Continuous matrix product state tomography of quantum transport experiments

In recent years, a close connection between the description of open quantum systems, the input–output formalism of quantum optics, and continuous matrix product states (cMPS) in quantum field theory has been established. The latter constitute a variational class of one-dimensional quantum field states and have been shown to provide an efficient ansatz for performing tomography of open quantum systems. So far, however, the connection between cMPS and open quantum systems has not yet been developed for quantum transport experiments in the condensed-matter context. In this work, we first present an extension of the tomographic possibilities of cMPS by demonstrating the validity of reconstruction schemes based on low-order counting probabilities compared to previous schemes based on low-order correlation functions. We then show how fermionic quantum transport settings can be formulated within the cMPS framework. Our procedure, via the measurements of low-order correlation functions only, allows us to gain access to quantities that are not directly measurable with present technology. Emblematic examples are high-order correlations functions and waiting time distributions (WTD). The latter are of particular interest since they offer insights into short-time scale physics. We demonstrate the functioning of the method with actual data, opening up the way to accessing WTD within the quantum regime.


Introduction
Continuous matrix product states (cMPS) have recently been recognized as powerful and versatile descriptions of certain one-dimensional quantum field states [1][2][3]. As continuum limits of the MPS-a well-established type of tensor network states underlying the density-matrix renormalisation group [4]-they introduce the intuition developed in quantum lattice models to the realm of quantum fields, offering similar conceptual and numerical tools. In the cMPS framework, interacting quantum fields such as those described by Lieb-Liniger models have been studied, both in theory [1,5,6] and in the context of experiments with ultra-cold atoms [7].
On a formal level, cMPS are intricately related to Markovian open quantum systems [1,2]: the open quantum system takes the role of an ancillary system in a sequential preparation picture of cMPS. Elaborating on this formal analogy, cMPS can capture properties of fields that are coupled to a finite dimensional open quantum system. This connection has been fleshed out already in the description of fermionic quantum fields [8] and of light emitted from cavities in cavity-QED [2,9] in the quantum optical context, under the keyword of the inputoutput formalism [10].
Another methodological ingredient to this work is that cMPS have been identified as tools to perform efficient quantum state tomography of quantum field systems [7,[11][12][13][14], related to other approaches of tensor network quantum tomography [12,15]. These efforts are in line with the emerging mindset that for quantum many-body and quantum field states, tomography and state reconstruction only make sense within a certain statistical model or a variational class of states. Importantly, in our context at hand, it turns out that cMPS can be  [3][4][5]. Previous works [7,11] have shown that measurements of low-order correlation functions C n with n = 2, 3 are sufficient to access higher-order correlations using cMPS tomography based on the cMPS matrices M and D (right side, black arrows). The first achievement of this work is the formal demonstration that measurements of low-order counting probabilities constitute an alternative to measurements of low-order correlation functions for carrying out cMPS tomography (blue arrow). We show that the probabilities to detect zero or one particle (P 0 and P 1 , respectively) are sufficient to reconstruct alternative cMPS parameter matrices  and ,  from which higher-order correlation functions can be computed. The second achievement of this work is to extend the applicability of this cMPS framework to quantum transport experiments. As a main illustration, we show that cMPS-based tomography provides an access to the distribution of waiting times .
 The according statistics are not directly measurable due to experimental limitations on single-particle detectors. However, we demonstrate with experimental data that they can in fact be reconstructed from the knowledge of low-order correlation functions (broad blue arrows). measured effectively: measuring WTDs requires the detection of single events while ensuring that no events have been missed-for instance, due to the dead time of the detector.
With present technologies, WTDs in transport experiments can be measured when the injection rate of electrons is within the kHz range as in the experiments of [26,27]. Indeed, at those frequencies, the current trace is resolved in time and the WTD can be directly deduced from it. As we will show in section 5, the WTD reflects the quantum statistics of the electrons. However, quantum coherence and entanglement cannot be detected at those frequencies. To observe these quantum effects, one needs to move to the GHz regime, which can be achieved either with DC sources with a typical bias of tens of meV, or with periodically driven sources at GHz frequencies [28][29][30][31][32]. In the GHz range, the current trace cannot be resolved in time so that the measurement of the WTD is not feasible at present. In contrast, second-and third-order correlation functions have been proven to be feasible [32,33].
With these experimental constraints in mind, we propose in section 5 an indirect way to access the WTD with methods that are within reach of the experimental state of the art. Namely, the dynamics of the full open quantum system is accessed from measurements of low-order correlation functions (typically second-or thirdorder). This is made possible with a cMPS formulation of the transport experiments as explained in the following section.
We illustrate this indirect path of accessing the WTD by considering real data obtained in the experiment of [27], where single electrons tunnel through a single-level quantum dot in the kHz regime. Both the current trace resolved in time and the two-and three-point correlation functions have been measured. The data allows us to demonstrate a very good agreement between the WTD deduced directly from the current trace and the WTD obtained via our reconstruction scheme based on the data of the correlation functions. This gives substance to our protocol based on cMPS to access the WTD with present technologies. We claim that this method remains valid in the GHz frequency range and for more complex systems such as a double quantum-dot coupled to two reservoirs-which would exhibit quantum coherence effects-and for quantum transport experiments with fermionic quantum gases.

Tomography of cMPS
In order to present a self-contained analysis, we start by reviewing the cMPS formulation of capturing a finite dimensional open quantum system [2] and the tomography procedure of reconstructing the relevant cMPS parameter matrices [11]. Consider an open quantum system (in cMPS terms the ancillary system) with dimension d (called bond dimension in that context) and interacting with one or more quantum fields that are described by field operators y â for different fields α. Its dynamics can in general be represented by different mathematical objects: (a) The master equation in Lindblad form, which governs the evolution of the ancillary system described by its state vector Yñ | defined on the Hilbert space  of dimension d × d. The degrees of freedom of the coupled fields are traced out in this approach.
(b) The set of n-point correlation functions of the coupled fields.
(c) The full counting statistics of the field system, i.e. the complete set of cumulants of the probability distribution of transferred particles. The nth cumulant of the generating function is linked to the n moments of this distribution, which correspond to the n-point correlation function.  Î´is known as the transfer matrix and the matrix Q is defined as Formally, the isomorphism introduced above is defined by the following relations for an operator and the product of operators Being closely connected to K and R a { } introduced above, the knowledge of the matrix and T and its components provides access to the dynamics of the open quantum system, and allows to directly derive the according Lindblad equation. The This expression is related to the path ordered exponential that arises when integrating the Lindblad equation. The embedding of the cMPS state vector cMPS y ñ | into Fock space becomes clear when expanding the path ordered exponential exp.
 For more details, we refer to [3] where the authors formulate the cMPS in different representations such as the Fock space and a path integral formulation. After integration, the ancillary system is traced out via Tr anc and the resulting term is applied to the vacuum state vector Wñ | , where 0 y Wñ = â | for each α.
Compared to the Lindblad equation, the main difference is that the degrees of freedom of the ancillary system are traced out such that its dynamics is mapped into the dynamics of the coupled quantum fields . y a {ˆ} The evaluation of expectation values of field operators leads to expressions that only contain quantities from the ancillary system, and information about the ancillary system can be inferred from according field operator measurements. For the sake of clarity, we restrict ourselves to the case where a single coupled quantum field, denoted as , y b is measured.
The density-like correlation functions of the measured quantum field y b then read where the columns of X represent the eigenvectors of T.
Let us mention that the knowledge of X is in principle not necessary to reconstruct the matrices Q and R and hence the according Lindblad equation [11]. Specifically, the second-and third-order correlation functions take the form with j l { } being the eigenvalues of T. Due to the translation invariance of the system, we can set x 0. 1 = The tomographic possibilities of the cMPS formalism can be understood from equations (10)

= ( )
Both numerator and denominator appear as coefficients in C 3 and can be determined with spectral estimation procedures. This means that in principle we just need to analyse a three-point function in order to obtain the building elements M and D of arbitrary-order correlation functions. This reconstruction scheme demonstrates the central role of the matrices M and D to derive the different equivalent objects that describe the dynamics of an open quantum system: the Lindblad equation, the set of npoint correlation functions, the full counting statistics of the number of transferred particles and the cMPS state vector. These matrices M and D can therefore be considered as the central quantities on which our reconstruction procedure is based; this is illustrated in figure 1.

Use of the thermodynamic limit
Intuitively, it is clear that the reconstruction of the matrices M and D should gain precision by increasing the number of correlation functions C n on which the reconstruction scheme is based. The same statement is valid when increasing the size of the set of available counting probabilities P n . But in general, experiments will only provide us measurements of low-order correlation functions, typically those of the second-and third-order [26,27,33]. A priori, this might render the reconstruction of the matrices M and D infeasible, but the work in [12] proved that this limitation can be circumvented by making use of the structure of the cMPS state vector combined with the thermodynamic limit.
For a given finite region I and a fixed bond dimension d, all expectation values can be computed from all correlation functions C x n ( ) taking values in the finite range I, This contrasts with the situation of having access to correlation functions C x n ( ) for arbitrary values of ) but for low n. Here, arbitrary values x imply the thermodynamic limit, i.e. the finite region I tends to infinity. Then indeed, low order correlation functions (typically C C C , , 3 ) are sufficient to reconstruct an arbitrary expectation value of an observable supported on I.

Reconstruction of cMPS from low-order counting probabilities
In this section, we extend the central role played by the matrices M and D for tomographic purposes by showing that they (more precisely: their equivalents  and ) are also accessible from low-number detector-click statistics, i.e. the idle time probability density function P 0 and the density function P 1 , which correspond to the detection of zero and one particle, respectively, within a certain time interval τ.
It is well-known that correlators and counting statistics are closely related. When assuming perfect detectors, the probability to observe n events in the time interval between t and t t + is given [35] by the expression where the correlation function C m has been introduced in equation (6). For a translationally invariant system, we can without loss of generality set t = 0. Furthermore, when changing the integration bounds and performing the limit L ,  ¥ we obtain where X diagonalizes T as defined in equation (8).With equations (13)-(15), the low-order counting probabilities P 0 t ( ) and P 1 t ( ) within a cMPS formulation are given by similar expressions to equations (10) and (11), namely -≕ (ˆ)). See appendix for details.
As a first step, we can extract from P 0 and P 1 the coefficients z z j j 1, ,1 {ˆ} and the eigenvalues ,  The matrix elements of  can then in principle be determined using gauge arguments and under the assumption that the additive components of P n are linearly independent. From  and ,  the cMPS matrices Q, R and K describing the dynamics of the open quantum system can be determined in a straightforward way (see appendix for details).
Let us comment on the feasibility of this reconstruction scheme with present technology. In order to measure P 0 and P 1 , efficient single-particle detectors without dark-counting and tiny dead-time are necessary. Dark-counting leads to detector output pulses in the absence of any incident photons while the dead-time is the time interval after a detection event during which the detector cannot detect another particle. Although significant experimental efforts have been made in order to improve single-photon [36] and single-electron detectors [37,38], the state-of-the-art for single-particle detection is not yet sufficient to perform a reliable measurement of P 1 . For the moment, these experimental constraints make the reconstruction scheme based on P 1 only valid on a formal, mathematical level. In the light of the recent experimental progress towards the reliable detection of single particles, we believe that this idea will become relevant in the future.

Application to fermionic quantum transport experiments
Very recent works have successfully formulated experimental setups in cavity QED and ultra-cold Bose gases as well as the corresponding measurements in terms of cMPS [7,9]. This allowed them to make predictions for higher-order correlation functions that are not accessible experimentally and to investigate the ground-state entanglement.
Here, we tackle the problem of formulating quantum transport experiments and the corresponding measurements (average charge current, charge noise) in cMPS terms To this end, we demonstrate that the field that is leaking out and is measured in a quantum transport experiment belongs to the cMPS variational class. We then provide an example to illustrate the equivalence between an Hamiltonian and a cMPS formulation by considering one of the simplest transport experiment, namely single electrons tunnelling through a single-level quantum dot. We derive the first-order and second-order correlation functions in cMPS terms, and show that we recover the well-known expression of the average current and charge noise, when writing the cMPS state equation (5) in terms of the parameters of the quantum system.

Quantum transport experiments in terms of cMPS
We now turn to a description of the physical setting under consideration. We assume here transport experiments, where single electrons transit through a scatterer coupled to fermionic reservoirs. The reservoirs, considered at equilibrium, are characterized by their chemical potential and their temperature via the Fermi distribution. The bias energy and the bias temperature between the different reservoirs will set the direction of the charge current. For the sake of simplicity, we restrict ourselves to two reservoirs, the source and the drain. This transport setting can be described by the Hamiltonian  In order to model a DC source, the energy levels in the left and right reservoirs are assumed to be densely filled up to the energies E eV F + and E F , respectively. Here, E F is the Fermi energy and V is the bias potential applied on the 'source' reservoir. At zero temperature, the bias energy eV enables uni-directional transport of electrons between the left and right reservoirs. It plays a similar role to the frequency bandwidth when, e.g., considering cavity QED setups, and fixes the energy domain over which electronic transport takes place.
With this assumption about the direction of propagation of the electrons (from left to right), we will see that equation (18) is equivalent to a generalized version of the cMPS Hamiltonian introduced in [1,2], where the matrices Q and R a { } and the quantum fields y a {ˆ} have been introduced in section 2. The cMPS Hamiltonian for quantum transport experiment reflects the direction of the current: a fermionic excitation present on the left of the scatterer is annihilated at the scatterer as described by the quantum field L ŷ (an electron jumps into the scatterer). Similarly, a fermionic excitation present on the right of the scatterer is created at the scatterer as described by the quantum field R ŷ † (an electron jumps out of the scatterer). The case of a multiterminal setup can be considered in a similar way. Showing that equations (18) and (21) are equivalent implicates that there is a fermionic quantum field leaking out of the scatterer to be measured and that it belongs to the cMPS variational class. Such a description of the transport experiment corresponds to a fermionic version of the inputoutput formalism of cavity-QED setups.
Using equation (20), the quantum field leaking out of the quantum system, t , The Fermi sea for the electrons is taken into account in the following way: on the right side of the scatterer, the quantum field satisfies t E 0,  This assumption is the so-called large-bias limit, which is considered in order to derive the master equation corresponding to the tight-binding Hamiltonian. In quantum optics, it corresponds to a finite frequency bandwidth, which allows the use of the rotating wave approximation [9,10]. In the following, we assume that the interaction amplitude is spin-and energy-independent within the interval E E eV , : Let us remark that the demonstration remains valid with an interaction amplitude that depends on spin and energy. Importantly, no assumption about the coupling strength is required here.
In a rotating frame with respect to the energies of the reservoirs and after a Jordan-Wigner transformation using the definitions of the quantum fields R,L ŷ given in equation (22), the Hamiltonian in equation (18)  in order to satisfy the Pauli principle. Equation (25) demonstrates that transport settings can be adequately formulated within the cMPS framework. This result is important as it clears the way for applying methods from cMPS tomography to fermionic quantum transport experiments.

Single energy-level quantum dot
To illustrate the input-output formalism and the cMPS formulation of quantum transport experiments, we consider one of the simplest setups, namely a single energy-level quantum dot, without spin-degree of freedom, weakly coupled to two fermionic reservoirs. Even though this experiment is characterized by Markovian dynamics, this example is of particular interest for this work as it has been widely investigated experimentally. In section 5, we will use real data obtained in [27] for this setup to show that cMPS tomography allows us to access the electronic distribution of waiting times.
This simple transport experiment is sketched in figure 2 and the corresponding Hamiltonian reads Assuming that we perform a measurement on the right of the scatterer, the first two correlation functions of the right quantum field t R ŷ ( ) read in terms of cMPS matrices The matrices R R L correspond to the operators d Inserting these expressions into equation (27), we recover the well-known expression for the steady-state current of a single-level QD coupled to biased reservoirs [39,40] Furthermore, we can derive the noise spectrum from equation (28) This example aims at bridging the gap between a more traditional Hamiltonian and the cMPS formulation, which allows to write these well-known expressions in terms of the parameter matrices Q, T, and R .

Reconstruction of waiting time statistics
In this section, we address the problem of accessing the distribution of waiting times in electronic transport experiments. As mentioned in the introduction, a direct measurement of the WTD in the GHz range is not yet possible due to the lack of single-particle detectors with sufficient accuracy at those frequencies. Here, we propose to reconstruct the WTD based on the experimental measurements of low-order correlation functions. The reconstruction is carried out using the cMPS framework presented in section 2 and the formulation of transport experiments in terms of cMPS as exposed in section 4.

Definitions
The statistics of waiting times can be expressed in terms of the probability density function P 0 , which-as a function of τ-expresses the probability of having detected zero particles in the interval 0, .
t [ ] In terms of P 0 , the WTD has first been derived in the context of quantum transport experiments in [20], Here, t á ñ denotes the mean waiting time. Inserting P 0 t ( ) in cMPS terms (equation (16)), we arrive at an expression for  in terms of the cMPS matrices D,  and Z defined in equations (8) and (15), The normalization factor c 0 > ensures that d 1.

Results based on experimental data
We demonstrate our novel approach to derive the WTD from the measurement of correlation functions using experimental data obtained in [27] for spinless electrons tunnelling through a single-level quantum dot. This system is also known as a single-electron transistor at the nanoscale and has been discussed in section 4.2. The experiment in [27] has been carried out in the kHZ frequency range, where a time-resolved measurement of the current trace is possible. Although all the statistics-including correlation functions of arbitrary order as well as the WTD-can directly be computed from this time-resolved current trace, this experiment provides an ideal test-bed for our proposal. We can compare the WTD obtained from our reconstruction scheme based on cMPS with the WTD directly deduced from the experimental current trace. Due to the simplicity of the setup, our proposed method to access the WTD only requires the two-point function C 2 . This one can directly be derived from the experimental spike train I (the time-resolved current trace) and is shown in figure 3 (red dots). The rates 13.23 kHz L G = and 4.81 kHz R G = have been determined experimentally and the corresponding C 2 -function agrees very well with the analytical expression when the detector rate is taken into account [27] C 1 e . 34 In our reconstruction scheme, the quantity L R G + G can be determined from the current spike train autocorrelation function I I  by least squares methods or spectral estimation procedures analogous to the procedure described in [11]. By requiring L R G > G and using the expression of the steady-state current (see The differences to the values from [27] are well within the range we would expect, regarding the time-resolution in the spike train data. The curve plotted from these reconstructed values of the parameters L,R G is shown in figure 3 in blue. The slight deviation between the experimental points and this reconstructed C 2 -function is due to the discretization of the counting time intervals used in the experiment: the size of each time bin is not much smaller than the time scale on which C 2 changes mostly. This leads to an error in the estimation of the damping factor L R G + G and explains the difference of the blue and the red dotted curves. Naturally, one could expect a more accurate reconstruction of the parameters L G and R G when increasing the time resolution of the current trace or of the measurement of C 2 . From L G and , R G the corresponding cMPS matrices R L and R R can be constructed, as well as the matrices M and D. In this simple case, we did not need to employ the whole reconstruction procedure from [11]. Indeed, it is clear from equation (34) that only two out of the four parameters that characterize the system appear: C n only depends on the tunnelling rates L G and R G -the eigenenergies 0 and ε of H syŝ do not contribute 5 . This will in general not be the case. The matrices R L and R R give access to the matrices D and  by direct computation. Inserting the latter into equation (33), the WTD can be reconstructed and the result is plotted in figure 4 (blue curve). In order to build confidence in our procedure, we compare this result with the experimentally accessible WTD (red dots). Let us recall that the transport rate is in the kHz range, hence the WTD can directly be extracted from the current spike train I: by sorting, counting all (discrete) waiting times between two consecutive incidents, and subsequently normalizing the resulting histogram, one obtains the red-dotted WTD in figure 4. The slight deviation between the WTD reconstructed via our proposal and the experimental one is again due to the discretization of the counting time intervals. One could expect a more accurate reconstruction of the WTD when increasing the time resolution of the current trace.
The WTD in figure 4 shows elementary transport properties of single independent fermions that cannot tunnel at the same time through the single-level quantum dot, which is consistent with the fact that 0  t  ( ) for 0. t  It is important to emphasize that it is the first time a WTD is extracted from experimental data, therefore bridging the gap between theoretical predictions and experiments. The good agreement of the two curves demonstrates the potential of our cMPS-based reconstruction procedure to access the WTD from the measurements of low-order correlation functions. This opens the route to access the WTDs in the highfrequency domain from low-order correlation-functions measurements. . WTD obtained from state-of-the-art experimental measurements with data from [27]. The reconstructed WTD using equation (33) is shown in blue. It matches the WTD obtained directly from the time resolved experimental current trace (red dots) well. The deviation is due to the finite-sized time bin corresponding to the resolution of the current trace. A more accurate reconstruction of the WTD is expected by increasing the time resolution of the current trace or of the measurement of C 2 .

Conclusion
In this work, we have taken an approach motivated by cMPS to perform tomographic reconstructions of quantum transport experiments. On a formal level, we have extended this formalism to perform a reconstruction of unknown dissipative processes based on the knowledge of low-order counting probabilities. We then demonstrated that cMPS is an adequate formalism to describe quantum transport experiments based on tight-binding Hamiltonians.
This work advocates a paradigm change in the analysis of transport experiments. The traditional method is to make explicit use of a model to put the estimated quantities into context, a model that may or may not precisely reflect the physical situation at hand. The cMPS approach is to not assume the form of the model, with the exception that the quantum state can be described by a cMPS. Such an approach is of particular interest as it opens the way to the access of quantities that are not measurable experimentally with current technologies, highorder correlation functions and distributions of waiting times.
To convincingly demonstrate the functioning of cMPS tomographic tools applied to quantum transport experiments, we presented a simple example that consists of electrons tunnelling through a single-level quantum dot. Making use of experimental data, we showed that we could successfully reconstruct the distribution of waiting times from the measurement of the two-point correlation function only. This work constitutes therefore a significant step towards accessing the waiting time distribution in the quantum regime experimentally, a challenge present for several years now. Importantly, the application of our reconstruction procedure goes beyond the interest in WTD: it also provides an access to higher-order correlation functions, which are key quantities to better understand interacting quantum systems.
In subsequent research, it would be desirable to further flesh out the statistical aspects of the problem. After all, the description in terms of cMPS constitutes a statistical model. It would constitute an exciting enterprise in its own right to identify region estimators that provide efficiently computable and reliable confidence regions [44] when considering the problem as a statistical estimation problem, related to the framework put forth in [45][46][47]. We hope that the present work inspires such further studies of transport problems in the mindset of quantum tomography.
here the integrand C n is altered to Note that in contrast to equation (7) where the propagating matrix is the transfer matrix T defined by equation (2) (or equivalently its diagonal representation D), the propagating matrix in the exponential terms between two measurement points now is the matrix S, which is defined by Equations (15) and (7) have a close structural resemblance: the matrices M and  are similar in the linear algebra sense, i.e., there exists a basis transformation from  to M. The matrices D and  are the diagonal matrices of the transfer matrix T and the matrix S, respectively. It is straightforward to transform M and D into  and  and vice versa: by subtracting M from D, we obtain S (up to similarity/basis transformation), whose diagonal matrix is .  Applying the same basis transformation (from D M -( )to ) to the matrix M results in the matrix .  For n = 0, the counting probability function then reads reordered such that it has the form is in general not diagonal. The symmetrized components of  can then be written as r r r r j k l m k j m l , , , , * * + and the constituents r j can be determined (up to a phase factor) by equating them with the coefficients in equation (A.10). The according equation system can then be solved.
The important point is that R rec and Q D Q rec ≔ are valid cMPS parameter matrices in the same gauge and hence are sufficient for reconstruction with the same argument as in [11,III.E]. Let us note that concrete values of the basis transformation matrices X and Y are in fact never used or needed in the reconstruction procedure. From R rec and Q rec , we can compute all quantities we need to establish the correlation and counting probability functions, in particular  and .
 Regauging R rec and Q rec such that the orthonormalization condition [1] is fulfilled, yields a reconstruction of the free Hamiltonian K rec of the ancillary system.