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Quantum State Fidelity

Closeness of two quantum states.

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Core idea

Overview

Quantum state fidelity is a metric used to determine the mathematical similarity between two quantum states, represented by their density matrices ρ and σ. It quantifies how indistinguishable the states are from one another, ranging from 0 for perfectly orthogonal states to 1 for identical states.

When to use: Use this formula when evaluating the accuracy of quantum state preparation or assessing the impact of decoherence in a quantum channel. It is the standard calculation for comparing a noisy experimental density matrix against a theoretical ideal state during quantum process tomography.

Why it matters: Fidelity serves as the primary benchmark for quantum hardware performance, providing a single scalar value that indicates how reliably a quantum computer preserves information. High fidelity is a prerequisite for the implementation of quantum error-correction and the realization of fault-tolerant quantum computation.

Symbols

Variables

F = Fidelity

Fidelity

Walkthrough

Derivation

Formula: Quantum State Fidelity

A measure of how close two quantum states are, used to verify quantum gate performance.

  • Pure state fidelity: |ψ⟩ is the target, |φ⟩ is the actual state.
  • For mixed states, use the generalised formula with density matrices.
1

Pure State Fidelity:

The fidelity is the square of the overlap (inner product) between the target and actual state. F = 1 means perfect match; F = 0 means orthogonal (maximally different).

2

Mixed State Fidelity:

For density matrices ρ and σ, fidelity generalises the pure state formula.

Result

Source: University Quantum Computing — Verification

Free formulas

Rearrangements

Solve for

Make F the subject

Start from the definition of Quantum State Fidelity. Since F is already the subject, no algebraic rearrangement is needed; simply identify F.

Difficulty: 2/5

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Visual intuition

Graph

Graph unavailable for this formula.

The graph depicts the fidelity F between two quantum states, typically represented as a linear transition from 0 to 1 as the overlap between state vectors increases. The curve is a straight line where a value of 0 indicates orthogonal quantum states and a value of 1 indicates perfectly identical states. This linear progression illustrates the degree of similarity or closeness between two density matrices within the Hilbert space.

Graph type: linear

Why it behaves this way

Intuition

Imagine two 'fuzzy' probability distributions (quantum states) in an abstract Hilbert space; fidelity measures the degree of their overlap, ranging from no overlap (0) to complete congruence (1).

F
A scalar measure quantifying the statistical distinguishability between two quantum states.
A value between 0 (perfectly distinguishable/orthogonal) and 1 (identical states), indicating how 'close' or 'similar' two quantum states are.
Density matrix representing the first quantum state.
Encapsulates all the measurable properties and probabilities associated with the first quantum system, potentially a mixed state.
Density matrix representing the second quantum state.
Encapsulates all the measurable properties and probabilities associated with the second quantum system, which is being compared against .
The trace operation, which sums the diagonal elements of a square matrix.
Collapses a matrix into a single scalar value, here representing a measure of 'overlap' or 'correlation' derived from the product of the state matrices.
The matrix square root operation, which for a positive semi-definite matrix A, yields a unique positive semi-definite matrix B such that B2 = A.
This operation is crucial for generalizing the concept of overlap from pure states (vector inner product) to mixed states (density matrices), ensuring the fidelity is well-defined and behaves consistently.

Signs and relationships

  • (\dots)^2: The final squaring ensures the fidelity value is a real number between 0 and 1, consistent with a 'probability-like' measure of similarity.

Free study cues

Insight

Canonical usage

Quantum state fidelity is a dimensionless quantity, representing a scalar value between 0 and 1 that quantifies the similarity between two quantum states.

Common confusion

A common mistake is to search for units for fidelity. As a measure of similarity or overlap between normalized states, fidelity is fundamentally a dimensionless quantity, similar to a correlation coefficient or a

Dimension note

Quantum state fidelity is inherently dimensionless because it quantifies the statistical overlap or similarity between two normalized density matrices (ρ and σ).

Ballpark figures

  • Quantity:

One free problem

Practice Problem

A quantum bit is intended to be prepared in the pure state |0⟩, but environmental noise results in a mixed state σ = 0.85|0⟩⟨0| + 0.15|1⟩⟨1|. Calculate the quantum state fidelity f between the ideal state and the actual state.

Fidelity0.85

Solve for:

Hint: When one state is pure, the fidelity is simply the expectation value of the other state's density matrix with respect to that pure state.

The full worked solution stays in the interactive walkthrough.

Study smarter

Tips

  • For pure states, the fidelity formula simplifies to the squared magnitude of the inner product overlap |⟨ψ|ϕ⟩|².
  • The fidelity function is symmetric, meaning f(ρ, σ) = f(σ, ρ).
  • Always ensure that your density matrices are properly normalized with a trace of 1 before calculating fidelity.
  • If one state is pure and the other is mixed, the calculation reduces to the expectation value of the density matrix with respect to the pure state.

Common questions

Frequently Asked Questions

A measure of how close two quantum states are, used to verify quantum gate performance.

Use this formula when evaluating the accuracy of quantum state preparation or assessing the impact of decoherence in a quantum channel. It is the standard calculation for comparing a noisy experimental density matrix against a theoretical ideal state during quantum process tomography.

Fidelity serves as the primary benchmark for quantum hardware performance, providing a single scalar value that indicates how reliably a quantum computer preserves information. High fidelity is a prerequisite for the implementation of quantum error-correction and the realization of fault-tolerant quantum computation.

For pure states, the fidelity formula simplifies to the squared magnitude of the inner product overlap |⟨ψ|ϕ⟩|². The fidelity function is symmetric, meaning f(ρ, σ) = f(σ, ρ). Always ensure that your density matrices are properly normalized with a trace of 1 before calculating fidelity. If one state is pure and the other is mixed, the calculation reduces to the expectation value of the density matrix with respect to the pure state.

References

Sources

  1. Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang
  2. The Theory of Quantum Information by John Preskill (Lecture Notes, Caltech)
  3. Wikipedia: Fidelity of quantum states
  4. IUPAC Gold Book: 'density matrix (density operator)'
  5. Nielsen, Michael A., and Isaac L. Chuang. 'Quantum Computation and Quantum Information.' Cambridge University Press, 2010.
  6. Wikipedia: 'Density matrix'
  7. Nielsen and Chuang Quantum Computation and Quantum Information
  8. Jozsa Fidelity for Mixed Quantum States