Mixed-state Quantum Phases: Renormalization and Quantum Error Correction

TL;DR

This work establishes a mixed state toric code phase using local channels obtained by truncating an RG-type decoder and the minimum weight perfect matching decoder, and discovers a precise relation between mixed state phase and decodability, by proving that local noise acting on tori code cannot destroy logical information without bringing the state out of the toriccode phase.

Abstract

Open system quantum dynamics can generate a variety of long-range entangled mixed states, yet it has been unclear in what sense they constitute phases of matter. To establish that two mixed states are in the same phase, as defined by their two-way connectivity via local quantum channels, we use the renormalization group (RG) and decoders of quantum error correcting codes. We introduce a real-space RG scheme for mixed states based on local channels which ideally preserve correlations with the complementary system, and we prove this is equivalent to the reversibility of the channel's action. As an application, we demonstrate an exact RG flow of finite temperature toric code in two dimensions to infinite temperature, thus proving it is in the trivial phase. In contrast, for toric code subject to local dephasing, we establish a mixed state toric code phase using local channels obtained by truncating an RG-type decoder and the minimum weight perfect matching decoder. We also discover a precise relation between mixed state phase and decodability, by proving that local noise acting on toric code cannot destroy logical information without bringing the state out of the toric code phase.

Figures (9)

• Figure 1: (a) Definition of mixed-state phase equivalence adopted in this work. Two many-body mixed states $\rho_1$ and $\rho_2$ are in the same phase if there is a pair of low-depth spatially local quantum channels $\mathcal{C}_1$ and $\mathcal{C}_2$ such that $\rho_2\approx\mathcal{C}_1(\rho_1)$ and $\rho_1\approx\mathcal{C}_2(\rho_2)$. (b) Illustration of the correlation-preserving criterion in Def.\ref{['def: correlation-preserving']}. For a given bipartite mixed state $\rho_{AB}$, a quantum channel $\mathcal{E}$ acting on one party is correlation-preserving if it leaves the mutual information between two parties invariant. Thm.\ref{['thm: thm2']} shows that $\mathcal{E}$ is correlation-preserving if and only if its action can be reversed by another channel $\mathcal{D}$. (c) Mixed-state RG consists of local channels ($\mathcal{E}$s) which coarse-grain degrees of freedom within a block. After iterating, all short-range correlations of the input state are discarded and only long-range ones remain. If all coarse-graining channels satisfy the correlation-preserving criterion, then the whole RG process can be reversed, by running from top to bottom and replacing each $\mathcal{E}$ with its recovery map $\mathcal{D}$. (d) Phase diagrams and RG flows of 4 exemplary mixed states studied in the Sec.\ref{['sec: examples']}. All 4 states come from perturbing a long-range entangled pure state in an incoherent way: In examples (i, ii) the pure state is the GHZ state, and in (iii, iv) it is the toric code state. In examples (i, ii, iv) the incoherent perturbation is a dephasing noise with strength $p$ acted upon the state, while in (iii) the perturbation is a non-zero temperature. The mixed-state phase corresponding to the GHZ state and the toric code state are denoted by [GHZ] and [T.C.], respectively.
• Figure 2: A circuit of local channel gates represented as an LC transformation.
• Figure 3: Real-space RG transformation of pure states-- Circuit representation of two iterations of pure state RG transformation. At the $\ell$-th iteration, the coarse-graining isometry $w^{(\ell)}$ is determined by the level's input state $\ket{\psi^{(\ell)}}$ using Eq.\ref{['eq: white_rule']}. By applying the circuit from bottom to top (red arrows), all the short-range features of the initial UV state are gradually discarded, and only long-range ones are kept in the IR state $\rho^{(\ell\rightarrow\infty)}$. By applying the circuit from top to bottom (blue arrows), the circuit generates the UV state $\ket{\psi^{(1)}}$.
• Figure 4: Tree tensor network of a GHZ state with $L=b^\ell=9$, $b=3$, $\ell=2$. Each triangle represents an isometry $w$ (Eq.\ref{['eq: ghz_tn_rep']}). By replacing the state at the top with a generic single qubit state $\ket{\psi}$, the same tensor network encodes $\ket{\psi}$ into a codeword state of the quantum repetition code.
• Figure 5: RG scheme for the thermal toric code state-- In all panels, a plaquette (vertex) is shaded (dotted) if has a non-zero probability of holding an $m$- ($e$-) anyon, and physical qubits are associated with edges of the lattice and drawn as circles. (left$\rightarrow$mid) $\mathcal{E}^X$ and $\mathcal{E}^Z$ (see Eq.\ref{['eq: theram_tc_E_X']} and Eq.\ref{['eq: theram_tc_E_Z']}) acts on each $2\times2$ block of plaquettes and vertices, respectively. The resulting state has anyons on one of its sublattices' plaquettes and vertices. (mid$\rightarrow$right) After disentangling with the unitary $\mathcal{U}$ depicted in Eq.\ref{['eq: thermal_tc_disentangle']} and discarding the decoupled qubits, the new state is still a toric code Gibbs state, but with renormalized temperature $p'$ (Eq.\ref{['eq: thermal_tc_p_iteration']}) supported on a coarse-grained lattice.

Theorems & Definitions (7)

• Definition 1: Local channel (LC) transformation
• Definition 2: Mixed-state phase equivalence
• Definition 3: correlation-preserving maps
• Theorem 1
• proof
• Theorem 2
• proof