Hardness of observing strong-to-weak symmetry breaking

TL;DR

The paper proves that observing strong-to-weak spontaneous symmetry breaking (SWSSB) in intrinsically mixed states is generically hard: no efficient state-agnostic protocol, given copies of an unknown mixed state, can reliably decide SWSSB for Abelian symmetries. The authors construct pseudo-SWSSB ensembles for $\mathbb{Z}_2$ and $U(1)$ using pseudorandom unitaries and symmetry-sector projections, and show these ensembles are computationally indistinguishable from SWSSB states while not exhibiting SWSSB themselves. Central results include rigorous bounds establishing indistinguishability under polynomial resources (with $\|\mathbb{E}\rho^{\otimes k}-\rho_0^{\otimes k}\|_{\rm tr} \le O(k^2/r)$ in the $\mathbb{Z}_2$ case and analogous arguments for $U(1)$), thereby ruling out efficient, hardware-agnostic detection of SWSSB in general. This frames a fundamental limitation on experimentally witnessing intrinsically mixed phases and highlights the need for additional structure or prior information to enable efficient SWSSB diagnostics.

Abstract

Spontaneous symmetry breaking (SSB) is the cornerstone of our understanding of quantum phases of matter. Recent works have generalized this concept to the domain of mixed states in open quantum systems, where symmetries can be realized in two distinct ways dubbed strong and weak. Novel intrinsically mixed phases of quantum matter can then be defined by the spontaneous breaking of strong symmetry down to weak symmetry. However, proposed order parameters for strong-to-weak SSB (based on mixed-state fidelities or purities) seem to require exponentially many copies of the state, raising the question: is it possible to efficiently detect strong-to-weak SSB in general? Here we answer this question negatively in the paradigmatic cases of $Z_2$ and $U(1)$ symmetries. We construct ensembles of pseudorandom mixed states that do not break the strong symmetry, yet are computationally indistinguishable from states that do. This rules out the existence of efficient state-agnostic protocols to detect strong-to-weak SSB.

Figures (1)

• Figure 1: Circuits for the preparation of pseudo-SWSSB states with (a) $\mathbb{Z}_2$ symmetry or (b) $U(1)$ symmetry. In (a), the encoder is a staircase circuit of CNOT gates mapping $X_1$ to the symmetry generator $\bar{X} = \prod_i X_i$. In (b), the charge measurement can be implemented with $O(N)$ ancilla qubits and $O(\log(N))$ depth; postselection of an outcome $Q$ is sample-efficient as long as $Q = N/2 + O(N^{1/2})$. To achieve pseudo-SWSSB the number of depolarized qubits $\log_2(r)$ must be superlogarithmic but subextensive.

Theorems & Definitions (10)

• Definition 1: Pseudo-SWSSB
• Definition 2: Pseudorandom unitary ji_pseudorandom_2018metger_simple_2024ma2024constructrandomunitarieschen_pseudorandom_2024
• Theorem 1: Pseudo-SWSSB for $\mathbb{Z}_2$ symmetry
• proof
• Lemma 1
• Theorem 2: Pseudo-SWSSB for $U(1)$ symmetry
• Lemma 2: absence of SWSSB in $\mathcal{E}(U(1))$
• proof
• Lemma 3: Statistical indistinguishability between $\rho_Q$ and $\mathcal{E}'(U(1))$
• proof