Decoherence-induced self-dual criticality in topological states of matter

TL;DR

This work unifies measurement-induced phase transitions and decoherence-driven mixed-state criticality by mapping 1D MIPTs to 2D mixed states via a PEPS construction, enabling a single diagnostic S_AC that captures both bulk and boundary critical data. It identifies a self-dual, decoherence-protected critical state described by a symmetry-class D nonlinear sigma model at Θ=π in the N→1 replica limit, with KW self-duality enforcing nontrivial criticality in the presence of Born-rule randomness. The authors quantify universal data at the self-dual point, reporting c_Casimir≈0.447 and c_ent^vN≈0.795 (with c_ent^(∞)≈0.484), and extract ν≈1.72 for the correlation-length, while demonstrating RG flows from self-dual to Nishimori and Ising+ universality under perturbations. The results connect multiple representations (MBQC-like circuits, 2D RBIM, and (2+1)D Chalker-Coddington networks) and reveal a rich landscape of mixed-state criticalities with potential implications for measurement-based quantum computation and topological quantum matter under decoherence.

Abstract

Quantum measurements performed on a subsystem of a quantum many-body state can generate entanglement for its remaining constituents. The whole system including the measurement record is described by a hybrid mixed state, which can exhibit exotic phase transitions and critical phenomena. We demonstrate that generic measurement-induced phase transitions (MIPTs) can be cast as decoherence-induced critical mixed states in one higher dimension, by constructing a projected entangled pair state (PEPS) prior to decoherence or measurement. In this context, a deeper conceptual understanding of such mixed-state criticality is called for, particularly with regard to algebraic symmetry as an advanced organizing principle for such entangled states of matter. Integrating these connections we investigate the role of self-dual symmetry -- a fundamental notion in theoretical physics -- in mixed states, showing that the decoherence of electric (e) and magnetic (m) vortices from the 2D bulk of the toric code, or equivalently, a 2D cluster state with symmetry-protected topological order, can leave a (1+1)D quantum critical mixed state protected by a weak Kramers-Wannier self-dual symmetry. The corresponding self-dual critical bulk is described by the N->1 limit of the 2D Non-linear Sigma Model in symmetry class D with target space SO(2N)/U(N) at $Θ$-angle $π$, and represents a "measurement-version" of the Cho-Fisher network model subjected to Born-rule randomness...

Figures (18)

• Figure 1: Equivalence between measurement induced phase transition (MIPT) and mixed state transition. (a) A general scheme to map a generic 1-dimensional MIPT to a 2-dimensional mixed state, by means of a tensor network. Each gray box denotes a generic weak 2-body measurement gate, by entangling the two qubits with a middle ancilla qubit and projectively measuring out the ancilla. Our discussion applies to a generic MIPT that may or may not contain unitary gates; the latter would be considered part of the measurement gates in our notation. Such a measurement gate can be equivalently viewed as a rank-5 tensor, where the measurement outcome is the last, "the physical" index. The circuit can be embedded in a 2-dimensional tensor network state, specifically, a projected entangled pair state (PEPS), where the measurement record is materialized as the physical degrees of freedom of the PEPS. The map can be generalized to higher dimensions. (b) The von Neumann entanglement entropy of the hybrid bulk-boundary system encodes the bulk and boundary conformal data of the underlying CFT. The PEPS can be written as a Rokhsar-Kivelson type state: $\sum_\mathbf{s} \ket{\mathbf{s}}\otimes \ket{\psi(\mathbf{s})}$ for the un-normalized state $\ket{\psi(\mathbf{s})}$, where $\mathbf{s}$ denotes the set of measurement ourcomes from the space-time history of the circuit.
• Figure 2: Bulk decoherence of surface code. (a) A pure surface code state with a rough boundary on the right. Here we attach a chain of boundary qubits that are entangled with the rough boundary of the surface code, satisfying the stabilizers illustrated in the figure. Such boundary coupling is a natural consequence of measuring a cluster state Raussendorf2001 (see Fig. \ref{['fig:classicquantumclusterroute']} for details). (b) The bulk qubits are collapsed (thus decohered) by measurements into classical snapshots of spin configurations. The site qubit at site $j$ collapses into the $X_j$ eigenstate $\ket{+}$, and is labeled by a right arrow in an orange disk, while $\ket{-}$ is labeled by a left arrow in a blue disk. The qubit on the bond between sites $i$ and $j$ collapses to one of the eigenstates for rotated Pauli operator $\sigma_{ij}^\theta \equiv \cos(\theta)Z_{ij}+\sin(\theta)X_{ij}$, denoted by the tilted up arrow in gray circle, or down tilted arrow in white circle. We label the snapshot by $e$ particle (for a left arrow at site) and $m$-string (for a tilted down arrow at bond). The endpoint of an $m$-string is an $m$-particle. The rightmost column of qubits remain the unmeasured quantum boundary, highlighted by purple disks. While the bulk becomes a classical mixed state, the boundary quantum state is a pure quantum state conditioned upon the bulk snapshot. (c) By fixing an arbitrary bulk snapshot, the boundary quantum state is effectively generated by a 1D deep monitored quantum circuit, where the measurement record corresponds to the bulk classical configuration. The derivation follows from the tensor network representation discussion in Eq. \ref{['eq:tnderivation']}. (d) Self-dual phase diagram by tuning the bond measurement angle $\theta$, shown in a circle. When the bond measurement angle $\theta$ is tuned to the symmetric angle $\pm\pi/4$ or $\pm 3\pi/4$, the system exhibits the weak self-dual criticality, described by the $SO(2N)/U(N)$ Non-linear Sigma Model (NLSM) at topological $\Theta$-angle $\pi$ in the $N\to 1$ limit. Changing the measurement angle $\theta$ at the microscopic level has the same effect as changing the topological $\Theta$-angle in the long-wave-length theory.
• Figure 3: Surface code with coherent rotation prior to measurement. (a) The quantum circuit. The combination of coherent rotation and $Z$-measurement is effective measurement of qubit in $\cos\theta Z + \sin\theta X$ basis. (b) The disk phase diagram parametrized by the measurement angle $\theta$ are separated into four quadrants, separated by self-dual critical points, where the snapshots are schematically shown. Measuring surface code in $Z$ basis results in a loop gas ensemble, where the blue shaded lines illustrate the loop defined by spin down. Measuring surface code in $X$ basis results in a dual loop gas ensemble, where the red shaded lines illustrate the dual loop defined by spin left as eigenstate of Pauli $X$ with eigenvalue $-1$. Measurement at the magic angle $\theta=\pi/4$ is self-dual, which not only excites the $e$ particle but also the $m$ particle, on equal footing. The Born's probability of the measurement outcomes is dictated by the statistical model in (c). (c) The statistical model is a random Ising model with both $e$ and $m$ particle disorder, in Eq. \ref{['eq:statmech']}. The schematic shows fluctuating strings (defined by $s_{ij}\sigma_i\sigma_j=-1$) that have to be closed or terminate at the $m$ particle. The Boltzmann weight of the string is given by the string tension $e^{-\beta}$ coupled to the length $l$ of the string. When the string passes through an $e$ particle, a $\pi$ phase is attached to the Boltzmann weight.
• Figure 4: 2D protocol and effective 1D circuit. (a) A cluster state on the Lieb lattice is prepared by controlled-Z gates for every nearest neighbor pair between the orange and green qubits. It has a $Z_2^{(0)}\times Z_2^{(1)}$ SPT, where $Z_2^{(0)}$ symmetry applies to the site (orange) qubits and $Z_2^{(1)}$ symmetry applies to the bond (gray) qubits. (b) The bulk qubits are collapsed (thus decohered) by measurements into classical snapshots of spin configurations. The site qubit at site $j$ collapses into the $X_j$ eigenstate $\ket{+}$, and is labeled by a right arrow in an orange disk, while $\ket{-}$ is labeled by a left arrow in a blue disk. The qubit on the bond between sites $i$ and $j$ collapses to one of the eigenstates for rotated Pauli operator $\sigma_{ij}^\theta \equiv \cos(\theta)Z_{ij}+\sin(\theta)X_{ij}$, denoted by the tilted up arrow in green circle, or down tilted arrow in red circle. We label the snapshot by $e$ particle (for a left arrow at site) and $m$-string (for a tilted down arrow at bond). The endpoint of an $m$-string is an $m$-particle. The rightmost column of qubits remain the unmeasured quantum boundary, highlighted by purple disks. While the bulk becomes a classical mixed state, the boundary quantum state is a pure quantum state conditioned upon the bulk snapshot. (c) By fixing an arbitrary bulk snapshot, the boundary quantum state is effectively generated by a 1D deep monitored quantum circuit, where the measurement record corresponds to the bulk classical configuration. The derivation follows from the tensor network representation discussion in Sec. \ref{['sec:derivetns']}.
• Figure 5: Decoherence schematic. A $Z_2^{(0)}\times Z_2^{(1)}$ SPT can be realized as a cluster state on a Lieb lattice (with site and bond qubits), which can be viewed as a lattice gauge state with matter electric charge $e$- and gauge magnetic flux $m$-vortices, that are dual to each other. When the site qubits are collapsed (decohered) / measured, $e$ is uncondensed, resulting in average or correctable topological order as a deconfined gauge state that spontaneously breaks the 1-form $Z_2^{(1)}$ symmetry. When the bond qubits are collapsed (decohered) / measured, $m$ is uncondensed and leads to the GHZ-like state as an average $Z_2^{(0)}$ spontaneous symmetry breaking order. When both $e$ and $m$ particles are uncondensed by measurement, it can leave a boundary critical state with weak self-dual symmetry and weak $Z_2^{(0)}$ symmetry.