---
title: Fidelity-Threshold Displacement Radii
url: https://www.emergentmind.com/topics/fidelity-threshold-displacement-radii
type: topic
---

# Fidelity-Threshold Displacement Radii

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" [2605.27660], 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 [2605.27660] [1106.0979].

## 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 \(\epsilon=|\alpha|\) is the displacement amplitude in the complex plane and \(\phi\) is the phase-space direction. The numerical convention is explicit: \(\epsilon\) denotes the complex displacement amplitude in \(D(\epsilon e^{i\phi})\), not the physical quadrature translation distance, so the resulting “radius” is defined in the \(\alpha\)-plane rather than directly in \(x\)- or \(p\)-units [2605.27660].

For pure states \(\ket{\psi}\), the radius-defining overlap is
\[
F_\psi(\alpha)=\left|\bra{\psi}D(\alpha)\ket{\psi}\right|^2.
\]
Given a fidelity threshold \(F_{\rm th}\), the fidelity-threshold displacement radius is
\[
R_F(\phi)=\max\left\{\epsilon: F_\psi(\epsilon e^{i\phi})\ge F_{\rm th}\right\}.
\]
Operationally, one fixes a direction \(\phi\), increases \(\epsilon\) from the origin, and records the largest value that still satisfies the threshold condition. The paper typically uses \(F_{\rm th}=0.90\). Special cases aligned with the laboratory quadratures are
\[
R_x=R_F(0),\qquad R_p=R_F(\pi/2),
\]
and anisotropy is summarized by
\[
\frac{R_{\max}}{R_{\min}}=\frac{\max_\phi R_F(\phi)}{\min_\phi R_F(\phi)}.
\]

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 [2605.27660].

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

The radius \(R_F(\phi)\) is tied directly to phase-space structure. With the standard Wigner normalization
\[
\int_{\mathbb{R}^2}W_\rho(x,p)\,dx\,dp=1,
\]
the origin value obeys
\[
W_\rho(0,0)=\frac{1}{\pi}\mathrm{Tr}[\Pi\rho],\qquad \Pi=(-1)^{a^\dagger a},
\]
and for Fock states
\[
W_{\ket{n}}(0,0)=\frac{(-1)^n}{\pi}.
\]
The displacement overlap is the squared magnitude of the symmetrically ordered characteristic function,
\[
\chi_\psi(\alpha)=\mathrm{Tr}[\rho D(\alpha)],\qquad
F_\psi(\alpha)=|\chi_\psi(\alpha)|^2,
\]
so, in phase-space terms, it measures the overlap of the Wigner function with itself after a shift by \(\alpha\) [2605.27660].

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 \(R_F(\phi)\) is therefore a directional “size” of the phase-space region over which the state is effectively invariant under coherent displacement.

For squeezed Fock states \(S(r)\ket{n}\), the directional asymmetry is controlled by quadrature variances:
\[
\mathrm{Var}(x)\propto \left(n+\frac{1}{2}\right)e^{-2r},\qquad
\mathrm{Var}(p)\propto \left(n+\frac{1}{2}\right)e^{+2r}.
\]
Near the origin, the fidelity decay obeys the small-displacement expansions
\[
1-F_x(\epsilon)=A\,\mathrm{Var}(p)\,\epsilon^2+O(\epsilon^4),
\]
for real displacements \(D(\epsilon)\), and
\[
1-F_p(\epsilon)=A\,\mathrm{Var}(x)\,\epsilon^2+O(\epsilon^4),
\]
for imaginary displacements \(D(i\epsilon)\), with a convention-dependent numerical factor \(A\). The conjugate variance controls the initial fidelity decay: displacement along \(x\) is governed by \(\mathrm{Var}(p)\), and displacement along \(p\) is governed by \(\mathrm{Var}(x)\). At fixed threshold this leads to the approximate scaling
\[
R_x\sim e^{-r},\qquad R_p\sim e^{+r},
\]
up to state-dependent constants [2605.27660].

The full angular dependence reflects these geometric mechanisms. Fock states \(\ket{1}\) and \(\ket{2}\) are isotropic, so \(R_F(\phi)\) is circular and \(R_{\max}/R_{\min}=1\). Cat states are anisotropic, with lobes aligned roughly with the coherent-state separation axis. Photon-conditioned squeezed states \(aS(r)\ket{0}\) and \(a^2S(r)\ket{0}\) 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 \(R_x\), \(R_p\), and \(R_{\max}/R_{\min}\) as functions of matched \(\langle n\rangle\), while Appendix Fig. 1 gives the polar contour \(R_F(\phi)\) at \(\langle n\rangle\simeq 3\) [2605.27660].

## 3. State-family comparisons at matched mean photon number

The task-oriented comparison is performed at matched mean photon number,
\[
\langle n\rangle=\langle a^\dagger a\rangle,
\]
so that differences in \(R_F(\phi)\) 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 [2605.27660].

Squeezed Fock states are defined by
\[
S(r,\theta)=\exp\!\left[\frac{1}{2}\left(r e^{-i\theta}a^2-r e^{i\theta}a^{\dagger 2}\right)\right],
\qquad
\ket{\psi_n(r,\theta)}=S(r,\theta)\ket{n},
\]
with mean photon number
\[
\langle n\rangle_{S\ket{n}}=n+(2n+1)\sinh^2 r.
\]
For example,
\[
\langle n\rangle_{S\ket{0}}=\sinh^2 r,\qquad
\langle n\rangle_{S\ket{1}}=1+3\sinh^2 r.
\]
For a target \(\langle n\rangle\), the squeezing \(r\) is chosen by solving this relation, the state is constructed in the Fock basis, and \(F_\psi(\alpha)\) and \(R_F(\phi)\) are then computed numerically.

Photon subtraction begins from squeezed vacuum. Single-photon subtraction obeys
\[
aS(r,\theta)\ket{0}=S(r,\theta)(\mu a+\nu a^\dagger)\ket{0}
=\nu\,S(r,\theta)\ket{1},
\]
with \(\mu=\cosh r\) and \(\nu=-e^{i\theta}\sinh r\). After normalization, the one-photon-subtracted squeezed state is exactly a squeezed single-photon state. Two-photon subtraction gives
\[
a^2S(r,\theta)\ket{0}
=
S(r,\theta)\left(\mu\nu\ket{0}+\sqrt{2}\nu^2\ket{2}\right),
\]
so the two-click state is an even-parity squeezed superposition of \(\ket{0}\) and \(\ket{2}\), not \(S(r,\theta)\ket{2}\). In the squeezed frame, the amplitude ratio is
\[
\frac{\sqrt{2}\nu^2}{\mu\nu}
=
-\sqrt{2}e^{i\theta}\tanh r,
\]
and in the large-squeezing limit \(\tanh r\to 1\), the inner state tends to
\[
\ket{0}-\sqrt{2}e^{i\theta}\ket{2}.
\]

Cat benchmarks are
\[
\ket{\mathcal C_\pm(\alpha)}=\mathcal N_\pm(\ket{\alpha}\pm\ket{-\alpha}),
\]
with mean photon numbers
\[
\langle n\rangle_{\mathcal C_-}=|\alpha|^2\coth |\alpha|^2,\qquad
\langle n\rangle_{\mathcal C_+}=|\alpha|^2\tanh |\alpha|^2,
\]
and overlap \(\langle \alpha|-\alpha\rangle=e^{-2|\alpha|^2}\). For larger \(|\alpha|\), the coherent components are nearly orthogonal, and even and odd cats have nearly identical displacement response.

At matched \(\langle n\rangle\), 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 \(R_{\max}/R_{\min}\). 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 \(\langle n\rangle\) [2605.27660].

## 4. Complementarity with integrated Wigner negativity

The same comparison is carried out using the integrated Wigner negativity
\[
\delta=\frac{1}{2}\left[\int_{\mathbb{R}^2}|W(x,p)|\,dx\,dp-1\right],
\]
together with the energy-normalized quantity \(\delta/\langle n\rangle\). The reported properties are sharp: \(\delta=0\) for Gaussian states such as squeezed vacuum, \(\delta>0\) quantifies non-Gaussianity or negativity, and \(\delta\) is invariant under Gaussian unitaries including squeezing and rotations [2605.27660].

This invariance is the key reason that \(\delta\) and \(R_F(\phi)\) encode different information. Squeezing a non-Gaussian state changes the geometry of the Wigner function without changing the total negativity volume, so \(\delta\) and \(\delta/\langle n\rangle\) 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 \(\delta\) and \(\delta/\langle n\rangle\) at matched \(\langle n\rangle\). Photon-conditioned squeezed states do not dominate in \(\delta\), but they can have larger \(R_F(\phi)\) in selected directions. In that sense, \(R_F(\phi)\) is sensitive to phase-space geometry, anisotropy, and interference structure, whereas \(\delta\) and \(\delta/\langle n\rangle\) 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 [2605.27660].

## 5. Relation to fidelity geometry and Bures structure

In Uhlmann’s treatment, transition probability \(\Pr(\rho,\sigma)\) and fidelity \(F(\rho,\sigma)\) for density operators are defined by
\[
F(\rho,\sigma)=\sqrt{\Pr(\rho,\sigma)}
=
\operatorname{Tr}\!\left(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right),
\]
and for pure states
\[
\Pr(\rho,\sigma)=|\langle\psi|\phi\rangle|^2,\qquad
F(\rho,\sigma)=|\langle\psi|\phi\rangle|.
\]
This means that the pure-state quantity used in the continuous-variable displacement setting, \(F_\psi(\alpha)=|\langle\psi|D(\alpha)|\psi\rangle|^2\), 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 [1106.0979].

Uhlmann’s broader framework supplies the geometry behind fidelity thresholds. For a fixed reference state \(\rho\) and threshold \(f\), the superlevel set
\[
\mathcal B_f(\rho)=\{\sigma:F(\rho,\sigma)\ge f\}
\]
is a Bures ball, with radius
\[
R(f)=\sqrt{2(1-f)}
\]
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 \(\rho=WW^\dagger\), together with the parallelity condition
\[
W_s^\dagger \dot W_s-\dot W_s^\dagger W_s=0
\]
for horizontal lifts of state-space curves. The Bures line element is
\[
\mathrm{length}_{\mathrm{Bures}}[\rho_s]
=
\int \sqrt{\operatorname{Tr}(G_s^\dagger G_s\,\rho_s)}\,ds
\]
with \(\dot\rho_s=G_s\rho_s+\rho_s G_s\), and for parallel lifts this becomes
\[
\mathrm{length}_{\mathrm{Bures}}[\rho_s]
=
\int \sqrt{\operatorname{Tr}(\dot W_s^\dagger \dot W_s)}\,ds.
\]

Channel monotonicity,
\[
F(\rho,\sigma)\le F(\Phi(\rho),\Phi(\sigma)),
\]
implies that fidelity-defined balls cannot shrink under the application of the same channel to both states. Tensor products obey
\[
F(\rho_1\otimes\omega_1,\rho_2\otimes\omega_2)
=
F(\rho_1,\rho_2)\,F(\omega_1,\omega_2).
\]
These statements are general state-space properties rather than specific results about phase-space displacements. This suggests that the directional radius \(R_F(\phi)\) can be viewed as the restriction of a fidelity superlevel set to the one-parameter family of displaced states \(D(\epsilon e^{i\phi})\ket{\psi}\), while retaining the warning already emphasized in the photonic context that no recovery map is assumed [1106.0979] [2605.27660].

## 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 \(\theta\), or equivalently the relative phase between squeezing and the homodyne local oscillator, to orient either a robust axis with large \(R_F(\phi)\) or a sensitive axis with small \(R_F(\phi)\). Large \(R_F(\phi)\) is advantageous for displacement-noise mitigation along a known direction; small \(R_F(\phi)\) is advantageous for directional sensing, where the same anisotropy is used in reverse as a sensitivity resource. The local sensitivity diagnostic is
\[
\Gamma(\phi)=
\left.\frac{d}{d\epsilon^2}\left[1-F_\psi(\epsilon e^{i\phi})\right]\right|_{\epsilon=0},
\]
and the abstract explicitly identifies directional sensing as a natural dual application [2605.27660].

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" [1108.2597], the displacement radius is defined by a threshold condition such as
\[
F(\lambda,\lambda+\delta_c)=F_c,
\]
and is obtained by inverting the scaling of \(-\ln F\). In the 1D Dirac example at the critical point, the thermodynamic law
\[
-\ln F\sim cL|m|
\]
implies
\[
|m_c|\sim \frac{-\ln F_c}{cL},
\]
so the allowable mass displacement scales as \(L^{-1}\). At the anisotropic quantum critical point of the Kitaev model, the thermodynamic scaling
\[
\ln F\sim -\delta^{3/2}L^2
\]
implies
\[
\delta_c\sim L^{-4/3},
\]
whereas in the non-thermodynamic regime
\[
\ln F\sim -\delta^2 L^{5/2}
\]
gives
\[
\delta_c\sim L^{-5/4}.
\]
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 [1108.2597].

## 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 \(n_{\max}=80\), using phase-space grids and linear interpolation at threshold crossings. Convergence was checked by increasing the cutoff above \(80\) and varying grid density and phase-space window; the reported changes in \(\delta\) and \(\delta/\langle n\rangle\) were small, on the order of \(10^{-3}\), and the displacement radii and state ordering were stable [2605.27660].

The open directions identified in the same work are correspondingly specific. They include incorporating realistic imperfections to determine how \(R_F(\phi)\) 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 \(\delta\), \(\delta/\langle n\rangle\), and \(R_F(\phi)\) 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 [2605.27660].

Source: https://www.emergentmind.com/topics/fidelity-threshold-displacement-radii