The trace distance is defined as half of the trace norm of the difference of the matrices:
where is the trace norm of , and is the unique positive semidefinite such that (which is always defined for positive semidefinite ). This can be thought of as the matrix obtained from taking the algebraic square roots of its eigenvalues. For the trace distance, we more specifically have an expression of the form where is Hermitian. This quantity equals the sum of the singular values of , which being Hermitian, equals the sum of the absolute values of its eigenvalues. More explicitly,
where is the -th eigenvalue of , and is its rank.
The factor of two ensures that the trace distance between normalized density matrices takes values in the range .
The trace distance can be seen as a direct quantum generalization of the total variation distance between probability distributions. Given a pair of probability distributions , their total variation distance is
Attempting to directly apply this definition to quantum states raises the problem that quantum states can result in different probability distributions depending on how they are measured. A natural choice is then to consider the total variation distance between the classical probability distribution obtained measuring the two states, maximized over the possible choices of measurement, which results precisely in the trace distance between the quantum states. More explicitly, this is the quantity
with the maximization performed with respect to all possible POVMs.
To see why this is the case, we start observing that there is a unique decomposition with positive semidefinite matrices with orthogonal support. With these operators we can write concisely . Furthermore , and thus . We thus have
This shows that
where denotes the classical probability distribution resulting from measuring with the POVM , , and the maximum is performed over all POVMs .
To conclude that the inequality is saturated by some POVM, we need only consider the projective measurement with elements corresponding to the eigenvectors of . With this choice,
By using the Hölder duality for Schatten norms, the trace distance can be written in variational form as [1]
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 or , each with probability and sends it to Bob who has to discriminate between the two states using a binary measurement.
Let Bob assign the measurement outcome and a POVM element such as the outcome and a POVM element to identify the state or , respectively. His expected probability of correctly identifying the incoming state is then given by
Therefore, when applying an optimal measurement, Bob has the maximal probability
of correctly identifying in which state Alice prepared the system.[2]
The fidelity of two quantum states is related to the trace distance by the inequalities
The upper bound inequality becomes an equality when and are pure states. [Note that the definition for Fidelity used here is the square of that used in Nielsen and Chuang]
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.