In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance 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.
The trace distance is defined as half of the trace norm of the difference of the matrices:
is the unique 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
This quantity equals the sum of the singular values of
Hermitian, equals the sum of the absolute values of its eigenvalues.
The factor of two ensures that the trace distance between normalized density matrices takes values in the range
The trace distance serves as a direct quantum generalization of the total variation distance between probability distributions.
, their total variation distance is defined as
When extending this concept to quantum states, one must account for the fact that for quantum states different measurement can produce different distributions.
A natural approach is to consider the (classical) total variation distance between the measurement outcomes produced by two states for a fixed choice of measurement, and then maximize over all possible measurements.
This procedure leads precisely to the trace distance between the quantum states.
with the maximization performed with respect to all possible POVMs
To understand why this maximum equals the trace distance between the states, note that there is a unique decomposition
positive semidefinite matrices with orthogonal support.
With these operators we can write concisely
denotes the classical probability distribution resulting from measuring
, 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
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
and sends it to Bob who has to discriminate between the two states using a binary measurement.
Let Bob assign the measurement outcome
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 trace distance has the following properties[1] For qubits, the trace distance is equal to half the Euclidean distance in the Bloch representation.
The fidelity of two quantum states
is related to the trace distance
[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.