--- title: Local One-Point Fidelity Correlator url: https://www.emergentmind.com/topics/local-one-point-fidelity-correlator type: topic --- # Local One-Point Fidelity Correlator 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 [2605.28967]. ## 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 \(y\). The local correlator replaces this nonlocal charge-transfer probe by a reduced-state comparison in a finite region around a single insertion point \(x\). The thermodynamic-limit order parameter is defined through a covering centered at \(x\): a nested sequence \(\{A_x^{(n)}\}\) with \(x\in A_x^{(n)}\subset A_x^{(n+1)}\) and \(\operatorname{dist}(x,\overline{A_x^{(n)}})\to\infty\). For any such covering, $$ \lim_{n\to\infty}F(\rho_{A_x^{(n)}};O_x)=F(\omega;O_x), $$ 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\(^*\)-algebraic Uhlmann fidelity \(F(\omega,\omega_O)\), with \(\omega_O(\cdot)=\omega(O_x^\dagger\,\cdot\,O_x)\). A closely related family of observables is given by the one-point Rényi-\(\alpha\) correlators, $$ R^{(\alpha)}(\rho_A;O_x)=\frac{\operatorname{Tr}\!\big(\rho_A^{\alpha/2}\,O_x\,\rho_A^{\alpha/2}\,O_x^\dagger\big)}{\operatorname{Tr}(\rho_A^\alpha)}. $$ Among these, the Rényi-1 case is singled out as a robust order parameter because it is monotone under enlarging regions: $$ R^{(1)}(\rho_A;O_x)\ge R^{(1)}(\rho_B;O_x)\qquad (A\subset B), $$ and is quantitatively equivalent to fidelity: $$ F(\rho_A;O_x)^2\le R^{(1)}(\rho_A;O_x)\le \|O_x\|_\infty\,F(\rho_A;O_x). $$ Hence, in infinite volume, $$ R^{(1)}(\omega;O_x)>0\iff F(\omega;O_x)>0. $$ ## 2. Symmetry-theoretic setting and motivation for locality The relevant symmetry distinction is between strong and weak symmetry for mixed states. A global symmetry \(U\) acts strongly on \(\rho\) if \(U\rho\propto \rho\), meaning the state lies in a definite charge sector. It acts weakly if \(U\rho U^\dagger=\rho\), 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 \(F(\rho;O_xO_y^\dagger)\) at long distance, whereas ordinary spontaneous symmetry breaking would be visible already in linear correlators \(\langle O_xO_y^\dagger\rangle\). 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 \(e^{\Omega(V)}\) resources in system volume \(V\), whereas the local correlator depends only on \(\rho_A\) and can be computed with resources \(e^{O(|A|)}=e^{O(\ell^d)}\). With a global resource budget \({\rm poly}(L)\), one can therefore probe $$ \ell_{\max}\sim \log^{1/d}(L), $$ 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 \(\rho\) 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, \(\rho_A\) 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 \(F(\rho_A;O_x)\) or \(R^{(1)}(\rho_A;O_x)\) by full tomography is exponential in \(|A|\sim \ell^d\). 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 \(\ell_0\) with \(\ell_00\). This suggests that local SW-SSB can be diagnosed directly through imaginary-time local correlators whenever a local reduced Gibbs description is valid [2605.28967]. ## 4. Thermodynamic limit, stability, and information-theoretic consequences In infinite volume, the state is represented as a positive functional \(\omega\) 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 \(F(\omega;O_x)>0\) together with decay of ordinary linear correlators, so that there is no ordinary SSB. Under translation invariance, \(F(\omega;O_x)\) is position independent. The local order is stable under strongly symmetric finite-depth channels. Such a channel \(E_{\rm SFD}\) is a CPTP map whose Kraus operators commute with the symmetry and that admits a Stinespring dilation $$ E_{\rm SFD}[\rho]=\operatorname{Tr}_a\!\Big(U^\dagger\,\rho\otimes|0\rangle\!\langle 0|_a\,U\Big), $$ with a finite-depth local unitary circuit \(U\) commuting with the strong symmetry on the physical space. If \(\rho\) has local SW-SSB, then \(E_{\rm SFD}[\rho]\) also has local SW-SSB. The proof enlarges the region to a light-cone-expanded neighborhood \(A^{+r}\) 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 \(\rho\), $$ I(A:C|B)_\rho\ge \left(\frac{F(\rho_{AB};O_A)}{c\,\|O_A\|_\infty}\right)^2. $$ 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 \(C\): $$ I(A:C|B)^{\rm Sym}_\omega\ge \left(\frac{F(\rho_{AB};O_A)}{c\,\|O_A\|_\infty}\right)^2. $$ 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, $$ F(\rho_A;O_x)\le \sum_\lambda \sqrt{p_A^{(\lambda)}\,p_A^{(\lambda\mu^*)}}, $$ and for \(G=\mathbb{Z}_2\), $$ F(\rho_A;O_x)^2+|\langle U_A\rangle|^2\le 1. $$ 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: $$ \|O_y\|_\infty\,F(\rho_A;O_x)\ge F(\rho;O_xO_y^\dagger). $$ 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: $$ \frac{1}{|\Omega|^2}\sum_{x,y\in\Omega}F(\rho_A;O_xO_y^\dagger) \ge \left(\frac{1}{|\Omega|}\sum_{x\in\Omega}F(\rho_A;O_x)\right)^2. $$ In infinite volume, if \(F(\omega;O_x)\ge F>0\) for infinitely many \(x\), then $$ \limsup_{|x-y|\to\infty}F(\omega;O_xO_y^\dagger)\ge F^2. $$ Equivalently, there exists an explicit subsequence \(\{y_n\}\) such that $$ \lim_{n\to\infty}F(\omega;O_xO_{y_n}^\dagger)\ge F^2. $$ The local two-point formulation itself is defined by $$ \lim_{|x-y|\to\infty}\lim_{|A|\to\infty}F(\rho_A;O_xO_y^\dagger) \equiv \lim_{|x-y|\to\infty}F(\omega;O_xO_y^\dagger), $$ 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 \(x\) becomes an average over random-bond Ising model partition functions with a defect line starting at \(x\) and ending on \(\partial A^c\): $$ F(\rho_A;Z_x)\propto \sum_{s,b}\mathcal Z_{s,b}\,e^{-\frac{1}{2}\beta\,\Delta F_{A,x}^{(s)}}. $$ Here \(\Delta F_{A,x}^{(s)}\) is the free-energy cost of inserting the bond-flux line. In the ferromagnetic phase \((pp_c)\) it saturates, yielding local SW-SSB. The Rényi-2 one-point correlator reduces to boundary magnetization, $$ R^{(2)}(\rho_{p,A};Z_x)\propto \langle \eta_x\rangle_{2K}. $$ For conformal field theories, the local Rényi-1 correlator has universal scaling. In the vacuum CFT, $$ R^{(1)}(\rho_A;O(x))\sim \frac{c_O}{\ell^{2\Delta_O}}, $$ while for a thermal cylinder the covering limit gives $$ R^{(1)}(\omega;O(x))=\left(\frac{\pi}{\beta}\right)^{2\Delta_O}. $$ Higher-dimensional CFTs obey the same scaling structure for ball regions, $$ R^{(1)}(\rho_A;O(x))=\frac{C_O}{\ell^{2\Delta_O}}. $$ For free fermions, the asymptotics are controlled by the single-particle correlator: $$ R^{(1)}(\rho_A;c_x)\sim \int_{\bar A} d^dy\,|C(x,y)|^2. $$ This leads to distinct universal regimes. Ballistic metals with a codimension-1 Fermi surface satisfy \(R^{(1)}\sim \ell^{-1}\); Dirac semimetals satisfy \(R^{(1)}\sim \ell^{-d}\); and diffusive metals obey the disorder-averaged law \(\overline{R^{(1)}}\sim \ell^{-2}\), independent of dimension. In one dimension, explicit half-line and interval formulas reproduce \(R^{(1)}(\rho_{[-\ell,\ell]};c_0)=1/(\pi\ell)\). Several noncritical examples clarify the scope of the notion. Pure ETH eigenstates have finite \(F(\rho_A;O_x)\) 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: $$ F(\rho_A;c_x)=\sqrt{\nu(1-\nu)}, $$ while \(\overline{|C_{xy}|^2}=\nu/N\to 0\) implies vanishing global two-point fidelity. These examples establish that locality is not merely a technical simplification but a distinct phase-diagnostic viewpoint [2605.28967]. A different usage of “one-point fidelity correlator” appears in fidelity out-of-time-order correlators in the spin-boson model. There, with \(V=|\psi\rangle\langle\psi|\) and \(W=e^{i\xi A}\), $$ \mathcal F(t)=|\langle \psi|e^{i\xi A(t)}|\psi\rangle|^2, $$ which is the squared modulus of a one-point function. In that work, however, the chosen generator \(A=\sum_k g_k(b_k+b_k^\dagger)\) 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 \(O=\sigma_z\) or \(O=\sigma_x\) [2303.12276]. ## 7. Limitations and open directions The robustness of the local one-point construction depends crucially on fidelity and on the Rényi-1 correlator. Only \(\alpha=1\) inherits the required DPI-like monotonicity and yields a covering-independent local order parameter; for \(\alpha\neq 1\), 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. Source: https://www.emergentmind.com/topics/local-one-point-fidelity-correlator