# 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:

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

(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,

${\displaystyle T(\rho ,\sigma )={\frac {1}{2}}\mathrm {Tr} \left[{\sqrt {(\rho -\sigma )^{2}}}\right]={\frac {1}{2}}\sum _{i}|\lambda _{i}|,}$

where the ${\displaystyle \lambda _{i}}$ are eigenvalues of the Hermitian, but not necessarily positive, matrix ${\displaystyle (\rho -\sigma )}$.

## Physical interpretation

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

${\displaystyle T(\rho ,\sigma )={\frac {1}{2}}\max _{\{P_{i}\}}\Sigma _{P_{i}}\mathrm {Tr} [P_{i}(\rho -\sigma )],}$

where the maximization can be carried out over all POVMs ${\displaystyle \{P_{i}\}}$. ${\displaystyle \mathrm {Tr} [P(\rho -\sigma )]}$ is the difference in probability that the outcome of the measurement be ${\displaystyle P}$, depending on whether the system was in the state ${\displaystyle \rho }$ or ${\displaystyle \sigma }$. 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 ${\displaystyle \rho }$ or ${\displaystyle \sigma }$, each with probability ${\displaystyle {\frac {1}{2}}}$ 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

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

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 ${\displaystyle T(\rho ,\sigma )=0\Leftrightarrow \rho =\sigma }$
• ${\displaystyle 0\leq T(\rho ,\sigma )\leq 1}$ and ${\displaystyle T(\rho ,\sigma )=1}$ if and only if ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ have orthogonal supports
• It is preserved under unitary transformations: ${\displaystyle T(U\rho U^{\dagger },U\sigma U^{\dagger })=T(\rho ,\sigma )}$
• It is contractive under trace-preserving CP maps, i.e. if ${\displaystyle \Phi }$ is a CPT map, then ${\displaystyle T(\Phi (\rho ),\Phi (\sigma ))\leq T(\rho ,\sigma )}$
• It is convex in each of its inputs. E.g. ${\displaystyle T(\sum _{i}p_{i}\rho _{i},\sigma )\leq \sum _{i}p_{i}T(\rho _{i},\sigma )}$

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 ${\displaystyle F(\rho ,\sigma )}$ is related to the trace distance ${\displaystyle T(\rho ,\sigma )}$ by the inequalities

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

The upper bound inequality becomes an equality when ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ 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 Nielsen, Michael A.; Chuang, Isaac L. (2010). "9. Distance measures for quantum information". Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180.
2. ^ S. M. Barnett, "Quantum Information", Oxford University Press, 2009, Chapter 4