Fidelity Zeros in Quantum Systems

• Fidelity zeros are points in parameter space where the overlap between quantum states becomes exactly zero, marking precise transitions between different phases.
• They play a crucial role in diagnosing quantum phase transitions, symmetry-breaking phenomena, and topological shifts by analyzing orthogonality in ground states.
• Analytical frameworks, including Lee–Yang theory and geometric formulations, provide scalable insights into spectral properties and finite-size effects in various quantum models.

Fidelity zeros are points or regimes in which a fidelity quantity vanishes exactly, or tends to zero in a thermodynamic or asymptotic limit. The expression appears in several related settings: ground-state fidelity in complexified parameter space for quantum many-body Hamiltonians, operation fidelity of a quantum channel, optimal fidelity for pure-state conversion under zero communication, and overlap-based diagnostics of impurity and lattice models. In the many-body setting developed for Lee–Yang-type analyses, a fidelity zero is a point in the complex parameter plane where

$$F(h,h')=\bigl|\langle \Psi_0(h')\mid \Psi_0(h)\rangle\bigr|=0,$$

so the two ground states are orthogonal (Gu et al., 24 Sep 2025).

1. Definitions and principal meanings

In the most general sense, fidelity measures the overlap between states, reduced density matrices, or channels. For positive semidefinite operators $R,S\ge 0$, the standard Uhlmann fidelity is

$$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$

and for pure states it reduces to

$$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$

In the zero-communication literature, a “fidelity zero” means that even the best protocol under the allowed free operations achieves fidelity $0$ for a given state transformation (George et al., 2023).

For quantum channels, the relevant quantity may be the operation fidelity for a pure input state $|\psi\rangle$,

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$

where $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$. In that setting, the central question is whether the minimum operation fidelity

$F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$

can reach zero, so that the channel admits zero fidelity states (Chełstowski et al., 2022).

A spectral formulation is obtained from the fidelity operator

$$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$

whose eigenvalues $R,S\ge 0$0 define the fidelity operator spectrum, while $R,S\ge 0$1 defines the fidelity spectrum. In this language, zeros or near-zeros of fidelity correspond to very small eigenvalues of $R,S\ge 0$2, and therefore to large or divergent entries of the fidelity spectrum (Sacramento et al., 2011).

2. Symmetry breaking, Lee–Yang theory, and quantum criticality

A central development is the formulation of fidelity zeros as the quantum analogue of Lee–Yang zeros. For the transverse-field Ising chain with complex transverse field,

$R,S\ge 0$3

the ground-state fidelity can be computed exactly, and fidelity zeros occur when the overlap between the ground states at $R,S\ge 0$4 and $R,S\ge 0$5 vanishes. In this framework, the fidelity is treated as an analogue of a partition function in the complex field plane, with complex fugacity

$R,S\ge 0$6

For the one- and two-dimensional ferromagnetic quantum Ising models considered, all fidelity zeros lie on the unit circle in fugacity, $R,S\ge 0$7, for each finite system size (Gu et al., 24 Sep 2025).

The microscopic mechanism is symmetry-sector switching. The models have a $R,S\ge 0$8 parity symmetry, and the Hilbert space decomposes into even- and odd-parity sectors. When the field acquires an imaginary part, the Hamiltonian becomes non-Hermitian but still parity symmetric, and the “ground state” is defined as the eigenstate with the lowest real part of the eigenvalues. As the complex field is varied, the lowest-real-part eigenstate can switch between parity sectors. Because states in distinct sectors are orthogonal, such sector transitions enforce exact zeros of the overlap. The paper identifies this phenomenon as non-Hermitian parity-symmetry breaking and shows that the exact locations of fidelity zeros coincide with the crossings between parity-sector ground states.

In the thermodynamic limit, finite-system zeros condense into arcs and define fidelity edges, which play the role of Yang–Lee edge singularities. For fixed modulus $R,S\ge 0$9, if $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$0 in one dimension or $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$1 in two dimensions, zeros are distributed uniformly along the entire unit circle; if $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$2 or $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$3, zeros occupy only part of the circle, and the endpoints define the fidelity edges. For the one-dimensional model,

$$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$4

and similarly in two dimensions,

$$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$5

As $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$6, $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$7, so the fidelity edges pinch the real axis at the quantum critical point.

The same framework yields a finite-size scaling relation for the zero closest to the real axis,

$$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$8

Fitting $$F(R,S)=\bigl\|\sqrt{R}\sqrt{S}\bigr\|_1^2 =\Bigl(\operatorname{Tr}\sqrt{\sqrt{S}R\sqrt{S}}\Bigr)^2,$$9 and $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$0 gives $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$1 and $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$2 in one dimension, and $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$3 with $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$4 in the square-lattice transverse-field Ising model. The distance of fidelity zeros from the real axis therefore scales as $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$5, and their condensation onto the real critical point encodes the universality class.

3. Topological models and complexified control parameters

The fidelity-zero framework has also been extended to topological quantum phase transitions in two-band models. For a generic Bloch Hamiltonian

$$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$6

with a complexified phase-driving parameter $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$7, the many-body ground-state fidelity is defined in a biorthogonal form as a product over momentum modes,

$$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$8

A fidelity zero occurs when any single mode’s factor vanishes. The decisive criterion is that a genuine fidelity zero appears precisely when there exists a momentum mode $$F(\ket{\psi},\ket{\phi})=|\langle\psi|\phi\rangle|^2.$$9 for which the real part of the energy gap closes,

$0$0

In finite-size systems the zeros form discrete lines parallel to the imaginary axis, while in the thermodynamic limit they accumulate into extended regions in the complex parameter plane (Lin et al., 19 Mar 2026).

For the Kitaev chain with complex chemical potential $0$1, the accessible interval of $0$2 supporting fidelity zeros is bounded by the topological critical points $0$3. For each discrete momentum,

$0$4

so finite-size zeros appear as vertical lines in the complex $0$5-plane. In the thermodynamic limit, the zero region satisfies

$0$6

For the Haldane model with complex sublattice mass $0$7, the real parts supporting fidelity zeros satisfy

$0$8

which coincides with the critical points of the Hermitian Haldane model. For the Qi–Wu–Zhang model with $0$9, the condition

$|\psi\rangle$0

implies $|\psi\rangle$1, identifying the transitions at $|\psi\rangle$2. The critical point at $|\psi\rangle$3 is not marked by the disappearance of zeros; instead, it is signaled by fidelity zeros crossing the real axis. This suggests two distinct complex-plane signatures of topological criticality: phase boundaries as endpoints of zero regions, and phase boundaries as points where zero lines pinch or cross the physical axis.

4. Finite-size exact zeros, orthogonality catastrophe, and impurity systems

A persistent complication in fidelity-based diagnostics is the Anderson orthogonality catastrophe: in general, the fidelity $|\psi\rangle$4 approaches zero in the thermodynamic limit no matter whether the parameters of two ground states belong to the same phase or different phases. To overcome this difficulty, finite-size systems with twist boundary conditions can be used. By tuning the magnetic flux, exact zeros of fidelity can be always accessed when $|\psi\rangle$5 and $|\psi\rangle$6 belong to different phases, whereas no exact zero can be observed if $|\psi\rangle$7 and $|\psi\rangle$8 are in the same phase (Zeng et al., 2023).

The thermodynamic scaling of these vanishing overlaps can be made explicit. For one-dimensional Dirac fermions, the fidelity between masses $|\psi\rangle$9 and $$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$0 obeys

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$1

in the thermodynamic limit, while in the non-thermodynamic regime

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$2

For the anisotropic quantum critical point of the Kitaev honeycomb model, the generic thermodynamic scaling takes the form

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$3

and for the specific anisotropic exponents of the model one obtains

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$4

Inside the gapless phase, the same model exhibits

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$5

showing that the overlap can remain strongly suppressed even away from a phase boundary (Mukherjee et al., 2011).

Impurity systems display a complementary distinction between vanishing and non-vanishing overlaps. For the Kondo impurity, the paper finds

$$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$6

so $$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$7 as $$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$8, a fidelity zero in the thermodynamic limit. For the symmetric Friedel–Anderson impurity, by contrast, the fidelity saturates once the level spacing at the Fermi level is smaller than the singlet–triplet excitation energy, so there is no fidelity zero. The paper attributes this difference to phase shifts at the Fermi level: when the fidelity path changes the scattering phase shift, the overlap vanishes; when the phase shifts remain aligned, the scalar product does not change (Bergmann et al., 2011).

A related result identifies a universal single-electron state $$F(\Phi,|\psi\rangle)=\langle\psi|\Phi(|\psi\rangle\langle\psi|)|\psi\rangle =\sum_{j=1}^m |\langle\psi|K_j|\psi\rangle|^2,$$9 at the Fermi level for the symmetric Friedel impurity, the symmetric Friedel–Anderson impurity, and the Kondo impurity. For sufficiently large volume,

$\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$0

Because the same universal state $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$1 appears for both the symmetric Friedel and symmetric Friedel–Anderson impurities, the fidelity between their ground states becomes independent of system size and approaches a finite constant instead of zero (Bergmann, 2012). This suggests that fidelity zeros in impurity problems are controlled not only by interaction strength but also by low-energy phase-shift structure and fixed-point equivalence.

5. Quantum channels, operation fidelity, and process-level constructions

In the channel setting, fidelity zeros are described geometrically through the Kraus operators. Writing each Kraus operator as

$\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$2

with $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$3 and $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$4 Hermitian, the operation fidelity is

$\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$5

The joint numerical range $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$6 and its Crawford number

$\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$7

determine the extreme fidelities: $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$8 A channel admits zero fidelity states iff $\Phi(\rho)=\sum_j K_j\rho K_j^\dagger$9, equivalently $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$0, so the joint numerical range touches the origin (Chełstowski et al., 2022).

This criterion is explicit in low-dimensional examples. For the qubit bit-flip channel,

$F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$1

the fidelity distribution is supported on $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$2, and $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$3. For the diagonal unitary channel and for the phase-damping projection channel,

$F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$4

so no fidelity zeros occur. In this language, fidelity zeros are geometric zeros of distance from the origin in the joint numerical range.

A distinct but related process-level construction is the $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$5-fidelity $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$6, the lowest member of the $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$7-fidelity hierarchy for an $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$8-qubit channel $F_{\min}=\inf_{|\psi\rangle}F(\Phi,|\psi\rangle)$9. It is defined by averaging survival probabilities over tensor products of single-qubit SIC-POVM states, and it satisfies the dimension-independent linear bounds

$$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$0

where $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$1 is the process fidelity. The lower bound was found numerically to be tight for $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$2, while the upper bound is not tight, though the SDP-derived maximum approaches $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$3 as the number of qubits increases (Mayer, 2021). In this usage, the “zero” refers to the lowest approximation in the hierarchy rather than to a vanishing overlap, but it belongs to the same family of fidelity-based diagnostics.

6. Resource-theoretic, geometric, and spectral generalizations

In zero-communication entanglement transformations, fidelity zeros mark strict impossibility under a specified operation set. For local unitaries, the optimal fidelity between pure bipartite states with Schmidt distributions $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$4 and $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$5 is

$$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$6

so

$$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$7

For LOSR or LO, fidelity zero means that even after tensoring the seed Schmidt spectrum with an arbitrary auxiliary probability distribution $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$8, the supports remain disjoint: $$\mathcal F=\sqrt{\sqrt{\rho_1}\,\rho_2\,\sqrt{\rho_1}},$$9 Catalysts can lift such zeros by enlarging the joint support, but the paper emphasizes that state-specific catalysts are much more resource-efficient than universal embezzling families (George et al., 2023).

A geometric extension is provided by the $R,S\ge 0$00-$R,S\ge 0$01-fidelity

$R,S\ge 0$02

which reduces to Uhlmann fidelity at $R,S\ge 0$03. For normalized projectors onto subspaces $R,S\ge 0$04 and $R,S\ge 0$05,

$R,S\ge 0$06

the fidelity depends on $R,S\ge 0$07, and when the subspaces are disjoint,

$R,S\ge 0$08

More generally, for all channels $R,S\ge 0$09,

$R,S\ge 0$10

so a true zero is possible iff $R,S\ge 0$11 is rank-deficient (Yan et al., 2024). In this setting, fidelity zeros correspond to orthogonal supports and to infinite $R,S\ge 0$12-$R,S\ge 0$13 Rényi relative entropy.

The fidelity-spectrum construction gives a further refinement. Because

$R,S\ge 0$14

with $R,S\ge 0$15 the eigenvalues of $R,S\ge 0$16, very small $R,S\ge 0$17 contribute little to the total overlap but strongly affect the fidelity spectrum $R,S\ge 0$18. In the $R,S\ge 0$19 chain, a magnetic impurity in a conventional superconductor, and a bulk superconductor at finite temperature, the paper finds that zeros or near-zeros in individual fidelity-spectrum levels provide a detailed characterization of different phases and phase transitions (Sacramento et al., 2011).

Taken together, these developments show that fidelity zeros are not a single invariant but a family of vanishing-overlap phenomena. In symmetry-breaking systems they arise from transitions between symmetry sectors and obey a Lee–Yang-type unit-circle theorem; in topological models they are controlled by the closure of the real part of the band gap in complexified parameter space; in finite-size and impurity problems they are shaped by orthogonality catastrophe, twist flux, and phase-shift structure; in channel and resource-theoretic settings they are geometric statements about support, numerical range, or factorization constraints. This suggests that fidelity zeros function as a unifying language for exact orthogonality, asymptotic orthogonality, and complex-plane singularity structure across quantum many-body physics and quantum information theory.