Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states

We develop a general theoretical framework for measurement protocols employing statistical correlations of randomized measurements. We focus on locally randomized measurements implemented with local random unitaries in quantum lattice models. In particular, we discuss the theoretical details underlying the recent measurement of the second Rényi entropy of highly mixed quantum states consisting of up to ten qubits in a trapped-ion quantum simulator [Brydges et al., Science 364, 260 (2019)]. We generalize the protocol to access the overlap of quantum states, prepared sequentially in one experiment. Furthermore, we discuss proposals for quantum state tomography based on randomized measurements within our framework and the respective scaling of statistical errors with system size.

Figure 1

Measurement of the second Rényi entropy using statistical correlations between randomized measurements. A quantum state $\rho$ of interest is prepared for instance via quench dynamics (see also Ref. [16]). A randomized measurement on a subsystem $A$ is performed by the application of (a) a global random unitary $U_A$ from a unitary 2 design on the entire Hilbert space or (b) a product of local random unitaries $U_A={⨂}_{i\in A}{U}_i$ sampled independently from a unitary 2 design on $\mathcal{h}$. Subsequently, a measurement in the computational basis is performed. From statistical correlations of the outcomes of such randomized measurements, the purity ${\mathrm{Tr}}_{}\left[{\rho }_A^2\right]$ of the reduced density matrix ${\rho }_A$ is estimated (see text).

Figure 2

(a) Graphical visualization of a pure ${\rho }_1$ (green) and a mixed ${\rho }_1$ (purple) singe qubit state with Bloch vectors (arrows) ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$, respectively. Points correspond to 50 randomly rotated states, generated via the application of random unitaries sampled from the CUE [40] to ${\rho }_1$ and ${\rho }_2$, respectively. (b) Histogram of the random variable $Z_U={\mathrm{Tr}}_{}\left[U\rho U^{†}{\sigma }_z\right]$ for pure ${\rho }_1$ (green) and mixed ${\rho }_2$ (purple) state, the indicated standard deviation (multiplied with a factor $\sqrt{3}$) corresponds to the length of the Bloch vectors.

Figure 3

(a) Graphical visualization of a pure, random (entangled) two-qubit state. The red arrows visualize the Bloch vectors $\mathbf{v}$ and $\mathbf{w}$ of reduced (mixed) single qubit states. Blue arrows correspond to the left $\sqrt{{\gamma }_i}{\mathbf{m}}_i$ and right $\sqrt{{\gamma }_i}{\mathbf{n}}_i$ singular vectors, rescaled with singular values ${\gamma }_i$, of the singular value decomposition of $R$ ($i=1,\cdots ,3$) [39]. The measurement of ${\sigma }_z\otimes {\sigma }_z$ after the application of a unitary $U_1\otimes U_2$ (rotations $Q_1$ and $Q_2$ of qubit 1 and 2) is visualized as a projection of the singular vectors onto the $z$ axis, its expectation value is given as ${\sum }_i{\gamma }_i{\left(Q_1{\mathbf{m}}_i\right)}_3{\left(Q_2{\mathbf{n}}_i\right)}_3$. (b) Histograms of random variables $Z_U^{\left(1\right)}={\mathrm{Tr}}_{}[U\rho U^{†}{\sigma }_z\otimes {\mathbb{1}}_2], Z_U^{\left(2\right)}={\mathrm{Tr}}_{}[U\rho U^{†}{\mathbb{1}}_2\otimes {\sigma }_z]$, and $Z_U^{\left(12\right)}={\mathrm{Tr}}_{}[U\rho U^{†}{\sigma }_z\otimes {\sigma }_z]$ generated using random unitaries of the form $U=U_1\otimes U_2$. The standard deviation (multiplied with a factor $\sqrt{3}$ for left and right and 3 for the middle panel) corresponds to the length of the Bloch vectors $\left|\mathbf{v}\right|$ and $\left|\mathbf{w}\right|$ (left and right) and the Hilbert-Schmidt norm $\parallel R\parallel$ of the correlation matrix $R$ (middle), see text.

Figure 4

Graphical dictionary: (a) Elementary diagrams, describing ket vectors [(0,1) tensors], bra vectors [(1,0) tensors], and operators [(1,1) tensors] on a single copy. (b) Graphical representation of permutation operators $\left[\right(k,k)$ tensors] acting on the $k$-fold copy space.

Figure 5

Bell states and the transpose matrix: (a) Definition of Bell states [(0,2) and (2,0) tensor] on two copies. (b) Definition of a diagram connecting two decorations of the same color, in the analogous way the diagram connecting to white decorations is defined. (c) Visualization of the transposed matrix using Bell states (see text).

Figure 6

Graphical evaluation of ${\mathrm{\Phi }}^{\left(2\right)}$. We evaluate graphically the matrix element $\langle i|\langle j|{\mathrm{\Phi }}^{\left(2\right)}\left(O\right)|i^{\prime }\rangle |j^{\prime }\rangle$. For clarity the boxes describing the basis states $\langle i|\langle j|, |i^{\prime }\rangle |j^{\prime }\rangle$ have been omitted, only their decorations are kept.

Figure 7

Graphical evaluation of ${\mathrm{\Phi }}_2^{\left(1\right)}$. We evaluate graphically the matrix element $\langle ij|{\mathrm{\Phi }}_2^{\left(1\right)}\left(U\right)|i^{\prime }{j}^{\prime }\rangle$ where only decorations of the same type, corresponding to the same random unitary, are connected. For clarity the boxes describing the basis states $\langle ij|, |i^{\prime }{j}^{\prime }\rangle$ have been omitted, only their decorations are kept.

Figure 8

Statistical errors of the estimated purity of a pure product state (PPS) and a random mixed state (RMS). (a) and (b) The statistical errors of the estimated purity using (a) global unitaries and (b) local unitaries in a system of $N_A=8$ qubits, as a function of $N_U$ for $N_M=16$ (dots) and $N_M=256$ (crosses). Hilbert space dimension is $\mathcal{D}=2^8$. (c) Global unitaries and (d) local unitaries, the number of unitaries $N_U=512$ is fixed, and the statistical error is shown as function of the number of measurements $N_M$. The Hilbert space dimension (number of qubits $N_A$) increases with darkness of the colors, ${\mathcal{D}}_A=2^4,2^6,2^8$. Solid lines are calculated from the given scaling laws. Random unitaries are sampled directly from CUE [40]. The RMS of $N_A=4,6,8$ qubits has been obtained by applying a Haar random unitary to a pure product state consisting of 12 qubits, and tracing out the residual 8,6,4 degrees of freedom.

Figure 9

Statistical errors of the reconstructed density matrix for a pure product state (PPS) and a random mixed state (RMS). (a) and (b) The average deviation of the reconstructed density matrix using (a) global unitaries and (b) local unitaries in a system of $N_A=8$ qubits (${\mathcal{D}}_A=2^8$), as a function of $N_U$ for $N_M=256$ (dots) and $N_M=\infty$, i.e., no projection noise, (crosses). (c) Global unitaries and (d) local unitaries, the number of unitaries $N_U=512$ is fixed, and the average statistical error is shown as function of the number of measurements $N_M$. The Hilbert space dimension (number of qubits $N_A$) increases with darkness of the colors, ${\mathcal{D}}_A=2^4,2^6,2^8$. Solid lines are calculated from the given scaling laws. Random unitaries are sampled directly from CUE [40]. The RMS of $N_A=4,6,8$ qubits has been obtained by applying a Haar random unitary to a pure product state consisting of 12 qubits, and tracing out the residual 8,6,4 degrees of freedom.