# Trace distance

In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.

## Definition

The trace distance is just half of the trace norm of the difference of the matrices:

$T(\rho,\sigma) := \frac{1}{2}||\rho - \sigma||_{1} = \frac{1}{2} \mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)} \right] .$

(The trace norm is the Schatten norm for p=1.) The purpose of the factor of two is to restrict the trace distance between two normalized density matrices to the range [0, 1] and to simplify formulas in which the trace distance appears.

Since density matrices are Hermitian,

$T(\rho,\sigma) = \frac{1}{2} \mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^2} \right] = \frac{1}{2} \sum_i | \lambda_i | ,$

where the $\lambda_i$ are eigenvalues of the Hermitian, but not necessarily positive, matrix $(\rho-\sigma)$.

## Physical interpretation

It can be shown that the trace distance satisfies the equation[1]

$T(\rho,\sigma) = \max_P \mathrm{Tr}[P(\rho-\sigma)],$

where the maximization can be carried either over all projectors $P$, or over all positive operators $P \leq I$, where $I$ is the identity operator. $\mathrm{Tr}[P(\rho-\sigma)]$ is the difference in probability that the outcome of the measurement be $P$, depending on whether the system was in the state $\rho$ or $\sigma$. Thus the trace distance is the probability difference maximized over all possible measurements: it gives a measure of the maximum probability of distinguishing between two states with an optimal measurement.

For example, suppose Alice prepares a system in either the state $\rho$ or $\sigma$, each with probability $\frac 12$ and sends it to Bob who has to discriminate between the two states. It is easy to show that with the optimal measurement, Bob has the probability

$p_{\text{max}} = \frac 12 (1 + T(\rho,\sigma))$

of correctly identifying in which state Alice prepared the system.[2]

## Properties

The trace distance has the following properties[1]

• It is a metric on the space of density matrices, i.e. it is non-negative, symmetric, and satisfies the triangle inequality, and $T(\rho,\sigma) = 0 \Leftrightarrow \rho=\sigma$
• $0 \leq T(\rho,\sigma) \leq 1$ and $T(\rho,\sigma)=1$ if and only if $\rho$ and $\sigma$ have orthogonal supports
• It is preserved under unitary transformations: $T(U\rho U^\dagger,U\sigma U^\dagger) = T(\rho,\sigma)$
• It is contractive under trace-preserving CP maps, i.e. if $\Phi$ is a CPT map, then $T(\Phi(\rho),\Phi(\sigma))\leq T(\rho,\sigma)$
• It is convex in each of its inputs. E.g. $T(\sum_i p_i \rho_i,\sigma) \leq \sum_i p_i T(\rho_i,\sigma)$

For qubits, the trace distance is equal to half the Euclidean distance in the Bloch representation.

### Relationship to other distance measures

#### Fidelity

The fidelity of two quantum states $F(\rho,\sigma)$ is related to the trace distance $T(\rho,\sigma)$ by the inequalities

$1-F(\rho,\sigma) \le T(\rho,\sigma) \le\sqrt{1-F(\rho,\sigma)^2} \, .$

The upper inequality becomes an equality when $\rho$ and $\sigma$ are pure states.

#### Total variation distance

The trace distance is a generalization of the total variation distance, and for two commuting density matrices, has the same value as the total variation distance of the two corresponding probability distributions.

## References

1. ^ a b M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000, Chapter 9
2. ^ S. M. Barnett, "Quantum Information", Oxford University Press, 2009, Chapter 4