Fidelity-Threshold Displacement Radii

• Fidelity-threshold displacement radii are defined as the maximum displacement in the complex plane for which a quantum state’s fidelity remains above a chosen threshold, capturing directional stability.
• They serve as an operational diagnostic by quantifying a state’s anisotropic response to coherent displacements in phase space, particularly in single-mode continuous-variable systems.
• This measure complements scalar indicators like Wigner negativity by revealing nuanced phase-space geometry, aiding in optimizing quantum sensing and error mitigation.

Fidelity-threshold displacement radii are direction-dependent overlap thresholds defined from the fidelity between a quantum state and a displaced version of itself. In the single-mode continuous-variable setting studied in "Photon-Conditioned Squeezed States for Directional Displacement Response in Continuous-Variable Photonics" (Kiefer et al., 26 May 2026), they are written $R_F(\phi)$ and specify, for each phase-space direction $\phi$, the largest complex-plane displacement amplitude $\epsilon$ for which the pure-state overlap $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$ remains above a chosen threshold. They are introduced as a direct displacement-response diagnostic rather than a quantum-error-correction distance, and they quantify anisotropic tolerance or sensitivity to coherent displacements at matched mean photon number. In a broader quantum-geometric sense, fidelity thresholds define superlevel sets in state space and are closely tied to Bures geometry, although the continuous-variable radii $R_F(\phi)$ restrict that general idea to a specific displacement orbit (Kiefer et al., 26 May 2026, Uhlmann, 2011).

1. Formal definition and single-mode continuous-variable conventions

The continuous-variable formulation uses the standard single-mode convention

$$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$

so that $[x,p]=i$. The displacement operator is

$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$

with coherent states $\ket{\alpha}=D(\alpha)\ket{0}$. The complex displacement argument is written as $\alpha=\epsilon e^{i\phi}$, where $\phi$0 is the displacement amplitude in the complex plane and $\phi$1 is the phase-space direction. The numerical convention is explicit: $\phi$2 denotes the complex displacement amplitude in $\phi$3, not the physical quadrature translation distance, so the resulting “radius” is defined in the $\phi$4-plane rather than directly in $\phi$5- or $\phi$6-units (Kiefer et al., 26 May 2026).

For pure states $\phi$7, the radius-defining overlap is

$\phi$8

Given a fidelity threshold $\phi$9, the fidelity-threshold displacement radius is

$\epsilon$0

Operationally, one fixes a direction $\epsilon$1, increases $\epsilon$2 from the origin, and records the largest value that still satisfies the threshold condition. The paper typically uses $\epsilon$3. Special cases aligned with the laboratory quadratures are

$\epsilon$4

and anisotropy is summarized by

$\epsilon$5

A central clarification accompanies the definition: this is not a quantum-error-correction distance, because no recovery map is assumed. It is instead a task-oriented diagnostic of how far a state can be displaced along a specified phase-space direction while still remaining above a prescribed overlap threshold (Kiefer et al., 26 May 2026).

2. Phase-space geometry, characteristic functions, and local scaling

The radius $\epsilon$6 is tied directly to phase-space structure. With the standard Wigner normalization

$\epsilon$7

the origin value obeys

$\epsilon$8

and for Fock states

$\epsilon$9

The displacement overlap is the squared magnitude of the symmetrically ordered characteristic function,

$$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$0

so, in phase-space terms, it measures the overlap of the Wigner function with itself after a shift by $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$1 (Kiefer et al., 26 May 2026).

This immediately gives the geometric content of the radii. If the Wigner function is broad along a given direction, larger displacements are needed before the overlap falls below threshold; if it is narrow or highly structured, small displacements suffice. The radius contour $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$2 is therefore a directional “size” of the phase-space region over which the state is effectively invariant under coherent displacement.

For squeezed Fock states $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$3, the directional asymmetry is controlled by quadrature variances: $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$4 Near the origin, the fidelity decay obeys the small-displacement expansions

$$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$5

for real displacements $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$6, and

$$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$7

for imaginary displacements $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$8, with a convention-dependent numerical factor $$F_\psi(\epsilon e^{i\phi}) = |\langle \psi | D(\epsilon e^{i\phi}) | \psi \rangle|^2$$9. The conjugate variance controls the initial fidelity decay: displacement along $R_F(\phi)$0 is governed by $R_F(\phi)$1, and displacement along $R_F(\phi)$2 is governed by $R_F(\phi)$3. At fixed threshold this leads to the approximate scaling

$R_F(\phi)$4

up to state-dependent constants (Kiefer et al., 26 May 2026).

The full angular dependence reflects these geometric mechanisms. Fock states $R_F(\phi)$5 and $R_F(\phi)$6 are isotropic, so $R_F(\phi)$7 is circular and $R_F(\phi)$8. Cat states are anisotropic, with lobes aligned roughly with the coherent-state separation axis. Photon-conditioned squeezed states $R_F(\phi)$9 and $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$0 are strongly anisotropic, and the two-photon-subtracted squeezed state displays an enlarged favorable-axis radius over a finite angular sector rather than only at a single special angle. In the reported numerics, Fig. 3 gives the Cartesian diagnostics $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$1, $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$2, and $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$3 as functions of matched $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$4, while Appendix Fig. 1 gives the polar contour $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$5 at $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$6 (Kiefer et al., 26 May 2026).

3. State-family comparisons at matched mean photon number

The task-oriented comparison is performed at matched mean photon number,

$$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$7

so that differences in $$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$8 reflect how the photon budget is distributed between squeezing and non-Gaussian excitation rather than simple energy increase. The benchmark families are squeezed Fock states, photon-subtracted squeezed states, and even/odd cat states (Kiefer et al., 26 May 2026).

Squeezed Fock states are defined by

$$[a,a^\dagger]=1,\qquad x=\frac{a+a^\dagger}{\sqrt{2}},\qquad p=\frac{a-a^\dagger}{i\sqrt{2}},$$9

with mean photon number

$[x,p]=i$0

For example,

$[x,p]=i$1

For a target $[x,p]=i$2, the squeezing $[x,p]=i$3 is chosen by solving this relation, the state is constructed in the Fock basis, and $[x,p]=i$4 and $[x,p]=i$5 are then computed numerically.

Photon subtraction begins from squeezed vacuum. Single-photon subtraction obeys

$[x,p]=i$6

with $[x,p]=i$7 and $[x,p]=i$8. After normalization, the one-photon-subtracted squeezed state is exactly a squeezed single-photon state. Two-photon subtraction gives

$[x,p]=i$9

so the two-click state is an even-parity squeezed superposition of $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$0 and $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$1, not $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$2. In the squeezed frame, the amplitude ratio is

$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$3

and in the large-squeezing limit $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$4, the inner state tends to

$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$5

Cat benchmarks are

$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$6

with mean photon numbers

$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$7

and overlap $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$8. For larger $D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a),$9, the coherent components are nearly orthogonal, and even and odd cats have nearly identical displacement response.

At matched $\ket{\alpha}=D(\alpha)\ket{0}$0, the one-photon-subtracted state behaves like a squeezed Fock-1 state. The two-photon-subtracted squeezed state is more strongly anisotropic and can have the largest favorable-axis radius while also having the largest $\ket{\alpha}=D(\alpha)\ket{0}$1. This is the basis for the statement that photon-conditioned squeezed states provide an origin-centered alternative with tunable anisotropic displacement response, while the two-photon-subtracted squeezed state shows favorable displacement-fidelity radii over selected quadrature directions at matched $\ket{\alpha}=D(\alpha)\ket{0}$2 (Kiefer et al., 26 May 2026).

4. Complementarity with integrated Wigner negativity

The same comparison is carried out using the integrated Wigner negativity

$\ket{\alpha}=D(\alpha)\ket{0}$3

together with the energy-normalized quantity $\ket{\alpha}=D(\alpha)\ket{0}$4. The reported properties are sharp: $\ket{\alpha}=D(\alpha)\ket{0}$5 for Gaussian states such as squeezed vacuum, $\ket{\alpha}=D(\alpha)\ket{0}$6 quantifies non-Gaussianity or negativity, and $\ket{\alpha}=D(\alpha)\ket{0}$7 is invariant under Gaussian unitaries including squeezing and rotations (Kiefer et al., 26 May 2026).

This invariance is the key reason that $\ket{\alpha}=D(\alpha)\ket{0}$8 and $\ket{\alpha}=D(\alpha)\ket{0}$9 encode different information. Squeezing a non-Gaussian state changes the geometry of the Wigner function without changing the total negativity volume, so $\alpha=\epsilon e^{i\phi}$0 and $\alpha=\epsilon e^{i\phi}$1 remain unchanged while the directional displacement response can change dramatically. The paper states this as a central message: scalar Wigner negativity and directional displacement-fidelity response do not rank the state families in the same order.

The comparative conclusions are correspondingly split. Cat states remain strong resources in $\alpha=\epsilon e^{i\phi}$2 and $\alpha=\epsilon e^{i\phi}$3 at matched $\alpha=\epsilon e^{i\phi}$4. Photon-conditioned squeezed states do not dominate in $\alpha=\epsilon e^{i\phi}$5, but they can have larger $\alpha=\epsilon e^{i\phi}$6 in selected directions. In that sense, $\alpha=\epsilon e^{i\phi}$7 is sensitive to phase-space geometry, anisotropy, and interference structure, whereas $\alpha=\epsilon e^{i\phi}$8 and $\alpha=\epsilon e^{i\phi}$9 are scalar summaries that are insensitive to Gaussian deformation. This is precisely why the directional radii are introduced as a task-oriented complement rather than a replacement for standard negativity measures (Kiefer et al., 26 May 2026).

5. Relation to fidelity geometry and Bures structure

In Uhlmann’s treatment, transition probability $\phi$00 and fidelity $\phi$01 for density operators are defined by

$\phi$02

and for pure states

$\phi$03

This means that the pure-state quantity used in the continuous-variable displacement setting, $\phi$04, coincides with transition probability in Uhlmann’s convention rather than with Uhlmann’s unsquared fidelity. The distinction is purely conventional but important when comparing formulas across subfields (Uhlmann, 2011).

Uhlmann’s broader framework supplies the geometry behind fidelity thresholds. For a fixed reference state $\phi$05 and threshold $\phi$06, the superlevel set

$\phi$07

is a Bures ball, with radius

$\phi$08

in the convention quoted in the paper’s discussion of Bures distance. The construction is supported by the amplitude formalism, in which a state is written as $\phi$09, together with the parallelity condition

$\phi$10

for horizontal lifts of state-space curves. The Bures line element is

$\phi$11

with $\phi$12, and for parallel lifts this becomes

$\phi$13

Channel monotonicity,

$\phi$14

implies that fidelity-defined balls cannot shrink under the application of the same channel to both states. Tensor products obey

$\phi$15

These statements are general state-space properties rather than specific results about phase-space displacements. This suggests that the directional radius $\phi$16 can be viewed as the restriction of a fidelity superlevel set to the one-parameter family of displaced states $\phi$17, while retaining the warning already emphasized in the photonic context that no recovery map is assumed (Uhlmann, 2011, Kiefer et al., 26 May 2026).

6. Operational roles and parameter-space analogues

The directional radius is introduced for explicitly operational reasons. In continuous-variable photonics, homodyne detection selects a quadrature axis through the local-oscillator phase. When the dominant displacement noise or the signal of interest is effectively aligned with a known quadrature, one can tune the squeezing phase $\phi$18, or equivalently the relative phase between squeezing and the homodyne local oscillator, to orient either a robust axis with large $\phi$19 or a sensitive axis with small $\phi$20. Large $\phi$21 is advantageous for displacement-noise mitigation along a known direction; small $\phi$22 is advantageous for directional sensing, where the same anisotropy is used in reverse as a sensitivity resource. The local sensitivity diagnostic is

$\phi$23

and the abstract explicitly identifies directional sensing as a natural dual application (Kiefer et al., 26 May 2026).

A distinct but closely related use of fixed-fidelity radii appears in many-body parameter space. In "Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit" (Mukherjee et al., 2011), the displacement radius is defined by a threshold condition such as

$\phi$24

and is obtained by inverting the scaling of $\phi$25. In the 1D Dirac example at the critical point, the thermodynamic law

$\phi$26

implies

$\phi$27

so the allowable mass displacement scales as $\phi$28. At the anisotropic quantum critical point of the Kitaev model, the thermodynamic scaling

$\phi$29

implies

$\phi$30

whereas in the non-thermodynamic regime

$\phi$31

gives

$\phi$32

These are parameter-space rather than phase-space radii, but they show the same general logic: a fixed fidelity threshold defines a displacement tolerance whose scaling reveals anisotropy, criticality, and orthogonality-catastrophe behavior (Mukherjee et al., 2011).

7. Scope, assumptions, and open directions

The continuous-variable study is deliberately restricted. It focuses on pure, idealized states; does not include loss, detector inefficiency, mode mismatch, or finite heralding probability; assumes no recovery map or explicit error-correction scheme; analyzes coherent displacements rather than loss or dephasing channels; and computes the radii numerically in a truncated Fock basis with cutoff $\phi$33, using phase-space grids and linear interpolation at threshold crossings. Convergence was checked by increasing the cutoff above $\phi$34 and varying grid density and phase-space window; the reported changes in $\phi$35 and $\phi$36 were small, on the order of $\phi$37, and the displacement radii and state ordering were stable (Kiefer et al., 26 May 2026).

The open directions identified in the same work are correspondingly specific. They include incorporating realistic imperfections to determine how $\phi$38 is degraded; connecting directional displacement-response analysis to explicit bosonic codes such as squeezed-Fock codes, cat codes, and GKP codes; exploring photon-conditioned squeezed states as seeds in grid-state or GKP-state generation protocols; and using resource maps based on $\phi$39, $\phi$40, and $\phi$41 to pre-evaluate candidate non-Gaussian state families before complex experimental generation. A plausible implication is that fidelity-threshold displacement radii are most informative when the displacement axis is known or controllable, because their principal advantage over scalar resource measures lies precisely in resolving directional structure (Kiefer et al., 26 May 2026).