Scalable Reconstruction of Density Matrices

Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The reconstruction scheme only requires local information about the state, giving rise to a reconstruction technique that is scalable in the system size. It is based on a constructive proof that generic matrix product operators are fully determined by their local reductions. We discuss applications of this scheme for simulated data and experimental data obtained in an ion trap experiment.

Figure 1

Definition of the sets $\mathcal{N}=\{1,\dots ,N\}$ and $\mathcal{I}$, $\mathcal{J}\subset \mathcal{N}$. The linear map $E_{\mathcal{I}}^{\mathcal{J}}(\widehat{X})={\mathrm{tr}}_{\mathcal{J}}[\widehat{X}{\widehat{O}}_{\mathcal{I}\cup \mathcal{J}}]$ maps operators $\widehat{X}$ (e.g., observables) living on set $\mathcal{J}$ into operators on set $\mathcal{I}$ by means of the reductions of $\widehat{O}$ (e.g., the state) to $\mathcal{I}\cup \mathcal{J}$.

Figure 2

Quality of our reconstruction scheme for thermal states of the Ising Hamiltonian in Eq. (9) for $\beta =5$ and $R=5$, i.e., the state is reconstructed from local expectation values on five consecutive sites. For each pair ($N$, $\sigma$), the plot shows the mean of the norm difference obtained from 100 realizations and renormalized by the purity of the exact state, i.e., $D(\widehat{\varrho },{\widehat{\varrho }}_{\mathrm{rec}})=\Vert {\widehat{\varrho }}_{\mathrm{rec}}-\widehat{\varrho }{\Vert }^2/\Vert \widehat{\varrho }{\Vert }^2$. This corresponds to 100 experiments, each of which carries an uncertainty of $\sigma$ about the local expectation values.

Figure 3

Absolute value $|{\widehat{\varrho }}_{\mathrm{rec}}|$ of the corresponding reconstructed density matrix of the experimentally realized $W$ state. (a) Reconstructed operator using the scheme described in this Letter where the reductions to $R=3$ sites are known. (b) Estimate with $R=5$ sites. (c) Maximum likelihood estimate of full quantum state tomography (see also Ref. 2). The numbers $1,2,\dots ,2^N$ denote the entries of the density matrix ${\widehat{\varrho }}_{\mathrm{rec}}$.