Fidelity-Based Analytical Lower Bounds

• Fidelity-based analytical lower bounds are rigorous, computable measures used to certify quantum operations and benchmark error correction in diverse quantum systems.
• They employ techniques such as state-dependent bounds, spectral expansion, and semidefinite programming to tighten estimates beyond traditional state-independent inequalities.
• These bounds enable efficient quantification of entanglement and coherence, supporting experimental validation in quantum process certification and resource theories.

Fidelity-based analytical lower bounds are rigorous, computable inequalities that provide guaranteed minimum values for fidelity or fidelity-derived quantities—such as quantum process fidelity, fidelity susceptibility, or resource-theoretic measures—in various quantum information processing, metrology, and quantum statistical physics contexts. These lower bounds allow for the certification of quantum operations, quantification of entanglement and coherence, and benchmarking of quantum computations and error correction schemes, often using limited or partial measurement data and with manageable analytical or computational effort.

1. Foundational Concepts of Fidelity and Its Bounds

In quantum information theory, the fidelity between density operators $\rho$ and $\sigma$ is defined by $F(\rho,\sigma) = \|\sqrt{\rho} \sqrt{\sigma}\|_1$, quantifying the overlap or distinguishability of quantum states. Fuchs–van de Graaf inequalities provide universal, state-independent upper and lower bounds for fidelity in terms of the trace norm: $$1 - \tfrac{1}{2}\|\rho-\sigma\|_1 \le F(\rho,\sigma) \le \sqrt{1-\tfrac{1}{4}\|\rho-\sigma\|_1^2}$$ (Zhang et al., 2014).

Recent research leverages additional properties, such as max-relative entropy, operator supports, or symmetry, to yield tighter, state-dependent lower bounds. In quantum process characterization, bounds are often derived for the process fidelity $F_\chi$—the overlap between the actual and ideal Choi operators—using subsets of experimentally accessible input states and corresponding output fidelities (Fiurasek et al., 2014, Mayer et al., 2018).

2. Analytical Lower Bounds: Generic Structures and Key Results

State-Dependent Bounds

State-dependent lower bounds improve upon state-independent Fuchs–van de Graaf results by incorporating information such as the max-relative entropy $$S_{\max}(\rho\|\sigma)=\inf\{\gamma: \rho\leq e^\gamma \sigma\}$$, resulting in

$$F(\rho, \sigma) \geq 1-\frac{1}{2} \left( \frac{e^{S_{\max}(\rho\|\sigma)}-1}{e^{S_{\max}(\rho\|\sigma)}+1} \right)\|\rho-\sigma\|_1$$

This form provides a strictly tighter bound when $S_{\max}$ is finite (Zhang et al., 2014). Such bounds have practical impact in error correction, hypothesis testing, and entanglement quantification.

Process Fidelity Certification

Certification of quantum operations with minimal measurement effort is addressed via analytical lower bounds derived from fidelity measurements of a carefully chosen set of input states. For a two-qubit unitary, if $F$ is the average state fidelity for computational-basis states and $G$ is the fidelity for a balanced superposition, the process fidelity lower bound is given by

$$\tilde{F}_\chi = \left[(2F-1)\sqrt{G} - \sqrt{(4F-1)(1-F)}\sqrt{1-G}\right]^2$$

(Fiurasek et al., 2014). More generally, with $d+1$ probe states in a Hilbert space of dimension $d$, nontrivial certification is possible, but the bound becomes increasingly loose as $d$ grows. The method is extensible to $N$-qubit operations, with bounds parameterized by the number of qubits and measured fidelities, but requires exponentially high average fidelity for a nonzero lower bound as $N$ increases.

Quantum Filters and Probabilistic Maps

For probabilistic processes (quantum filters), generalizations of the Hofmann bound are obtained. When probing with two mutually unbiased bases, the process fidelity is lower bounded by a weighted average of the output state fidelities, with weights corresponding to success probabilities and intrinsic properties of the filter's Kraus operator. Analytical bounds are tight if the probe basis is chosen as the right eigenbasis of the filter's Kraus operator; for arbitrary probe bases, the bounds can be further tightened via semidefinite programming (SDP) postprocessing (Sedlak et al., 2015).

Direct Resource-Centric Bounds

Experimental quantification of multipartite entanglement and coherence is facilitated by fidelity-based lower bounds on convex-roof and geometric measures. Given a pure reference state $|\phi\rangle$ and measured fidelity $\langle\phi|\rho|\phi\rangle$, a normalized overlap $S$ is defined; then, entanglement or coherence measures can be lower bounded in closed form solely as functions of $S$ (Dai et al., 2018). This enables resource quantification without full state tomography, scaling efficiently to many-body systems.

3. Methodological Frameworks and Mathematical Techniques

Spectral Expansion and Operator Inequalities

Many analytical lower bounds derive from Taylor expansions (e.g., of Gibbs states with respect to an external parameter), spectral decomposition in the eigenbasis of the unperturbed Hamiltonian, and systematic use of operator inequalities. In fidelity susceptibility analysis, this yields bounds directly in terms of the thermodynamic susceptibility and fluctuation-dissipation relations, incorporating quantum corrections via double commutator terms (Brankov et al., 2011): $$X_F(\rho) \geq (\delta S; \delta S)_0 - \frac{1}{3}\langle[[S,T],S]\rangle_0$$

Semidefinite Programming (SDP)

SDP plays a central role in both validating analytical bounds and optimizing process fidelity subject to experimental constraints. By expressing the Choi operator of a quantum channel and fidelity constraints as semidefinite constraints, it becomes possible to numerically (and in some cases analytically) prove the tightness of lower bounds for process fidelity and to capture the best-possible bound compatible with observed data (Mayer et al., 2018, Mayer, 2021). SDP-based approaches underpin recent advances in device-independent quantum key distribution by enabling convex relaxations of fidelity optimization problems (Hahn et al., 2021).

Combinatorial and Ensemble Averaging Techniques

For quantum coding, analytical lower bounds on code fidelity are obtained by relating the uncorrectable error probability to enumerator polynomials (weight spectra) and averaging over code ensembles. The key quantity is the difference $B_j - B_j^\perp$ between the weight enumerators of the code and its dual (Ashikhmin, 2017). Expurgation methods further improve these bounds by removing codes with poor low-weight spectra from the ensemble under consideration.

4. Physical Examples, Applications, and Experimental Relevance

Quantum Criticality and Phase Transitions

Fidelity susceptibility provides an alternative, geometric indicator of phase transitions. Analytical lower bounds show that, in many systems exhibiting criticality (such as the Dicke or Kondo models), the divergent part of the fidelity susceptibility matches that of the conventional thermodynamic susceptibility, and the scaling behavior near the critical point is mirrored (Brankov et al., 2011).

Quantum Gate and Process Certification

Process fidelity lower bounds are experimentally valuable for certifying gate implementations in multi-qubit processors without performing full process tomography. When using symmetric sets of inputs (such as those forming a symmetric POVM or mutually unbiased bases), the lower bounds become nearly linear in average state fidelity error and are efficiently attainable (Fiurasek et al., 2014, Mayer et al., 2018).

Quantum Error Correction and Noise Benchmarking

In approximate quantum error correction, lower bounds on the infidelity after a noise channel can be computed efficiently using subfidelity and superfidelity, allowing closed-form and scalable benchmarks for logical state recovery in $N$-qubit systems. These alternatives are especially relevant as direct computation of the Uhlmann fidelity becomes intractable for large $N$ (Fiusa et al., 2022).

Entanglement and Coherence Certification

Experimentally accessible fidelity measurements with respect to reference states enable lower bounds on genuine multipartite entanglement and various coherence measures, applied in practice to multi-photon GHZ/W states, cluster states, and high-dimensional qudit states (Dai et al., 2018).

5. Scaling, Limitations, and Tightness of Bounds

Scaling with System Size

For many certification protocols using $d+1$ probe states, the lower bound on process fidelity becomes exponentially loose with increasing Hilbert space dimension—substantially limiting their utility in large systems. Methods using more probe states (such as the Hofmann bound) maintain better scaling properties but at the expense of increased experimental overhead (Fiurasek et al., 2014).

Tightness of Bounds

State-dependent lower bounds are often maximally tight for pure states or for processes/channels close to the identity. Analytical work is frequently supplemented (and validated for small $d$ or $N$) by semidefinite programming. In certain scenarios, such as the lower bound on process fidelity in terms of "0-fidelity," numerics confirm that the analytical lower bound is closely achievable (Mayer, 2021).

Trade-Offs

There exists a clear trade-off between experimental and computational efficiency and achievable tightness of the lower bound. Fewer measurement settings or reduced computation generally entail a looser lower bound. Convex-optimization-based post-processing can mitigate this to a degree by extracting tighter bounds from the same measurement data (Sedlak et al., 2015, Mayer et al., 2018, Hahn et al., 2021).

6. Implications for Protocol Design, Resource Theory, and Future Directions

Fidelity-based analytical lower bounds provide a foundational layer for uncertainty and performance guarantees in quantum protocols, enabling resource-efficient verification of entanglement, error correction, process implementation, and cryptographic security. Their explicit forms facilitate practical benchmarking and error estimation in large quantum systems.

Current trends seek to further refine these bounds in terms of state-dependent parameters (such as entropic measures), optimize probe state sets for improved scaling, and fully characterize the gap between analytic lower bounds and achievable performance in high-dimensional settings. For resource theories, such bounds unify entanglement and coherence quantification under experimentally tractable protocols, while in quantum communication and cryptography, they underpin device-independent security claims when only limited measurement data is available.

Analytical lower bounds on fidelity thus form a core toolset for modern quantum information science, linking experimental observables, theoretical performance, and computational tractability.