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 half of the trace norm of the difference of the matrices:

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

Here, for a positive semidefinite matrix A, ${\sqrt {A}}$ denotes a positive semidefinite matrix B such that $B^{2}=A$. Note that B is a unique matrix so defined.

(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

By using the Hölder duality for Schatten norms, the trace distance can be written in variational form as

$T(\rho ,\sigma )={\frac {1}{2}}\sup _{-\mathbb {I} \leq U\leq \mathbb {I} }\mathrm {Tr} [U(\rho -\sigma )]=\sup _{0\leq P\leq \mathbb {I} }\mathrm {Tr} [P(\rho -\sigma )].$

As for its classical counterpart, the trace distance can be related to the maximum probability of distinguishing between two quantum states:

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 using a binary measurement. Let Bob assign the measurement outcome $0$ and a POVM element $P_{0}$ such as the outcome $1$ and a POVM element $P_{1}=1-P_{0}$ to identify the state $\rho$ or $\sigma$, respectively. His expected probability of correctly identifying the incoming state is then given by

$p_{\text{guess}}={\frac {1}{2}}p(0|\rho )+{\frac {1}{2}}p(1|\sigma )={\frac {1}{2}}\mathrm {Tr} (P_{0}\rho )+{\frac {1}{2}}\mathrm {Tr} (P_{1}\sigma )={\frac {1}{2}}\left(1+\mathrm {Tr} \left(P_{0}(\rho -\sigma )\right)\right).$

Therefore, when applying an optimal measurement, Bob has the maximal probability

$p_{\text{guess}}^{\text{max}}=\sup _{P_{0}}{\frac {1}{2}}\left(1+\mathrm {Tr} \left(P_{0}(\rho -\sigma )\right)\right)={\frac {1}{2}}(1+T(\rho ,\sigma ))$

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

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

The upper bound inequality becomes an equality when $\rho$ and $\sigma$ are pure states. [Note that the definition for Fidelity used here is the square of that used in Nielsen and Chuang]

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.