a non-equilibrium magnetisation state ($N_j$) by an initial sequence of radiofrequency (RF) pulses. From the reconstructed quantum map, we infer both longitudinal ($T_1$) and transversal ($T_2$) relaxation times, and introduce the $J$-coupling relaxation times ($T^J_1,T^J_2$), which are relevant for quantum information processing simulations. We show that the map associated to the relaxation process cannot be assumed approximated unital and trace-preserving for times greater than $T_2^J$.

where δ ij is the Kronecker's delta in i and j and H is an effective Hamiltonian for a single molecule. The wave function and the density matrix corresponding to the N-particle Hamiltonian are assumed to be separable, namely: where ψ i and ρ i are the wave function and density matrix for a single molecule, respectively, where we shall work only in the spin degrees of freedom. * thiago@fisica.ufmg.br In an NMR experiment, one does not have access to the particular state of a distinct molecule in the preparation procedure, all one has is the average state of a representative molecule of the ensemble. Therefore, we can write: where ρ stands for this representative state, which we shall discuss further later. Our observables are related to the resultant macroscopic magnetisation induced by an external static magnetic field, and we assume they are operators of the form: where δ ij is the Kronecker's delta. Therefore, we have expectation values of the kind: This rationale, assuming Eqs. (1)(2)(3)(4) as premisses, formalises a model which one has equally prepared, but independent, black boxes and the same measurement "buttons" are pressed in all of them when the measurement occurs. Thus, we have a reducible to a single box model and we can generalise and specialise our arguments back and forth, i.e., what happens in a single box, will happen in the other ones and vice versa. We shall use this freely. This model fits the common knowledge that measurements in NMR have contributions of all the molecules in the ensemble [1,2], justifying the detected macroscopic signals.
The observables of interest in the NMR experiment are generally the components of the macroscopic nuclear magnetisation, which are proportional to the ensemble average values of the nuclear spin operators. The matrix representation of these operators in the |S, m basis (with S = 1/2) reads S x = σ x /2, S y = σ y /2, S z = σ z /2, with the usual Pauli matrices given by: For example, one can calculate the magnetisation in the z-direction of an ensemble of nuclear spins using the following relation: whereÎ z is the empirical value associated with the observation of the magnetisation in the z-direction and ρ is the density matrix of the system of a single specimen. Remember that the left-hand side of this equation represents the statistical average over the entire ensemble of N molecules, and not the expectation value for a given particular member of the ensemble. The relevant effective two-spin Hamiltonian in Eq. (1) is given by (7) where the labels H and C stand for hydrogen and carbon, respectively, as the first and second particles. The resonant frequencies ω H and ω C include the effects of isotropic chemical shift σ iso in acetone solution for each nucleus, i.e., ω = ω L (1 − σ iso ). The Larmor frequency is given by ω L ≡ γ n B 0 , where γ n is the gyromagnetic ratio of the nucleus, and B 0 is the static magnetic field in z-direction.
At thermal equilibrium, the density matrix obtained from the Hamiltonian in Eq.(7) is given by the maximum entropy principle [3]: where k B is the Boltzmann constant and T is the temperature. In the high temperature limit, one can expand this exponential as (9) Now, we are able to calculate the total magnetisation of the system in the z-direction, using the equilibrium state in Eq.(9), the relation in Eq.(6) and Curie's law, which leads to where n H and n C are the number of hydrogen and carbon nuclei per unit volume, respectively, which are equal and correspond to the number of chloroform molecules per unit volume (n H = n C = n); n J stands for the number of responsive pairs of J-coupled nuclei per unit volume, and ι HC its "gyromagnetic ratio". These responsive pairs of spins correspond to a small fraction of chloroform molecules (N J /N ≪ 1), for which the spins of 1 H and 13 C are in a chosen coherent state prepared by a certain sequence of radiofrequency pulses (RF). The first issue on normalising the experimental data arises due to the fact that nor N J , neither its density n j , are straightforwardly revealed by the magnetisation measurements. We shall discuss this later. Figure 1. A schematic picture of the dynamics of the experiment. Top: A sequence of radiofrequency pulses prepares the system in some desired non-equilibrium state at t=0, where we have the maximum number of responsive J-coupled spins. Middle: The system is allowed to relax, and the fraction of molecules that responded to the preparation starts to recover its equilibrium state. Bottom: t → ∞, the system reaches the equilibrium and there are no molecules in the initial nonequilibrium spin state.
Let us focus on the experiment now. We know that interesting NMR quantum information processing experiments happen in non-equilibrium, where the responsive J-coupled pairs of spins play an important role. In our experiment, we prepare the system in a desired nonequilibrium magnetisation state in time t = 0 and allow it to relax. A schematic picture of the experiment is presented in Fig. 1. First, at t = 0, a certain sequence of RF pulses prepares the system in some desired nonequilibrium state, where we have the maximum number of responsive J-coupled spin pairs. This results in a small macroscopic non-equilibrium magnetisation. The second part of the experiment happens in the time win-dow which we can use the small fraction of responsive molecules in the non-equilibrium state to simulate the dynamics of quantum parameters (e.g., [4]). The system is allowed to relax, the molecules that responded to the preparation start to recover their equilibrium state, diminishing N J , and finally (t → ∞ ) the system reaches the equilibrium with vanishing N J .
The relaxation is such that, after some time has elapsed, the magnetisation returns completely to the initial z-direction, satisfying again the thermal equilibrium requirements. It is worth emphasising that the exact meaning of "some time" is largely dependent on the details of each particular nuclear spin system and its environment, ranging typically from microseconds to several hours. The relaxation consists of two different processes, occurring simultaneously but, in general, independently, namely: the transverse relaxation and the longitudinal relaxation [2]. We shall characterise, by the quantum process tomography, a net effect of these two phenomena, that is paramount to the problem of NMR simulation of quantum information processing. For reasons to become clear soon, we shall refer to this net phenomenon as the J-coupling relaxation.
The longitudinal relaxation is the process that leads the longitudinal component of the nuclear spin magnetisation to recover its equilibrium value. The recovery of the M z component of the magnetisation is related to transitions between the nuclear spin levels. The natural tendency is the system to give up its excess of energy by effecting transitions from the upper to the lower energy level. After some time, which is commonly named T 1 , the Boltzmann distribution is reestablished.
Simultaneously to the longitudinal relaxation, the transverse relaxation is the process that leads to the disappearance of the components of the nuclear magnetisation M that are perpendicular to the static field B 0 . The origin of the transverse relaxation relies on the loss of coherence in the precession motion of the spins (or dephasing of the spins), caused by the existence of spread in precession frequencies for the collection of nuclear spins. This spread progressively results in a reduction of the resultant transverse components (e.g., M x and M y ). After some time, which is commonly named T 2 , the spins distribute randomly in a precession cone around B 0 and the transverse magnetisation is again zero.
The J-coupling relaxation, which we introduce here, can be considered as the net effect of both longitudinal and transverse relaxations. In NMR quantum information processing simulation experiments, correlated spins in non-equilibrium states are indispensable to simulate quantum algorithms. We name T J 1 and T J 2 the characteristic times in which the coupled magnetisations M ij (∀ i, j ∈ {x, y, z}) decay, T J 1 for the longitudinal, or M zz component, and T J 2 for the transverse components. Note that what we call coupled-magnetisation refers specifically to expectation values of two-spin operators (tr(ρσ H i ⊗ σ C j )), while uncoupled-magnetisation is related to expectation values of one-spin operators (tr(ρσ H i ⊗ 1) or tr(ρ1 ⊗ σ C j )). In order to reconstruct the density matrix of the system along the time, we need to normalise the experimental data, such that we can relate the measured nonequilibrium magnetisations to expectation values. Here, we are not concerned about the simulated (pseudopure) states, we deal only with the real non-equilibrium NMR states, although it is worth to emphasise that, in this preparation procedure, we also assume ergodicity, as we collect the whole set of measurements using temporal labeling [1,2,5].
The normalisation issue is a daunting problem in NMR experiments. Firstly, by the data acquisition itself, since involves taking the intensity of the peaks from a Fourier transform spectra of an oscillating signal. Secondly, because we are forced to assume the representative state ρ in Eq. (3). It would be preferable if we could address each molecule individually like If it were the case, we would be able to count how many spins are still coherent and responding accordingly, therefore it would be possible to know exactly the number of responding molecules as in Eq. (5) (e.g., if we were interested in knowing N J ) . But such a case is not possible and we shall stick with the weaker premise of the representative state, losing track of this counting. Thirdly, as consequence of the traceless nature of the operators corresponding to the magnetisation components, we do not possess any direct experimentally measured data related to the trace of the non-equilibrium density matrix. If we were measuring the eigenprojectors (P k ) of these operators, this problem would not appear, as the completeness relation (1 = ∑ k P k ) would give us a trivial recipe for normalisation.
As we shall see, the quantum tomography techniques we employ here, allow us to circumvent this problem, yielding proper inferences of the normalised nonequilibrium quantum states and quantum maps.
We have three different kinds of signals corresponding to our measurements, which we have to deal with in order to normalise the experimental data. One may grasp this in Fig. 1. We have, responding to RF pulses in the resonant frequency ω H of the hydrogen, both coherently coupled and uncoherently coupled hydrogen nuclei; the same happens to carbon, for RF pulses in the resonant frequency ω C . As the gyromagnetic ratio of the hydrogen is almost 4 times larger than that of the carbon, the hydrogen species is expected to have a larger magnetisation in equilibrium. The last kind of signal, which is very small compared to the first two, corresponds to that coherently coupled magnetisation of the N j chloroform molecules. To remedy the lack of some relation tying the intensity of measured magnetisation components, like the completness relation we have mentioned in the previous paragraph, we normalise all measured signals [5] corresponding to nonequilibrium magnetisations backwards, such that the expected value of the magnetisation in equilibrium is equal to one. This rescaling of the magnetisation signals is also convenient in order to perform our numerical optimisations with good precision. For the hydrogen and carbon signals, we havê and for the coupled signals we havê where (t) are the empirical values associated with the observation of the magnetisation in the i, j ∈ {x, y, z} directions relative to the hydrogen, carbon and J-coupled signals respectively.
The equilibrium (Eq. (9)) and non equilibrium states dealt with in the NMR experiment are of the kind where ǫ ≈ 10 −5 and ∆ is a traceless matrix, known as the deviation matrix, corresponding to the last three terms of Eq. (9). Though the experimental magnetisation signals come just from the deviation matrix, due to the traceless nature of the spin operators (for example: tr(ρσ i ⊗ σ j ) = ǫtr(∆σ i ⊗ σ j )), these are macroscopic signals, for there are still a huge number (O(ǫ × 10 24 ) = O(10 19 )) of molecules contributing to the magnetisation. The reason why we have the trouble to deal with the density matrix instead of the deviation matrix, is that we are interested in reconstructing a quantum map that is non-unital and affects the identity operator. If we ignored this fact, and adopted some arbitrary normalisation for the experimental data, we would not be able to compare density matrices reconstructed in different times during the relaxation process, and therefore could not infer physical parameters characterising the phenomenon, let alone the quantum map.
Since we have a large ensemble and sharp peaked signals, we can considerÎ {H,C,J} ex p i,j (t) gaussianly distributed with standard deviation σ 2 = 1. Thus, we can deal with the experimental errors with a simple ℓ 2 -norm minimisation, which is equivalent to a loglikelihood method assuming gaussian noise (cf. [6,7]).
The standard quantum process tomography (SQPT) [1,8] consists in obtaining information about the map, by means of quantum tomographies of states modified by the action of the unknown process map. If the density matrices of the tomographed states span the Hilbert-Schmidt space, then it is possible to reconstruct the quantum map. Our experiment consists in preparing ̺ l k : 20 different non-equilibrium magnetisation ensemble states (l = 1, 2, . . . , 20), which can be used to simulate 20 different two-qubit pure states, where the density matrices of 16 of them span the Hilbert-Schmidt space L(H d ), with d = 4. We also have 51 observations of these states in the time instants t k (k = 0, 1, . . . , 50).
First, we perform quantum state tomographies in order to find the initial states. We use a variation of [9,10] also equipped with nuclear norm minimisation [11] as follows: where · * stands for the nuclear norm or trace norm. This type of Semidefinite Problem (SDP) in the secondorder cone can be efficiently solved using [12][13][14]. We can read Eq.(14) as follows. As we do not know how to explicitly normalise the experimental data (Î We shall express the output states resulted from the action of the unknown map Λ(t k ) as where ̺ l 0 are the point estimates of the system initial states in Eq. (14),ρ l k ∈ {ρ l k |ρ l k ∈ L(H d ),ρ l k 0, l = 1, . . . , 20, k = 0, . . . , 50} and the {A m } is a complete set of operators forming a basis in the Hilbert-Schmidt space L(H d ). The {χ ij (t k )} defines the superoperator χ(t k ) which has all the information about the process.
It is a d 2 × d 2 positive operator in the Hilbert-Schmidt space with d 4 independent real parameters (or d 4 − d 2 in the trace preserving case), cf. [1,8].
At each time t k , the quantum process tomography is obtained using a variation of the method in [15] also equipped with a matrix recovery technique like nuclear norm minimisation, in order to get the information about the trace: The reason behind the choice of the objective function in Eq. (16) being the minimisation of the trace, relies on the fact that we do not know the number of coherently magnetised nuclei at each time. Therefore, after running the optimisation, we know that the trace must be at least tr ρ l k ∀ l = 1, . . . , 20, k = 0, . . . , 50.
In order to compare the states at each time t k , we assume that the process acts homogeneously in the trace, thus we define N(t k ) = max tr ρ l k , l = 1, . . . , 20. (17) Finally, we have everything to formulate our last procedure in order to perform the quantum process tomography of the relaxation. Since we are dealing with a toward-equilibrium experiment, inferences based on maximum entropy are reasonable. We shall choose the maximisation of the linear-entropy [16] as the objective function, namely, The choice of S L is due to the fact that we can formulate the minimisation of it as an SDP in the second-order cone, which makes it efficient. min  Figure 2. The relaxation of the states toward equilibrium. The figure shows the trace distance between the states ρ l 0 under the action of the tomographed quantum map Λ(t k ) and the equilibrium thermal state. Note that the states in t = 0 are perturbations of the thermal state by different sequences of radiofrequency pulses. These perturbed thermal states are the ones useful for quantum information processing simulations.
After running the program in Eq. (18), we plot in Fig.  2 the trace distance between the states ρ l 0 under the action of the map Λ(t k ) and the rescaled equilibrium state 1 M ρ eq . We note that the functional form of the curves are like an exponential decay, as expected.
As a byproduct of the process tomography, the map must give us information about the relaxation times . We define the quantity  i, j-direction (i, j ∈ {x, y, z}) related to the equilibrium value and it gives us the functional behaviour of the magnetisation during the time. We could fit a decay model like where T * will be the aimed T C,H,J The results are summarised in Table I, in which we are able to conclude that, if one desires to simulate quantum parameters using the chloroform molecule, the time window where it can be done is 0.17s, which is  as a function of time. Note that these "magnetisations" are relative to the equilibrium magnetisation, according to Eq.(19). the lifetime of the coherent coupling of the spins (T J 2 ) in the chloroform molecule. Otherwise, we can consider the system (partially) relaxed and unable to simulate all parameters properly.
Summarising, we have performed a quantum process tomography of the relaxation of a two-spin system in an NMR liquid-state experiment. As the experimental data correspond to expectation values of traceless operators, the use of informational incomplete quantum tomography techniques to handle the missing quantum state normalisation information was paramount to reconstruct the time dependent non-unital quantum map. The successful approach we have employed, clearly illustrated in Fig.2, which shows the dynamics of the ensemble state towards equilibrium, allowed the recovery of the characteristic relaxation times (Tab. I) directly from the tomographed quantum map, confirming our heuristic for normalisation, which consists in arbitrarily assuming the equilibrium magnetisation as unity, in conjunction with Eqs. (16) and (17). Although the system is initially in the thermal equilibrium state (Eq.(8)), one should remember that in quantum information processing simulations (QIPS), a sequence of radiofrequency (RF) pulses prepares a non-equilibrium state that will relax in a time T 1 . It is this non-equilibrium state, which is a small perturbation of the thermal state, that is useful in QIPS. In Fig.2, the dynamics of the prepared non-equilibrium states is shown. From the quantum process tomography, we have learned that the J-coupling relaxation times are the relevant temporal parameters for QIPS. For t < T J 2 (Tab. I), which corresponds to the time window where the transverse coupled magnetisation survives (Fig.5), we have coherent magnetisation in all directions, revealed by the quantum tomography of ensemble states that can be linearly transformed to access all the points of a two-qubit Hilbert space, allowing QIPS involving states of the form |Ψ = α| ↑↑ + β| ↑↓ + γ| ↓↑ + δ| ↓↓ . For T J 2 < t < T J 1 , one can see in Figs. 4 and 5, that we still have transverse magnetisation, but only of the uncoupled kind, and the quantum tomography reveals that the ensemble states can be linearly transformed to simulate, with good fidelity, only statistical mixtures of two-qubit basis states, ρ = p 1 | ↑↑ ↑↑ | + p 2 | ↑↓ ↑↓ | + p 3 | ↓↑ ↓↑ | + p 4 | ↓↓ ↓↓ |. Finally, for t > T J 1 , there is no useful transverse magnetisation, and only trivial two-qubit states can be simulated, ρ = (p 1 1 | ↑ ↑ | + p 1 2 | ↓ ↓ |) ⊗ (p 2 1 | ↑ ↑ | + p 2 2 | ↓ ↓ |). Note that this association of the QIPS with the J-coupling is what is schematically represented in Fig.1.
In conclusion, we hoped we were able to cast some light on the post-processing problem in NMR quantum information processing experiments: not just in the daunting problem of the normalisation of the quantum states during the time, but also in considering the real non-equilibrium states, instead of the simulated (pseudo-pure) (pseudo-)states. Note that we have reconstructed the quantum map of the real relaxation process taking place in the NMR experiment. The understanding of this dynamics is paramount for the proper simulation of quantum algorithms with pseudo-pure states. An interesting aspect to be explored in future works, is the characterisation of the (non-)markovianity of the dynamics, shown in Fig.2.