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

edit

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  .

Connection with the total variation distance

edit

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, where   are the eigenvalues of  .

Physical interpretation

edit

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]

Properties

edit

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  
  •   and   if and only if   and   have orthogonal supports
  • It is preserved under unitary transformations:  
  • It is contractive under trace-preserving CP maps, i.e. if   is a CPT map, then  
  • It is convex in each of its inputs. E.g.  
  • On pure states, it can be expressed uniquely in term of the inner product of the states:   [3]

For qubits, the trace distance is equal to half the Euclidean distance in the Bloch representation.

Relationship to other distance measures

edit

Fidelity

edit

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]

Total variation distance

edit

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

edit
  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
  3. ^ Wilde, Mark (2017). Quantum Information Theory. arXiv:1106.1445. doi:10.1017/9781316809976. ISBN 9781107176164. S2CID 2515538.