Trace Distance

Formula

D(\rho,\sigma) =\frac12\lVert\rho-\sigma\rVert_1 =\frac12\operatorname{Tr}|\rho-\sigma|.

For equal prior probabilities, optimal binary discrimination gives

P_{\mathrm{guess}}^{\mathrm{opt}} =\frac12[1+D(\rho,\sigma)].

Useful special cases are

D(\rho,\sigma)=\frac12\sum_i|p_i-q_i| \quad\text{for commuting states},

D(|\psi\rangle,|\phi\rangle) =\sqrt{1-|\langle\psi|\phi\rangle|^2},

and, for qubit Bloch vectors,

D(\rho,\sigma)=\frac12\lVert\mathbf r-\mathbf s\rVert_2.

Assumptions and Conventions

$\rho$ and $\sigma$ are normalized states on the same Hilbert space.
$\|A\|_1=\operatorname{Tr}\sqrt{A^\dagger A}$ .
The displayed guessing formula assumes equal priors. For priors $p,1-p$ , use $\frac12[1+\|p\rho-(1-p)\sigma\|_1]$ .

Symbols

Symbol	Meaning
$D(\rho,\sigma)$	trace distance, between $0$ and $1$
$\\|\cdot\\|_1$	Schatten trace norm
$P_{\mathrm{guess}}^{\mathrm{opt}}$	optimal one-shot guessing probability
$F(\rho,\sigma)$	squared Uhlmann fidelity on this site

Validity and Warnings

Trace distance is contractive under a common channel: $D(\mathcal N(\rho),\mathcal N(\sigma))\le D(\rho,\sigma)$ .
It equals the maximum difference in probability assigned to one measurement event.
For squared fidelity,

1-\sqrt F\le D\le\sqrt{1-F}.

State trace distance is not the diamond distance between channels.
Unequal priors require the Helstrom operator $p\rho-(1-p)\sigma$ ; do not insert priors into the equal-prior shortcut.
Numerical negative eigenvalues from an unphysical state estimate must be diagnosed, not silently accepted as a distance calculation.

Canonical Treatment

Operational discrimination, variational and observable bounds, commuting, pure-state and qubit examples, channel behavior, exercises, and references are at Density Operators for Quantum Information.