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

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

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 ${\frac {1}{2}}$ 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

$p_{\text{max}}={\frac {1}{2}}(1+T(\rho ,\sigma ))$

of correctly identifying in which state Alice prepared the system.

Properties

The trace distance has the following properties

• 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

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

The upper bound 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.