Entanglement Negativity at Measurement-Induced Criticality

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their “mutual negativity” and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of “hybrid” circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension $3$ across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

Popular Summary

One key feature of quantum systems is entanglement, which involves correlations between two or more microscopic particles generated after they interact. Being fundamentally different from any classical correlation in the macroscopic world, quantum entanglement gives rise to many of the peculiarities of quantum mechanics. Quite generally, an isolated quantum system will become highly entangled, due to the interactions between its constituent particles. However, recently researchers discovered that such proliferation of entanglement can be hindered by frequently measuring the system. In particular, the amount of entanglement in the system will go through a sudden change when the frequency of measurement exceeds some threshold, a phenomenon named measurement-induced criticality (MIC).

In this work, we study detailed aspects of entanglement at MIC, with an emphasis on distinguishing bipartite and multipartite entanglement. We analyze a series of models with progressive levels of complexity, and identify “generic” entanglement properties shared by the models at MIC.

Figure 1

Illustration of the decomposition of a pure stabilizer state $|{\mathrm{\Psi }}_{Q=A\cup B\cup C}⟩$ in Eq. (11) for the tripartition $(A,B,C)$. For the ease of graphical presentation, we choose $A$, $B$, and $C$ to be contiguous segments; however, the state decomposition works generally for an arbitrary tripartition, and we are mostly interested in the case when $A$ and $B$ are two contiguous, distant segments separated by $C=\overline{A\cup B}$ [see inset of Fig. 3].

Figure 2

(a) An instance of the measurement-only circuit (Sec. 3) in the Majorana-loop representation while (b) is the same circuit in the qubit representation. In both figures the red part represents the initial product state as defined in Eq. (15), and each rectangular corresponds to an operation happening at that space-time location, which can either be an identity operator [see Eq. (17)] or a measurement [see Eq. (18)]. The possible operations in two representations are one to one related by the Jordan-Wigner transformation, as shown in (c). In (a), the output state can be determined by the pairing pattern of the endpoints on the upper boundary. By “fusing” Majorana pairs on odd bonds (dashed lines), pairing arcs form several disjoint loops with disjoint supports, each of which corresponds to a GHZ cluster in the output state in qubit representation. The Majorana loops are cluster boundaries in percolation, and we provide several examples in (d)–(f). Each tick on the upper boundary represents a qubit. Two qubits on the boundary (highlighted $A$ and $B$ here) belong to the same GHZ cluster if and only if they are connected to the same percolating cluster (colored regions) of the bulk. In (d), $A$ and $B$ are disconnected, and $\frac{1}{2}{I}_{A,B}=N_{A,B}=0$. In (e), $A$ and $B$ belong to the same GHZ cluster, which also touches the boundary at other qubits, and $\frac{1}{2}{I}_{A,B}=\frac{1}{2}\ln 2$, $N_{A,B}=0$. In (f), $A$ and $B$ belong to the same GHZ cluster, which does not contain other boundary qubits, and $\frac{1}{2}{I}_{A,B}=N_{A,B}=\ln 2$.

Figure 3

Numerical simulation results for the measurement-only circuit with periodic boundary condition. (a) The equilibrium length distributions of Majorana arcs and stabilizers. (b) Mutual information and mutual negativity between two disjoint, distant intervals $A$ and $B$ (separated by $C=\overline{A\cup B}$) as a function of the cross ratio $\eta =w_{12}{w}_{34}{w}_{13}^{-1}{w}_{24}^{-1}$.

Figure 4

(a) A circuit instance of the completely packed Majorana-loop model with crossings, and (b) its counterpart in the qubit representation after the Jordan-Wigner transformation.

Figure 5

The phase diagram for the completely packed Majorana loop model with crossings (CPLC), which is composed of a Goldstone phase, an topological insulating phase, and a trivial insulating phase. Red dots mark the points that we studied numerically in Sec. 4a. The way we identity Majorana pairs into qubits breaks the full lattice translation symmetry as well as the equivalence of the two short loop phases. Particularly, the loop configurations correspond to points $(p,q)=(0,0)$ and $(p,q)=(0,1)$ have the same bulk pattern, while the $(p,q)=(0,1)$ one has an additional large macroscopic loop (“edge mode”) on the its boundary.

Figure 6

(a) The arc length distribution $P_{\mathrm{arc}}$ for CPLC model in the Goldstone phase $(p,q)=(0.5,0.5)$. Here we fit $P_{\mathrm{arc}}$ in two ways, and notice that $K(\ell ){\ell }^{-2}$ [see Eqs. (41), (42)] works slightly better than ${\ell }^{-2}$ for the system size $L=2^{15}$ we have here. Their difference is made clearer in (b), where we plot the $L$ dependence of $K(\ell =L/8)$ at different points in the phase diagram. Within the Goldstone phase at $(p,q)=(0.5,0.5)$, $K(\ell =L/8)$ depends logarithmically on $L$. While on the critical line $(p,q)=(0.5,0.823)$, $K(\ell =L/8)$ is a universal constant proportional to the critical spanning number [46] [see also Eq. (44)].

Figure 7

Mutual information and mutual negativity between two disjoint segments $A$ and $B$ for several choices of $(p,q)$ in the Goldstone phase of the CPLC model.

Figure 8

Numerical results for CPLC model on critical lines. (a) Entanglement entropy of a contiguous segment $A=[x_1,x_2]$ within the Goldstone phase and on two critical lines. (b) Mutual information and mutual negativity between two disjoint segments $A$ and $B$ on the Goldstone-trivial critical line. (c) Mutual information and mutual negativity between two disjoint segments $A$ and $B$ on the Goldstone-topological critical line.

Figure 9

An instance of the hybrid random circuits. Two-site unitary gates (blue box) are arranged in a brick-wall pattern. On each vertical link a single-site measurement of $X$ (green circle) may be applied independently with probability $p_{\mathrm{meas}}$. The unitaries are either sampled uniformly from the two-qubit Clifford at $d=2$ [results in Fig. 10], or from the Haar measure on the two-qubit unitary group $\mathsf{S}\mathsf{U}(4)$ [results in Fig. 10].

Figure 10

Mutual information and mutual negativity between two disjoint segments $A$ and $B$ for hybrid Clifford and Haar random unitary circuits.

Figure 11

Numerical results for the random Clifford circuit in its mixed phase. (a) Comparison between $\frac{1}{2}{I}_{A,\overline{A}}$ and $N_{A,\overline{A}}$ for a varying $A$, where we find they differ by at most an $O(1)$ constant. (b) The halfcut mutual negativity of $N_{A,\overline{A}}$ for $|A|=|\overline{A}|=L/2$ as a function of $L$, where we find $N_{A,\overline{A}}\propto L^{0.36}$. (c) The quantity $\widetilde{d_{\mathrm{cont}}}$, defined as $|A|$ where $N_{A,R}=ϵ(\ln 2)$ [compare Eq. (49)]. We see that $\widetilde{d_{\mathrm{cont}}}$ is proportional to $d_{\mathrm{cont}}$ [7].

Figure 12

An example of a bipartite mixed stabilizer state. Here we have $|A\cup B|=13$ lattice sites, where $|A|=7$ and $|B|=6$. There are $m=9$ stabilizers in the basis of the stabilizer group, represented by each row of $|A\cup B|$ squares. Blue squares represent local Pauli $Z$ operators, red squares represent local Pauli $X$ operators, and white or gray squares represent identity operators. Stabilizers in this basis can be classified into four sets, as detailed in Eq. (A50). When a stabilizer is supported nontrivially only on $A$ or $B$, they are called “local,” and we can disregard their (trivial) content on the other subsystem (represented with gray color).

Figure 13

Numerical results for MI and MN at a few more points in the CPLC Goldstone phase; compare Figs. 5, 7.

Figure 14

Graphical illustration, phase diagram, and numerical results for the ${\mathbb{Z}}_2$ hybrid circuit model.