Fidelity and Trace Distance

Trace distance measures the largest probability difference achievable by a bounded measurement effect. Fidelity compares the overlap of purifications. Both remain well-defined for infinite-dimensional density operators, and dimension-independent inequalities relate them. The definitions require care because “fidelity” is used for both a quantity and its square.

Helpful background. Trace-norm duality supplies norm estimates; purification in infinite dimensions supplies the ancillary construction.

Distance and fidelity conventions

For density operators $\rho,\sigma$ on a separable Hilbert space, define

$$D(\rho,\sigma)=\tfrac12\|\rho-\sigma\|_1, \qquad f(\rho,\sigma)=\|\sqrt\rho\sqrt\sigma\|_1, \qquad F(\rho,\sigma)=f(\rho,\sigma)^2.$$

Here $f$ is root fidelity and $F$ is squared fidelity. Both conventions occur in the literature; every quoted bound must be translated accordingly.

The product $\sqrt\rho\sqrt\sigma$ is trace class because both square roots are Hilbert–Schmidt. The product bound gives $f\leq\|\sqrt\rho\|_2\|\sqrt\sigma\|_2=1$. Both $D$ and $f$ lie in $[0,1]$. Equality $D=0$ or $f=1$ means $\rho=\sigma$; $f=0$ means their supports are orthogonal. Trace distance is a metric, whereas fidelity itself is a similarity measure.

Optimizing over measurement effects

Set $\Delta=\rho-\sigma$. It is self-adjoint, trace class, and traceless. Its Jordan decomposition is $\Delta=\Delta_+-\Delta_-$ with orthogonal positive supports. Trace zero implies

$$\operatorname{Tr}\Delta_+ =\operatorname{Tr}\Delta_- =\tfrac12\|\Delta\|_1.$$

For an effect $0\leq E\leq I$,

$$-\operatorname{Tr}\Delta_- \leq\operatorname{Tr}(E\Delta) \leq\operatorname{Tr}\Delta_+.$$

The positive spectral projection of $\Delta$ attains the upper bound. Therefore

$$D(\rho,\sigma) =\sup_{0\leq E\leq I} |\operatorname{Tr}(E(\rho-\sigma))|.$$

This proof applies in infinite dimension because the positive and negative parts are trace class and the optimizing projection is bounded. For equal prior probabilities, the optimal binary discrimination success probability is $(1+D)/2$. No finite-dimensional matrix diagonalization is assumed by the theorem, although finite approximations may be needed to construct a measurement computationally.

Purification overlap and bounds

Uhlmann’s characterization identifies $f(\rho,\sigma)$ with the supremum of absolute overlaps between purifications in a sufficiently large common ancillary space. The freedom is an ancillary isometry or partial isometry on occupied supports; it should not be replaced indiscriminately by a unitary on an arbitrarily fixed infinite-dimensional ancilla.

The Fuchs–van de Graaf bounds, written in the conventions above, are

$$1-f(\rho,\sigma) \leq D(\rho,\sigma) \leq\sqrt{1-f(\rho,\sigma)^2}.$$

The upper bound follows by choosing purifications with overlap approaching $f$, using the pure-state distance formula below, and applying partial-trace contractivity. The lower bound can be obtained from the positive-operator square-root inequality

$$\|\sqrt\rho-\sqrt\sigma\|_2^2 \leq\|\rho-\sigma\|_1.$$

Indeed, its left side is $2-2\operatorname{Tr}(\sqrt\rho\sqrt\sigma)$, and $f\geq\operatorname{Tr}(\sqrt\rho\sqrt\sigma)$. This gives $2(1-f)\leq2D$. The cited fidelity treatments provide the square-root inequality and the infinite-dimensional extension.

Consequently $D(\rho_n,\rho)\to0$ exactly when $f(\rho_n,\rho)\to1$. This equivalence controls bounded-observable statistics. It does not imply entropy or energy convergence without further assumptions.

Worked pure-state and commuting examples

For $\rho=P_\psi$ and $\sigma=P_\phi$, let $c=|\langle\psi,\phi\rangle|$. If $c<1$, the difference $P_\psi-P_\phi$ has eigenvalues $\pm\sqrt{1-c^2}$ on their two-dimensional span; if $c=1$, the projectors coincide and their difference is zero. Thus

$$f(P_\psi,P_\phi)=c,\qquad D(P_\psi,P_\phi)=\sqrt{1-c^2}.$$

The upper bound is saturated. A pure state compared with a mixed state instead has $F(P_\psi,\sigma)=\langle\psi,\sigma\psi\rangle$, but generally does not saturate that bound.

For commuting diagonal states with probabilities $(p_n)$ and $(q_n)$,

$$D=\tfrac12\sum_n|p_n-q_n|, \qquad f=\sum_n\sqrt{p_nq_n}.$$

The quantum quantities reduce to total variation distance and the classical overlap coefficient. Countably infinite sums are justified by absolute summability and Cauchy–Schwarz.

Processing cannot improve distinguishability

For a completely positive trace-preserving map on trace-class operators, the adjoint sends effects to effects. Applying the variational formula shows $D(\Phi(\rho),\Phi(\sigma))\leq D(\rho,\sigma)$. The corresponding dilation and purification argument gives $f(\Phi(\rho),\Phi(\sigma))\geq f(\rho,\sigma)$. Discarding information makes states harder to distinguish and can increase their overlap.

Watrous gives detailed finite-dimensional proofs and operational applications. Hou and Qi establish infinite-dimensional fidelity statements; the trace-class hypotheses here identify the setting in which those extensions apply.

Exercises

1. Compare $\rho=P_0$ with $\sigma=(1-\epsilon)P_0+\epsilon P_1$ for orthogonal pure states. Which fidelity bound is saturated?

Solution

Here $D=\epsilon$ and $f=\sqrt{1-\epsilon}$. For $0<\epsilon<1$, neither bound is generally saturated: $1-\sqrt{1-\epsilon}<\epsilon<\sqrt\epsilon$. This distinguishes a classical mixture from the pure-state overlap calculation.

2. If a normalized spectral truncation discards weight $t$, compute its root fidelity with the original state.

Solution

The states commute. Summing the retained contributions gives $f=\sum_{n\leq N}p_n/\sqrt{1-t}=\sqrt{1-t}$. Their trace distance is $t$. This supplies complementary error measures for density-matrix approximations.

References

• Fuchs, Christopher A., and Jeroen van de Graaf, “Cryptographic distinguishability measures for quantum-mechanical states,” IEEE Transactions on Information Theory 45, 1216–1227, 1999 — comparison inequalities for distinguishability measures. Preprint.
• Hou, Jinchuan, and Xiaofei Qi, “Fidelity of states in infinite-dimensional quantum systems,” Science China Physics, Mechanics & Astronomy 55, 1820–1827, 2012 — infinite-dimensional Uhlmann theorem and fidelity properties. Preprint; DOI.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, Chapter 3 — discrimination, fidelity, and norm inequalities in finite dimension. Author manuscript.