Quantum State Fidelity

• Quantum state fidelity is a quantitative metric that measures the closeness between quantum states, essential for evaluating state preparation, transmission, and error correction.
• Advanced estimation protocols such as classical shadow tomography, variational methods, and QSVT enable efficient and accurate fidelity assessments in high-dimensional quantum systems.
• Research explores both geometric and operational facets of fidelity while addressing limitations through noise-aware training and resource-theoretic analysis.

Quantum state fidelity is a quantitative metric that assesses the similarity or "closeness" between two quantum states. Fidelity plays a central role across quantum information science, functioning both as a mathematical distance measure in Hilbert space and as an operational benchmark for tasks like state preparation, transmission, error correction, and resource certification. Beyond pure states, fidelity extends to mixed states and can be adapted to channels and ensembles, underpinned by a range of analytical formulas, estimation protocols, and performance bounds. Recent research has systematically analyzed the structural, geometric, and practical facets of quantum state fidelity, revealing both its utility and fundamental limitations.

1. Formal Definitions and Analytical Frameworks

Quantum state fidelity is rigorously defined for pure and mixed quantum states, with the most common form for pure states |ψ₁⟩ and |ψ₂⟩ given by

$F = |\langle \psi_1 | \psi_2 \rangle|^2.$

For mixed states ρ and σ (density operators), the Uhlmann–Jozsa fidelity is employed:

$$F(\rho, \sigma) = \left[\operatorname{Tr}\left(\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}\right)\right]^2,$$

which satisfies symmetry, reduction to transition probability for pure states, invariance under unitaries, and joint concavity (Liang et al., 2018). Extensions to norm-based (Schatten-p), arithmetic/geometric mean, and Chernoff-type fidelities have also been proposed, each fulfilling subsets of required axioms.

For families like coherent or squeezed states, exact analytical relations connect fidelity to physically relevant observables—most notably in terms of energy. For example, for coherent states with relative energy difference $y$,

$F_\mathrm{max}^{(\mathrm{coh})} = \exp(-y^2/2),$

and for squeezed vacuum states,

$$F_\mathrm{max}^{(\mathrm{sqz})} = [1 + (y^2/4)]^{-1/2}$$

(Dodonov, 2011). Binomial and negative binomial states reveal even more restrictive bounds, with closed-form expressions depending on energy difference and state parameters.

Fidelity for Gaussian states—ubiquitous in continuous-variable quantum systems—is characterized via first and second statistical moments (displacement vector and covariance matrix). For mixed Gaussian state ρ₀ and pure σ₁,

$$F(\rho_0, \left|\phi\right\rangle) = \frac{2^n}{\sqrt{\det(V_0 + V_1)}} \exp\left\{ -\frac{1}{2} (x_0 - x_1)^{\mathrm{T}}(V_0 + V_1)^{-1}(x_0 - x_1) \right\}$$

(Spedalieri et al., 2012).

2. Fidelity Estimation: Protocols and Algorithms

Direct estimation of fidelity is often a key practical challenge, addressed by a variety of protocols:

• Classical shadow tomography & amplitude estimation:

Efficient protocols combine classical shadow tomography (sampling Clifford unitaries and outcomes) with quantum amplitude estimation, enabling fidelity estimation with resources scaling as $O(\sqrt{d})$ for Hilbert space dimension $d$—offering a quadratic speedup over standard DFE methods (Vairogs et al., 10 Dec 2024).

• Variational quantum fidelity estimation:

For scenarios where one state is approximately low-rank, fidelity bounds can be systematically tightened using the truncated eigenbasis of ρ, coupled with swap tests or similar routines performed variationally. This protocol achieves efficient lower and upper bounding, scales polynomially for low-rank states, and provides an exponential quantum advantage in specific regimes (Cerezo et al., 2019).

• Quantum algorithms leveraging block-encodings and singular value transformation (QSVT):

Algorithms exist to estimate

$$F(\rho, \sigma) = \mathrm{Tr} \big[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\big]$$

in time $O(\operatorname{poly}(\log d, r, 1/\varepsilon))$, with $r$ the lower rank and $\varepsilon$ the target precision, using quantum block-encoding described by oracles for the purifications (Wang et al., 2021, Gilyén et al., 2022). These protocols are provably more efficient than full quantum tomography for low-rank states.

• Machine-learning-based fidelity estimation:

Artificial neural networks can map minimal sets of measurement outcomes to fidelity predictions, requiring only a small (system-size–independent) number of measurement settings in nonadversarial scenarios (Zhang et al., 2021).

• Optimal convex approximations and geometric protocols:

For certain operational settings, the optimal convex approximation of a target state in a feasible set (e.g., mixtures of available preparations) can be obtained by maximizing fidelity—as contrasted with minimizing trace norm distance (Zhou et al., 2021).

• Resource-aware fidelity estimation:

In the context of the resource theory of nonstabilizerness, the sample complexity of phase-space–based fidelity estimation protocols is governed by the “Wigner rank” or “mana,” quantifying the non-classicality/magic of the state; protocols for Clifford-stabilizer states are exponentially more efficient (Liu et al., 15 Jun 2025).

3. Operational and Resource-Theoretic Significance

Fidelity functions as both a geometric and operational measure of quantum state similarity:

• Hilbert space and geometric interpretations:

Fidelity quantifies the "distance" in Hilbert space, with geometric interpretations connecting it to principal angles between subspaces, and to the extremal points (minimal eigenvalues) of convex sets of states or channels (Li et al., 2015). For convex combinations, fidelity minimizes to the square root of the minimal eigenvalue of ρ under arbitrary quantum channels.

• Physical and observable-sensitive considerations:

Sole reliance on fidelity can be misleading. For instance, two quantum states may have high fidelity even if they differ significantly in observable properties such as mean energy or entanglement (Dodonov, 2011, Bina et al., 2013). To guarantee, e.g., a difference in energy less than 10% for coherent states, fidelity must typically exceed 0.995, and the threshold is even more stringent for squeezed and binomial-type states (Dodonov, 2011). Similarly, high-fidelity regions on the Bloch sphere can span both separable and entangled states, or classical and nonclassical regimes in continuous-variable systems (Bina et al., 2013).

• Resource theory links:

Recent research connects sample complexity in fidelity estimation directly to resource-theoretic quantities like the logarithmic Wigner rank and mana (Liu et al., 15 Jun 2025).

• Purity, coherence, and correlation:

Fidelity-based definitions of state purity

$$\mathcal{P}_F(\rho) = -\log_d \mathcal{F}(\rho, I/d)$$

and coherence monotones allow a one-to-one relationship between maximal coherence and purity, demonstrating that maximal coherence is strictly determined by the state's purity (S et al., 2021). These relations extend to quantifying measurement-induced quantum correlations.

4. Benchmarking, State Transfer, and Experimental Applications

• Quantum information benchmarks:

Fidelity plays a decisive role in characterizing teleportation (with classical and quantum thresholds), quantum memory, gate calibration, and quantum communication performance, especially when output states are noisy or imperfect (Liang et al., 2018). In process verification, fidelity lower bounds (e.g., Hofmann bounds, and more recent minimal probe-state protocols) are tied to experimentally accessible state fidelities while optimizing measurement resources (Fiurasek et al., 2014).

• Fidelity distributions in protocols:

Assessing an average fidelity may conceal performance disparities. A full probability distribution of fidelity, pdf(F), reveals protocol-specific features like worst-case performance and tail probabilities. Two quantum state transfer protocols can be engineered to produce the same ⟨F⟩ but result in fundamentally different likelihoods of low fidelity, impacting reliability in quantum computation or communication tasks (Lorenzo et al., 4 May 2024).

• Experimental implementation and noise-aware training:

State-of-the-art platforms, including silicon photonic chips, adopt fidelity-optimized measurement protocols for efficient characterization with minimized error bars compared to traditional direct estimation (Wollmann et al., 2023). Methods like RobustState, combining noise-aware updates from real hardware with classical simulation, have demonstrated up to 96% fidelity in 4-qubit state preparation and substantial improvement in noise robustness (Wang et al., 2023).

• Functional graph inference:

Quantum state fidelity is being actively explored as an alternative metric in high-dimensional graph inference. Embedding classical data as quantum states, then constructing distance matrices via quantum fidelity circuits, can reveal functional modularity and subtle dependencies beyond standard metrics like Pearson correlation or Euclidean distance (Chan et al., 23 Aug 2025).

5. Limitations, Cautions, and Future Perspectives

Using fidelity as a proxy for operational quantum resources (entanglement, magic, nonclassicality) is fundamentally limited, as demonstrated by counterexamples where high-fidelity pairs span distinct domains of operational utility (Bina et al., 2013). In state engineering, quantum certification, and tomography, fidelity should thus be combined with constraints on additional observables or accompanied by full tomographic characterization.

Open research avenues highlighted in recent literature include:

• Extension of analytic fidelity–observable relations to general mixed and entangled states (Dodonov, 2011).
• Development of fidelity metrics for channels, ensembles, or time-varying states with operational guarantees.
• Algorithms that further reduce experimental overhead (sample complexity, state copies, or communication) for fidelity estimation in large-scale systems (Vairogs et al., 10 Dec 2024, Gilyén et al., 2022).
• Noise-aware and error-mitigated strategies for variational algorithm training (Wang et al., 2023).
• Explicit resource-theoretic accounting in the fidelity estimation tasks, mapping required resources to structural properties of the state (Liu et al., 15 Jun 2025).

6. Summary Table: Fidelity Analytical Formulas for State Families

State Family	Fidelity–Energy Relation	Small y Expansion	Key Threshold Example
Coherent	$F=\exp(-y^2/2)$	$1 - y^2/2$	$F > 0.995$ for $y < 0.1$
Squeezed vacuum	$F=[1 + y^2/4]^{-1/2}$	$1 - y^2/8$	Stricter than coherent
Negative binomial	$F=[1 + y^2/4]^{-1}$ (p=1: phase state)	$1 - y^2/4$	Strongest restriction
Binomial	Closed form; recovers coherent for large M	$1 - y^2$ (for large M)	Matches coherent as $M\to\infty$

For these families, achieving an energy discrepancy less than 10% between states requires fidelity greater than 0.995 for coherent states, and even higher for others (Dodonov, 2011). This quantitative table underscores the need for very high fidelity, above conventional operational thresholds, to ensure genuine physical similarity regarding observables.

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Quantum state fidelity occupies a foundational position in quantum information science, but must be evaluated and applied judiciously, incorporating both analytical understanding and operational context. Its geometric, resource-theoretic, algorithmic, and experimental dimensions continue to evolve with the broader development of quantum technologies.