quantum fidelity Quantum fidelity is a measure of similarity between two quantum states, quantifying how close they are in terms of distinguishability and overlap. It is widely used in Quantum information theory and Quantum metrology to assess state preparation, gate performance, and decoherence effects. Fidelity matters because it provides an operationally meaningful scalar that connects theoretical models such as the density matrix formalism to laboratory protocols in platforms like superconducting qubits and trapped ions.
In physical terms, fidelity expresses the probability that a system prepared in one state will pass a test for being the other state. For pure states |ψ〉 and |φ〉 in a Hilbert space, the fidelity is |〈ψ|φ〉|^2, directly interpretable as a transition probability under projective measurement. For mixed states described by density matrixes ρ and σ, fidelity generalizes this overlap to account for classical and quantum uncertainty, reflecting the maximal achievable overlap after purifications and unitary operations. Operationally, fidelity connects to hypothesis testing tasks studied by researchers such as Helstrom and to measures of quantum channel performance in quantum error correction and quantum computing.
The most common definition for mixed states is the Uhlmann fidelity, defined as F(ρ,σ) = (Tr √(√ρ σ √ρ) )^2, with an equivalent expression via Uhlmann's theorem that relates fidelity to maximal overlap between purifications in an enlarged Hilbert space. For pure states the expression reduces to F(|ψ〉,|φ〉)=|〈ψ|φ〉|^2. Alternative related quantities include the trace distance D(ρ,σ) = (1/2)Tr|ρ−σ| and the quantum relative entropy S(ρ||σ), which are linked to fidelity through inequalities such as the Fuchs–van de Graaf relations. In channel contexts, process fidelity compares quantum channels via their Choi–Jamiołkowski states, using the Choi–Jamiolkowski isomorphism to translate channel performance into state fidelity.
Fidelity satisfies symmetry F(ρ,σ)=F(σ,ρ), bounds 0≤F≤1, and equals 1 iff ρ=σ. Monotonicity under completely positive trace-preserving maps (CPTP maps) guarantees that fidelity cannot decrease under quantum operations, a property used in proofs of data-processing inequalities. Uhlmann's theorem provides the purification-based characterization; the Fuchs–van de Graaf inequalities relate fidelity to trace distance. Connections to Bures metric and quantum Fisher information show fidelity's role as a local metric: infinitesimal changes in fidelity define the Bures line element and hence determine quantum Cramér–Rao bounds in parameter estimation problems.
Computing fidelity for large systems often requires numerical linear algebra: eigenvalue decompositions, singular value decompositions, and matrix square roots. Algorithms exploit sparsity and tensor structure in systems studied at Los Alamos National Laboratory, IBM Quantum, and Google Quantum AI. For Gaussian states in continuous-variable systems, closed-form expressions use covariance matrices and symplectic eigenvalues via Williamson's theorem. Monte Carlo techniques and randomized benchmarking adapt fidelity estimation into scalable protocols, while semidefinite programming (SDP) formulations enable convex optimization for fidelity-constrained problems in quantum control and verification.
Fidelity is central to assessing quantum gate fidelity in quantum computing experiments, quantifying the performance of error-correcting codes like the surface code and the Shor code. In quantum communications it rates the quality of teleportation and entanglement distribution protocols, such as those tested in experiments by Haroche and Zeilinger groups. In quantum metrology, fidelity-based distances determine sensitivity limits via the quantum Fisher information and underpin protocols for magnetometry and frequency estimation in systems including NV center (diamond) sensors. Fidelity thresholds define fault-tolerance criteria and benchmark standards in initiatives like the Quantum Economic Development Consortium.
Experimental estimation of fidelity uses state tomography, direct fidelity estimation, and randomized benchmarking. Full quantum state tomography reconstructs ρ and σ to compute fidelity but scales poorly with system size. Direct fidelity estimation reduces cost by sampling specific measurement settings, a technique applied in ion-trap experiments at institutions such as NIST and in superconducting circuits at Rigetti and IBM. Randomized benchmarking reports average gate fidelity by fitting decay curves from random Clifford sequences, while interleaved benchmarking isolates specific gate performance. Error bars are analyzed using statistical tools from classical estimation theory and bootstrap methods.
Various generalizations extend fidelity to specialized settings: the superfidelity and subfidelity provide computationally cheaper bounds; fidelity for quantum channels compares completely positive maps via process matrices; and entanglement fidelity measures how well a channel preserves entanglement with an ancillary system. In many-body physics, fidelity susceptibility probes quantum phase transitions as introduced in studies of the XY model and Ising model. Extensions to non-Hermitian and open quantum systems connect fidelity notions to Loschmidt echoes, decoherence rates, and studies of quantum chaos at institutions researching many-body localization, such as the Perimeter Institute and Max Planck Institute for Quantum Optics.
Category:Quantum information theory Category:Quantum measurement