  Symmetry in mixed quantum states can manifest in two distinct forms:
\textit{strong symmetry}, where each individual pure state in the quantum
ensemble is symmetric with the same charge, and \textit{weak symmetry}, which
applies only to the entire ensemble. This paper explores a novel type of
spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak
one. While the SSB of a weak symmetry is measured by the long-ranged two-point
correlation function $\mathrm{Tr}(O_xO^{\dagger}_y\rho)$, the strong-to-weak
SSB (SW-SSB) is measured by the fidelity $F(\rho, O_xO^{\dagger}_y\rho
O_yO^{\dagger}_x)$, dubbed the \textit{fidelity correlator}. We prove that
SW-SSB is a universal property of mixed-state quantum phases, in the sense that
the phenomenon of SW-SSB is robust against symmetric low-depth local quantum
channels. { We also show that the symmetry breaking is "spontaneous " in the
sense that the effect of a local symmetry-breaking measurement cannot be
recovered locally.} We argue that a thermal state at a nonzero temperature in
the canonical ensemble (with fixed symmetry charge) should have spontaneously
broken strong symmetry. Additionally, we study non-thermal scenarios where
decoherence induces SW-SSB, leading to phase transitions described by classical
statistical models with bond randomness. In particular, the SW-SSB transition
of a decohered Ising model can be viewed as the "ungauged" version of the
celebrated toric code decodability transition. We confirm that, in the
decohered Ising model, the SW-SSB transition defined by the fidelity correlator
is the only physical transition in terms of channel recoverability. We also
comment on other (inequivalent) definitions of SW-SSB, through correlation
functions with higher R\'enyi indices.
