Measuring Entanglement Negativity
Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose an operational scheme to measure the entanglement — as given by the negativity — between arbitrary subsystems of size and , with measurements, and without any prior knowledge of the state. We propose how to experimentally measure the partially transposed moments of a density matrix, and using just the first few of these, extract the negativity via Chebyshev approximation or machine learning techniques. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many body systems in both equilibrium and non-equilibrium regimes.
Entanglement has emerged as a key quantum property in multiple fields. For quantum technologies, it is a resource that allows enhanced sensing rao1945information (), faster computation shor1999polynomial (); harrow2009quantum () and new paradigms of communication bennett1993teleporting (); bennett1992communication (); ekert1991quantum (). In condensed matter, entanglement is essential for understanding the structure of strongly correlated many-body systems amico2008entanglement () and has allowed the development of efficient classical algorithms for simulating many-body systems schollwock2011density (). Despite its paramount importance, only for the very limited case of a bipartition of a pure state can the entanglement, quantified by subsystem entropy, be measured in an efficient and state-independent way horodecki2002method (). There have been multiple theoretical proposals to carry out such a scheme in various physical systems, such as optical lattices daley2012measuring (); alves2004multipartite (), quantum dot arrays banchi2016entanglement () and Gaussian systems weedbrook2012gaussian (). Recently, some of these have also been experimentally realised in simulated spin chains, for example in cold atoms islam2015measuring () and photonic chips pitsios2016photonic (). Nonetheless, pure states are very rare: they are not only difficult to prepare in realistic situations, but also difficult to maintain in the presence of an environment. For example, just consider the entanglement between: (i) two optical modes traversing a fibre, crucial for quantum communication; (ii) spatially separated parts of an extended many-body pure state, important for determining how much long range entanglement can be extracted as a resource reznik2003entanglement () and for shedding light on the entanglement structure of varied many-body models bayat2010negativity (); alkurtass2016entanglement (); gray2017many (); banchi2009finite (); roscilde2004studying (); (iii) two systems in a thermal state – in none of the above cases, ironically, can the entanglement entropy quantify the entanglement. While witnesses HORODECKI19961 (); terhal2000bell (); lewenstein2000optimization () exist for specific forms of entanglement, these are state-dependent and provide only a simple yes/no answer without quantifying the amount of entanglement. It is thus crucial to be able to measure entanglement for mixed states in an experimental setting.
While for pure states bipartite entanglement is uniquely defined by the entropy of the subsystems, this ceases to quantify entanglement for the more generic case of mixed states. Here, the landscape of bipartite entanglement measures is far more complex horodecki2009quantum (); plenio2007introduction (), and aside from isolated special cases such as two qubit states wootters1998entanglement () and bosonic Gaussian states weedbrook2012gaussian (); wolf2004gaussian (), only the (logarithmic) negativity is a computationally tractable quantity zyczkowski1998volume (); lee2000partial (); vidal2002computable (). It has the operational characteristic of being a bound on distillable entanglement and teleportation capacity vidal2002computable (), and is a pivotally important quantity to estimate for both quantum technologies plenio2007introduction (); horodecki2009quantum (); eisler2014entanglement (); lanyon2016efficient (), and condensed matter systems calabrese2012entanglement (). Nonetheless, there is no state-independent observable that can measure the logarithmic negativity, and thus it is impossible to experimentally measure it without reconstructing the state using quantum tomography d2003quantum () - a generically demanding process with an exponential number of measurements in the system size. An exception is for low-entangled states, for which an efficient tomography scheme does exist cramer2010efficient () and has recently been experimentally demonstrated for estimating negativity in quench dynamics lanyon2016efficient (). In this case however, the scheme is limited to short time scales well before the saturation of entanglement growth.
Here, we put forward a practical proposal for accurately estimating the logarithmic negativity in a completely general and realistic setting, using an efficient number of measurements – scaling polynomially with system size. Our method is based on measuring a finite number of moments of a partially transposed density matrix, from which we extract the entanglement negativity using either a Chebyshev functional approximation or machine learning. This represents a new front in the emerging arena of classical machine learning methods applied to quantum estimation and measurement problems carrasquilla2017machine (); carleo2017solving (); schollwock2011density (); hentschel2010machine (); hentschel2011efficient (); aaronson2007learnability (); lloyd2013quantum (); banchi2016quantum (); wang2017experimental (). Moreover, our method is experimentally feasible, since the individual building blocks have already been demonstrated in solid state petta2005coherent (); schuld2015introduction () and cold atoms trotzky2008time ().
Logarithmic Negativity. — Logarithmic negativity zyczkowski1998volume (); lee2000partial (); vidal2002computable () for a generic mixed state quantifies the quantum entanglement between subsystems and . It is defined as:
where denotes the trace norm, is the partial transpose with respect to subsystem , and are the eigenvalues of . Because of the non-trivial dependence of on , there is no state-independent observable that can measure it — one generally needs to reconstruct the full density matrix via state tomography. The are the roots of the characteristic polynomial, , where each coefficient is a polynomial function of the partially transposed moments:
In this way, full information about the spectrum is contained in , with the first few moments carrying the most significance. Indeed, since for all rana2013negative () and , generically, the magnitude of the moments quickly decreases with . We will show firstly that can be measured using copies of the system, and secondly that can be accurately estimated using only , with as low as 3. Note that is simply the dimension of the systems Hilbert space, while in all cases. Additionally, it can be easily shown that is equal to the purity of the state , and as such, is needed to extract any information about .
Measuring the Moments of a Partially Transposed Density Operator. — We now show that the moments in Eq. (2) can be measured via copies of the state , namely . This general set-up is shown in Fig. 1(a), where the mixedness of arises from possible entanglement with a third system , such that with being a pure tripartite state. The first step is to write the matrix power as an expectation of a permutation operator, similar to Ref. ekert2002direct (); horodecki2002method (), but here on the partially transposed copies:
where is any linear combination of cyclic permutation operators of order and the second line makes use of the identity , valid for any operator . A schematic of the equality in Eq. (3) for is shown in Fig. 1(b). In the appendix we provide a choice of with a neat operational meaning, both for spin and bosonic systems. For spin lattices, our choice of to measure the moments results, in practice, to the following steps: (i) prepare copies of the state ; (ii) sequentially measure a ‘forward’ sequence of adjacent swaps, between neighbouring copies of system from to ; (iii) sequentially measure a ‘backward’ sequence of adjacent swaps, between neighbouring copies of system from to ; (iv) repeat these steps in order to yield an expectation value. This procedure is also depicted for in Fig. 1(a). For bosonic lattices, our procedure corresponds to the following steps: (i) prepare copies of the state ; (ii) Perform ‘forward’ Fourier transforms between modes in different copies for each site in – this can be achieved using a series of beam splitters reck1994experimental (); (iii) Perform ‘backwards’ (reverse) Fourier transform between modes in different copies for each site in , via reverse beam splitter transformations; (iv) Measure the boson occupation numbers on all sites and all copies to compute . (v) Repeat these steps to obtain the expectation value as an average of . Both procedures require measurements for each between and , and are explained in detail in the appendix.
It is worth emphasizing the difference between our procedure, and recently proposed operational methods for measuring Renyi entropies daley2012measuring (); banchi2016entanglement (); abanin2012measuring (). First of all, Renyi entropies only quantify entanglement for pure states, and cannot be used in the more general mixed state scenario. Secondly, while for entropies the operations are only performed on a single subsystem, here, one performs both ‘forward’ and ‘backward’ operations on two subsystems at once, as explained above. Remarkably, even though partially transposed density matrices are generically un-physical, measurement of their moments is possible. Indeed, we note a previously suggested scheme involving multi-qubit controlled generalized swaps carteret2005noiseless () – an experimentally challenging prospect in many-body systems. Here, thanks to the forward and backward operations outlined above, measurement of the moments ought to be possible in current experiments.
Estimating Entanglement. — In this section, we demonstrate two different methods for estimating the logarithmic negativity from the information contained in the moments, . The first method is based on functional approximation, and the second on machine learning.
Logarithmic negativity, , is a function of the eigenvalues , as shown in Eq. (1). One could try to directly reconstruct the main features of the spectrum using a few measured moments – an approach closely related to general Hausdorff moments problem in statistics mead1984maximum (). However, it is known from a numerical perspective that this problem is unstable viano1991solution (). To avoid such instabilities, we take an alternative approach based on functional approximation. Considering that with , if we can find a polynomial expansion , then by linearity of the trace, , with as given in Eq. (2). In other words, given a polynomial expansion of the absolute function, , – i.e. the coefficients – one can approximate the entanglement using a finite number of moments. A naive choice for this would be a Taylor expansion, but the non-analyticity of at prevents convergence. On the other hand, a nearly optimal choice for approximating a function throughout an interval rather than around a point, is a Chebyshev expansion han2016approximating (). On the interval , this yields where the Chebyshev polynomials are known -th order polynomials. The coefficients are given, via the orthogonality of , as , where are the Chebyshev nodes. When is defined on a different interval, one can simply linearly transform the Chebyshev points and polynomials. Although in principle the spectrum of a generic state lies between and , in practice it is often much more tightly clustered. Decreasing the window size significantly improves the approximation for a fixed . Therefore, we need to find the minimal sized window such that all of the spectrum of a given is contained. A tight guess for such a window can be found since when is even, with being the eigenvalue with largest absolute value. Thus in our numerics, we define the window as , with . The quality of the Chebyshev approximation rapidly increases with and becomes exact in the limit .
An alternative approach for computing negativity from only the even moments has been proposed in quantum field theory literature calabrese2012entanglement (); de2015entanglement () by exploiting numerical extrapolation. However, this method neglects the odd moments which are utilised in the Chebyshev expansion. These indeed carry additional information that can improve the accuracy of the logarithmic negativity estimation in the systems studied here.
The above Chebyshev expansion is analytically tractable, and becomes very accurate for large enough , as we will see in later numerical results. Nonetheless, this expansion is based on a linear mapping between the moments and the negativity, although this relationship is inherently non-linear. Therefore it is natural to think that a non-linear transformation could be more optimal, and thus more efficient for smaller – namely fewer copies. Machine learning has recently emerged as a key tool for modelling an unknown non-linear relationship between sets of data. In the supervised learning paradigm, one trains a model with a set of known inputs and their corresponding outputs. Once trained, the model can then be used to predict the unknown output of new input data. Here, we take the moments as the input and the logarithmic negativity as the output. Training is performed by taking a large set of states for which and can be computed on a classical computer. This model can then be used to predict from a set of experimentally measured moments. The experimental system under study motivates the choice of which training states to use, so that they share, for example, similar entanglement features. Among the most successful machine learning algorithms for non-linear regression are supervised vector machines cristianini2000introduction (), random decision forests ho1998random (), and deep neural networks rojas2013neural (); schmidhuber2015deep (). However, we have found that using the same training set for each, neural networks are superior when it comes to predicting logarithmic negativity for a wide range of states beyond the training set. As we show with our numerical results, neural networks provide a very accurate method for extracting the logarithmic negativity with as few as copies. The details of our neural network construction can be found in the appendix.
Entanglement Features of Random States. — In order to understand the capabilities and limitations of the above two methods, in this section we consider the entanglement properties of relevant states in relation with the spectral properties of . This will also inform our choice of states for numerical study. As mentioned before, we can always think of the mixed state as being the reduced density matrix of a tripartite pure state . Without loss of generality, we take each to be qubit system of , and sites respectively – a geometry shown in Fig. 1(a).
From an entanglement perspective, relevant states in condensed matter physics can be classified as either area-law, or volume-law. In the first case, the entanglement of a subsystem with the rest is proportional to the number of qubits along their boundary. In the second, this entanglement is instead proportional to , the number of qubits in . Area-law states arise as low energy eigenstates of local gapped Hamiltonians, with logarithmic corrections in critical systems. Volume-law states however, are associated with the eigenstates found in the mid-spectrum, and as such arise in non-equilibrium dynamics, e.g. quantum quenches barmettler2009relaxation (); nanduri2014entanglement ().
Rather than concentrate on a specific model system, we initially consider the very general case of random states. To encompass both area- and volume-law states, we consider two classes of states : (i) random generic pure states (R-GPS), e.g. sampled from the Haar measure, which typically have volume-law entanglement popescu2006entanglement (); hamma2012quantum (); (ii) random matrix product states (R-MPS) with fixed bond dimension, which satisfy an area-law by construction schollwock2011density (). Further details for generating these states are given in the appendix. These two classes of random states yield very different eigenvalue distributions of , and, as such, different sets of moments . A detailed analysis of the above is provided in the appendix. The difference in the distribution of affects our entanglement estimation analysis. In the Chebyshev expansion, it sheds light on the source of errors and provides insight into the required number of copies for an accurate estimation. On the other hand, as far as machine learning is concerned, it implies that the model must be trained with states from both classes, as well as varying choices for , and , in order to predict entanglements for a wide range of quantum states.
Numerical Results for Random States. — In this section we numerically study the performance of the two methods introduced above – Chebyshev approximation and machine learning – for extracting logarithmic negativity, , from a finite set of moments, . In order to keep the discussion model independent, we analyse random states , taking the two classes also discussed above, namely, R-GPS and R-MPS, representing volume- and area-law states respectively.
In Fig. 2(a) we plot the relationship between the real logarithmic negativity, , and the approximated value, , calculated using the Chebyshev expansion for copies. In order to generate states with a wide range of entanglement features, subsystem sizes, and mixedness, we perform the following procedure. (i) For a fixed number of qubits , take either a R-GPS, or R-MPS with bond dimension . (ii) Take different tri-partitions such that , and for each calculate and for . (iii) Repeat for different random instances, while separately varying and . As the figure shows, typically underestimates , especially for MPS states. As we discuss in the appendix, we attribute this to the fact that the distribution of is peaked around zero, particularly for MPS, where the Chebyshev approximation error is always negative. In the inset we plot a histogram of the errors which clearly shows this negative bias. As shown in Fig. 2(b) and its inset, by doubling the number of copies to , the accuracy of this method is significantly improved.
From an experimental point of view however, it would be highly desirable to use as few copies as possible, both in terms of preparation, and ease of moment measurement, without sacrificing the accuracy of the estimation. We now show that machine learning can indeed satisfy both of these requirements. We take the same set of states as above, but split the data into two halves, one for training the neural network model, and the other as ’unseen’ test data. In Fig. 3(a) we plot the machine learning model’s predictions, , for the test data, using only copies, in which a high degree of accuracy is achieved. In the inset of Fig. 3(a), we plot a histogram of the errors , which displays a very sharp peak at zero error. A further improvement, particularly in outliers, can be achieved by increasing the number of copies to 10, as shown in Fig. 3(b). However, the machine learning method is already very accurate for extracting logarithmic negativity, using as few as three copies. As with the Chebyshev technique, the machine learning approach works particularly well for large bond dimension and volume-law like states – an important fact given that these are the exact cases where efficient tomography fails.
Numerical Results for Physical States. — In order to show that the techniques introduced here transfer well to a more realistic setting, we now consider the case of the quench dynamics of a many-body system. We take a system of spin-1/2 particles with the nearest neighbour Heisenberg Hamiltonian where is the exchange coupling strength and is the vector of Pauli matrices acting on site . The system is initialised in the (separable) Neel-state . As the chain unitarily evolves in time as , it becomes entangled, with an effective MPS description whose bond dimension increases until the state is essentially volume-law barmettler2009relaxation (); nanduri2014entanglement (). In a recent experiment using ion traps lanyon2016efficient (), very similar evolution has been implemented, for which the negativity between different spin blocks has been quantified using an efficient form of state tomography cramer2010efficient (), only valid for low-entangled states. As such, the negativity could only be computed for very short times () while the state was still area-law like. Using our scheme, is tractable for all times, and has no trouble characterising the highly entangled states that naturally arise at long time scales.
In Fig. 4 we plot the evolution of and three approximation methods, as functions of time for four different choices of subsystems. The three methods are the Chebyshev approximation with and and machine learning with , with respective approximate entanglements , and . In Fig. 4(a), we consider a specific partition with small subsystems, namely two neighbouring spins in a chain of length . The logarithmic negativity displays oscillatory dynamics, which is well captured by all three approximation methods throughout the evolution. In Fig. 4(b) we show a slightly larger case where the machine learning and Chebyshev approximation methods are comparable. For larger subsystems, the dynamics are quite different, with saturating after some time, as shown in Fig. 4(c) and (d). Here, the machine learning approximation, using only copies, significantly outperforms the Chebyshev approximations, which use either or copies. Nonetheless the Chebyshev approximation still captures all of the qualitative features of the evolution and improves by increasing the number of copies . It is remarkable that despite being trained on a arbitrary set of random states with no knowledge of the underlying physical system, the evolution of is accurately captured by the neural network estimator for all partitions and times, with as few as copies.
Conclusions. — The measurement of bipartite entanglement in generic multi-particle mixed states has so far relied on the complete reconstruction of a quantum state, which in general requires an exponential number of measurements, and is thus limited to small system sizes. In this work, we have devised an alternative strategy based on measuring a finite number of moments of the partially transposed density matrix, using a few copies of the system.
From the experimental perspective, in a spin system, the moments can be obtained by measuring two counter-propagating series of swap operators – techniques for achieving this have already been demonstrated in a number of physical set-ups ranging from quantum dot arrays petta2005coherent (); schuld2015introduction () to cold atoms in optical lattices trotzky2008time (). In a bosonic set-up the moments can be obtained by performing Fourier and inverse-Fourier transformations among different modes, e.g. by using a series of beam splitter transformations, followed by particle counting – similarly to what has been used in a recent experiment in optical lattices islam2015measuring ().
The logarithmic negativity is then estimated from these moments using either Chebyshev functional approximation or machine learning methods. The total number of measurements required in our scheme scales only polynomially in the system size. Additionally, the precision of the estimated logarithmic negativity can be arbitrarily enhanced by increasing the number of copies. For the machine learning method, we trained a neural network using random states with no further assumptions regarding the underlying physics. Remarkably, this method is already very accurate for as few as three copies – making it very resource efficient and desirable for practical applications – even for estimating the entanglement of highly entangled physical states, such as those arising in quantum quenches.
JG acknowledges funding from the EPSRC Centre for Doctoral Training in Delivering Quantum Technologies at UCL. SB acknowledges the EPSRC grant EP/K004077/1. LB, AB and SB acknowledge financial support by the ERC under Starting Grant 308253 PACOMANEDIA.
.1 Measuring Moments in spin- Systems
Here we show that the operational procedure described in the main text allows us to measure the moments for spin-1/2 systems. Let be the operator that swaps copies and on subsystem , which can be written as where . The projective measurement of corresponds to a singlet triplet measurement (ST-measurement) between spins sitting at the same site , but different copies and . Indeed, has an outcome for the singlet state and for the triplet states. In view of this we write where correspond to the eigen-projections with corresponding eigenvalues .
We now consider the case and then generalise to arbitrary . We first perform a sequential set of ST-measurements on copies , with outcome and then do the same measurement on copies , with outcome . We introduce the notation to describe this process. After the first measurement, the (non-normalized) state of the system will be , while after the two sets of measurements it is . Therefore,
where we used the identity .
We now generalize the above argument for higher values of . We apply sequential ST-measurements on neighbouring copies, using the notation , meaning that we first perform and so forth. Taking the averages one then finds that
where . We define the operators recursively as: . Then, using the cyclic property of the trace, one finds that where the are different cyclic permutation of the elements . For instance, for one has . In view of the above, this sequential set of ST-measurements corresponds to the measurement of the operator .
In summary, when is equal to both and , one obtains the operator, defined in Eq. (2). As proven above, such an operator is defined according to the following recursion relations banchi2016entanglement (): Moreover, the effect of the partial transpose on the recursion relation is as follows: We take as an example:
From which it can be seen, as described in the main text, that the order of measurements on and is reversed. Indeed, for , the ST-measurement is performed between copies 1 and 2, then 2 and 3, whereas for , the ST-measurements is performed for copies 2 and 3, then 1 and 2. Because of the non-commutative nature of these measurements, this ordering is crucial in order to yield moments of the partially transposed state as in Eq. 2.
.2 Measuring Moments in Bosonic Systems
We show here how to measure the moments as given in Eq. (2) for a bosonic system. Unlike for the case of spin systems, we directly choose the operator as a product of specific, non-hermitian, permutations, , such that
where , and labels the number of bosons in copy and physical site . We can write this operator in second quantized form as
where denotes the normal ordering of the operator , and denotes the annihilation operator acting on site and copy . We choose as the shift permutation such that . Note that . In order to diagonalise we introduce the Fourier transform, which acts independently on each site as
The expectation value in Eq.(2) can thus be measured in three steps. First, perform the Fourier (inverse Fourier) transform between copies at the sites belonging to (), as written in Eq. (12). Second, measure the bosonic occupation number with outcome at every site and compute the outcome of the permutation operator as . Finally, compute the expectation value as the average over many repeatations of the above steps: .
.3 Generation of random states
Random generic pure states R-GPS have been obtained by generating random vectors with complex elements distributed according to the normal distribution. R-GPS obtained by sampling from the Haar measure have also been considered, though they are numerically more demanding. Nonetheless, they provide the same results. Random Matrix Product States, R-MPS, have been obtained by writing for random tensors , where , , and , being the bond dimension, with complex elements drawn from a normal distribution.
.4 Chebyshev Approximation Error analysis
In this section we study in detail where the error in the Chebyshev approximation stems from and how it is related to the spectral properties of . It is convenient to consider the thermodynamic limit (), where one can describe the spectrum of as a probability density function, , and write the logarithmic negativity of Eq. (1) as
Therefore, this underlying function determines , though it is not directly accessible from measurements of the moments . Nonetheless, from a theoretical perspective, the study of for certain classes of states yields insight into the performance of the Chebyshev approximation and machine learning approaches.
For R-GPS, , as proven in aubrun2012partial (); fukuda2013partial (), the spectral distribution tends towards a shifted Wigner semi-circle law in the limit of large , and . This is described by the continuous distribution where and , and . An instance of the spectral distribution is shown in Fig. 5(a) for finite values of , and , where one can already see a clear semi-circular shape. As is also evident from the figure, the support of the distribution is far smaller than the theoretical interval , yet the bound established above as quite tightly captures the real interval.
On the other hand, random states constructed using a MPS ansatz schollwock2011density () with fixed bond dimension – which inherently obey an area-law – show a significantly different distribution , with a high concentration around 0 but long tails on either side. This can be seen in Fig. 5(b) for a single R-MPS instance of . Nonetheless, the support of this type of distribution is even more tightly bounded by .
In this thermodynamic limit the Chebyshev approximation can be understood from Eq. (15), as . The support of for the two classes of states discussed above is typically very different, being much wider for R-MPS. The effect of this wider range can be seen if we consider the error in the Chebyshev approximation, , as a function of . By construction this error is spread roughly throughout the interval, with alternating sign. In the case of for R-MPS, the peak at zero, together with the large support, concentrate a large number of eigenvalues into a small region with the same signed error. This gives a negative bias that does not exist when is more even throughout the interval, as for random pure states. Therefore, Chebyshev methods are expected to have a larger error for area law states.
.5 Extracting Negativity with Neural Networks
In deep neural networks, the unknown function mapping inputs to outputs is approximated with a directed graph organized in layers, where the first layer is the input data and the last one is the output. In our case, the input data consists of the numbers , namely, the number of spins in each subsystem and the non-trivial moments. The value of the -th node in layer is updated via the equation where is an appropriate (typically non-linear) activation function and is the weight between node in layer and node in . The training procedure consists in finding the optimal weights by minimizing a suitable cost function.
In our numerical investigations, we use the Hyperopt bergstra2013making () and Keras chollet2015keras () packages to find the optimal network structure, including the number of hidden layers. For example, for , the resulting network consists of two hidden layers, both with rectified linear unit (ReLU) activation functions, with and neurons respectively. For however, the resulting network consists of three hidden layers, with exponential linear unit (ELU), ReLU and linear activation functions, with , and neurons respectively.
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