Local One-Point Fidelity Correlator

• Local One-Point Fidelity Correlator is a diagnostic tool that quantifies local order in mixed states by comparing reduced states via fidelity and Rényi-1 correlators, ensuring thermodynamic consistency.
• It utilizes a covering-limit approach and finite-region bounds, enabling scalable detection of symmetry breaking without needing full global state tomography.
• The correlator bridges local and global formulations by revealing critical scaling and stability under finite-depth symmetric channels, applicable to systems like CFTs and free fermions.

The local one-point fidelity correlator is a local diagnostic for strong-to-weak spontaneous symmetry breaking (SW-SSB) in mixed states, introduced as

$$F(\rho_A;O_x)\equiv F\!\big(\rho_A,\,O_x\rho_A O_x^\dagger\big), \qquad F(\rho,\sigma)=\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}},$$

where $A$ is a finite spatial region containing $x$, $\rho_A$ is the reduced state on $A$, and $\ell=\operatorname{dist}(x,\partial A)$ is the point-to-boundary distance. In this formulation, local SW-SSB is defined by the nonvanishing covering-limit condition

$\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$

Compared with the earlier global two-point fidelity correlator, the local formulation is easier to detect in large systems and remains well defined in the thermodynamic limit, where a global density matrix is not well defined (Divi et al., 27 May 2026).

1. Definition and equivalent formulations

The local one-point fidelity correlator arose as a local counterpart of the previous global two-point fidelity correlator

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$

which quantifies the similarity between the mixed state $\rho$ and the state obtained by moving a charge from $x$ to $A$0. The local correlator replaces this nonlocal charge-transfer probe by a reduced-state comparison in a finite region around a single insertion point $A$1.

The thermodynamic-limit order parameter is defined through a covering centered at $A$2: a nested sequence $A$3 with $A$4 and $A$5. For any such covering,

$A$6

and the limit is independent of the covering. The construction relies on the data processing inequality for fidelity and monotonicity under partial trace. The resulting infinite-volume quantity coincides with the C$A$7-algebraic Uhlmann fidelity $A$8, with $A$9.

A closely related family of observables is given by the one-point Rényi-$x$0 correlators,

$x$1

Among these, the Rényi-1 case is singled out as a robust order parameter because it is monotone under enlarging regions:

$x$2

and is quantitatively equivalent to fidelity:

$x$3

Hence, in infinite volume,

$x$4

2. Symmetry-theoretic setting and motivation for locality

The relevant symmetry distinction is between strong and weak symmetry for mixed states. A global symmetry $x$5 acts strongly on $x$6 if $x$7, meaning the state lies in a definite charge sector. It acts weakly if $x$8, meaning only the ensemble is invariant. This distinction collapses for pure states but is fundamental for mixed states and open systems.

SW-SSB denotes a mixed-state phase in which a strong symmetry is spontaneously reduced to a weak symmetry without fully breaking the symmetry. In the global formulation, the phase is detected by the persistence of $x$9 at long distance, whereas ordinary spontaneous symmetry breaking would be visible already in linear correlators $\rho_A$0. The local one-point formulation isolates the same phenomenon at the level of reduced states.

The motivation for locality is twofold. First, reconstructing the global density matrix is tomographic and costs $\rho_A$1 resources in system volume $\rho_A$2, whereas the local correlator depends only on $\rho_A$3 and can be computed with resources $\rho_A$4. With a global resource budget $\rho_A$5, one can therefore probe

$\rho_A$6

which is sufficient to test whether the local correlator saturates, decays exponentially, or shows a critical power law. Second, in infinite systems the Hilbert space and global $\rho_A$7 are ill defined, but local reduced states remain meaningful, so the covering-limit definition gives a thermodynamically sharp notion of SW-SSB.

The construction is conceptually analogous to local thermalization. In eigenstates obeying the eigenstate thermalization hypothesis, $\rho_A$8 is thermal on small regions and thus exhibits nonzero local one-point fidelity, while the global two-point fidelity reduces to the ordinary two-point function and decays. This establishes a precise distinction between local and global SW-SSB.

3. Detection, computability, and finite-region control

Even without additional structure, evaluating $\rho_A$9 or $A$0 by full tomography is exponential in $A$1. Nevertheless, the local formulation makes scalable diagnosis possible because only a finite region needs to be reconstructed. The operational criteria are explicit: saturation of the local correlator beyond a length $A$2 with $A$3 supports local SW-SSB, while exponential decay $A$4 with $A$5 supports its absence. For critical states, accessible scales can reveal a power law $A$6.

A central finite-region bound relates the asymptotic local correlator to the conditional mutual information (CMI). For a tripartition $A$7,

$A$8

When the CMI decays as $A$9, the finite-region value is exponentially close to the infinite-volume limit once $\ell=\operatorname{dist}(x,\partial A)$0. This yields a direct measurement protocol: choose a local operator $\ell=\operatorname{dist}(x,\partial A)$1, estimate $\ell=\operatorname{dist}(x,\partial A)$2 on an expanding tripod $\ell=\operatorname{dist}(x,\partial A)$3, and increase $\ell=\operatorname{dist}(x,\partial A)$4 until the value stabilizes within the predicted $\ell=\operatorname{dist}(x,\partial A)$5 tolerance.

The same local structure also appears in thermal states. For Gibbs states,

$\ell=\operatorname{dist}(x,\partial A)$6

so the one-point correlator is finite at any $\ell=\operatorname{dist}(x,\partial A)$7. This suggests that local SW-SSB can be diagnosed directly through imaginary-time local correlators whenever a local reduced Gibbs description is valid (Divi et al., 27 May 2026).

4. Thermodynamic limit, stability, and information-theoretic consequences

In infinite volume, the state is represented as a positive functional $\ell=\operatorname{dist}(x,\partial A)$8 on the quasi-local algebra rather than by a global density matrix. Weak symmetry and local reduced states remain well defined, whereas strong symmetry in the global density-matrix sense does not. Within this framework, local SW-SSB is defined by $\ell=\operatorname{dist}(x,\partial A)$9 together with decay of ordinary linear correlators, so that there is no ordinary SSB. Under translation invariance, $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$0 is position independent.

The local order is stable under strongly symmetric finite-depth channels. Such a channel $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$1 is a CPTP map whose Kraus operators commute with the symmetry and that admits a Stinespring dilation

$\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$2

with a finite-depth local unitary circuit $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$3 commuting with the strong symmetry on the physical space. If $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$4 has local SW-SSB, then $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$5 also has local SW-SSB. The proof enlarges the region to a light-cone-expanded neighborhood $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$6 and uses the data processing inequality to propagate the nonvanishing one-point fidelity through the channel.

The same framework yields a lower bound on long-range CMI. For a strongly symmetric $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$7,

$\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$8

Thus a finite local one-point fidelity implies non-Markovianity. In the infinite system, a symmetry-averaged local CMI restores an analogous statement under local indistinguishability of isotypic components for large $\lim_{|A|\to\infty}F(\rho_A;O_x)>0.$9:

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$0

This identifies long-range CMI as a persistent information-theoretic hallmark of local SW-SSB.

An additional order–disorder inequality constrains the one-point fidelity by symmetry-sector weights. In general,

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$1

and for $$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$2,

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$3

These bounds formalize the competition between local fidelity order and conventional symmetry order.

5. Relation to the two-point fidelity correlator

The local one-point fidelity correlator was introduced as a replacement for, not a rejection of, the earlier two-point fidelity formulation. The two notions are connected by a hierarchy of results.

The basic implication is global-to-local:

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$4

Therefore global SW-SSB implies local SW-SSB. The converse does not hold in full generality, but several averaged and asymptotic equivalence statements are available.

On finite regions, averaged one-point order implies averaged two-point order:

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$5

In infinite volume, if $$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$6 for infinitely many $$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$7, then

$$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$8

Equivalently, there exists an explicit subsequence $$F(\rho;O_xO_y^\dagger)\equiv F\!\big(\rho,\,O_xO_y^\dagger\,\rho\,O_yO_x^\dagger\big),$$9 such that

$\rho$0

The local two-point formulation itself is defined by

$\rho$1

and under the appropriate limits it is equivalent to the one-point formulation. A plausible implication is that the one-point version isolates the minimal local ingredient required for SW-SSB, while the two-point version packages the same phenomenon into a nonlocal transfer process.

6. Critical scaling, model realizations, and scope of the concept

For critical states, the local one-point fidelity correlator defines a defect problem. In the ZZ-decohered Ising paramagnet, the one-point fidelity at $\rho$2 becomes an average over random-bond Ising model partition functions with a defect line starting at $\rho$3 and ending on $\rho$4:

$\rho$5

Here $\rho$6 is the free-energy cost of inserting the bond-flux line. In the ferromagnetic phase $\rho$7 this cost grows $\rho$8, so the fidelity decays exponentially; in the paramagnetic phase $\rho$9 it saturates, yielding local SW-SSB. The Rényi-2 one-point correlator reduces to boundary magnetization,

$x$0

For conformal field theories, the local Rényi-1 correlator has universal scaling. In the vacuum CFT,

$x$1

while for a thermal cylinder the covering limit gives

$x$2

Higher-dimensional CFTs obey the same scaling structure for ball regions,

$x$3

For free fermions, the asymptotics are controlled by the single-particle correlator:

$x$4

This leads to distinct universal regimes. Ballistic metals with a codimension-1 Fermi surface satisfy $x$5; Dirac semimetals satisfy $x$6; and diffusive metals obey the disorder-averaged law $x$7, independent of dimension. In one dimension, explicit half-line and interval formulas reproduce $x$8.

Several noncritical examples clarify the scope of the notion. Pure ETH eigenstates have finite $x$9 locally but decaying global two-point fidelity, so they exhibit local but not global SW-SSB. Random Gaussian states from GOE quadratic Hamiltonians give an explicit realization of the same separation:

$A$00

while $A$01 implies vanishing global two-point fidelity. These examples establish that locality is not merely a technical simplification but a distinct phase-diagnostic viewpoint (Divi et al., 27 May 2026).

A different usage of “one-point fidelity correlator” appears in fidelity out-of-time-order correlators in the spin-boson model. There, with $A$02 and $A$03,

$A$04

which is the squared modulus of a one-point function. In that work, however, the chosen generator $A$05 acts on the bath, so the correlator is one-point but not spin-local; the same formal definition would become a local one-point fidelity correlator on the impurity only for a spin-local choice such as $A$06 or $A$07 (Chen, 2023).

7. Limitations and open directions

The robustness of the local one-point construction depends crucially on fidelity and on the Rényi-1 correlator. Only $A$08 inherits the required DPI-like monotonicity and yields a covering-independent local order parameter; for $A$09, the covering limit can depend on the sequence, with a dimer counterexample given in the source work. This sharply delimits which “one-point Rényi correlators” are genuine thermodynamic diagnostics.

Local SW-SSB also does not imply global SW-SSB in general. ETH eigenstates and pseudo-SWSSB ensembles provide explicit counterexamples. At the same time, the local and global phase diagrams can coincide in many physical settings with decaying conditional mutual information, such as finite-time or finite-decoherence transitions. This suggests that the distinction is physically meaningful but model dependent.

The framework is developed primarily for onsite Abelian symmetries, although the source extends to non-Abelian groups by optimizing over irrep multiplets in the one-point correlator. Non-onsite symmetries, gauge symmetries, and higher-form symmetries require adapted local formulations. Additional open directions identified in the literature include defect interpretations beyond CFTs and Fermi metals, relations to local thermalization and hydrodynamics of SW-SSB, charge scrambling and sharpening, and efficient experimental detection based on local Rényi-1 correlators and finite Markov length.

Taken together, these results place the local one-point fidelity correlator at the intersection of mixed-state order, locality, and quantum information. Its defining feature is not merely that it is a one-point observable, but that it remains monotone under region enlargement, thermodynamically well defined, stable under finite-depth strongly symmetric channels, and sensitive to universal defect scaling across critical and noncritical systems.