Trace distance explained

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 defined as half of the trace norm of the difference of the matrices:$$T(\backslash rho,\backslash sigma)\; :=\; \backslash frac\backslash |\backslash rho\; -\; \backslash sigma\backslash |\_\; =\; \backslash frac\; \backslash mathrm\; \backslash left[\backslash sqrt\{(\backslash rho-\backslash sigma)^\backslash dagger\; (\backslash rho-\backslash sigma)\}\; \backslash right],$$where

\|A\| _{1\equiv} \operatorname{Tr}[\sqrt{A ^{\dagger} A}] is the trace norm of A , and \sqrt A is the unique positive semidefinite B such that B ^{2=A} (which is always defined for positive semidefinite A ). This can be thought of as the matrix obtained from A taking the algebraic square roots of its eigenvalues. For the trace distance, we more specifically have an expression of the form |C|\equiv \sqrt{C ^{\dagger} C} = \sqrt{C ^{2}} where C = \rho - \sigma is Hermitian. This quantity equals the sum of the singular values of C , which being C Hermitian, equals the sum of the absolute values of its eigenvalues. More explicitly,$$T(\backslash rho,\backslash sigma)\; =\; \backslash frac12\; \backslash operatorname|\backslash rho-\backslash sigma|\; =\; \backslash frac12\backslash sum\_^|\backslash lambda\_i|,$$where λ _{i\inR} is the i -th eigenvalue of \rho - \sigma , and r is its rank.

The factor of two ensures that the trace distance between normalized density matrices takes values in the range

[ 0 , 1 ] .

Connection with the total variation distance

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

P,Q , their total variation distance is$$\backslash delta(P,Q)\; =\; \backslash frac12\backslash |P-Q\backslash |\_1\; =\; \backslash frac12\; \backslash sum\_k\; |P\_k-Q\_k|.$$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$$\backslash max\_\backslash Pi\; \backslash frac12\backslash sum\_i\; |\backslash operatorname(\backslash Pi\_i\; \backslash rho)\; -\; \backslash operatorname(\backslash Pi\_i\backslash sigma)|,$$with the maximization performed with respect to all possible POVMs \{\Pi _{i\}} _{i} .

To see why this is the case, we start observing that there is a unique decomposition

\rho - \sigma = P - Q with P,Q \ge 0 positive semidefinite matrices with orthogonal support. With these operators we can write concisely |\rho - \sigma| = P + Q . Furthermore \operatorname{Tr}(\Pi _{i} P),\operatorname{Tr}(\Pi _{i} Q)\ge 0 , and thus |\operatorname{Tr}(\Pi _{iP)-\operatorname{Tr}(\Pi} _{i} Q))| \le \operatorname{Tr}(\Pi _{iP)+\operatorname{Tr}(\Pi} _{i} Q)) . We thus have$$\backslash sum\_i\; |\backslash operatorname(\backslash Pi\_i\; (\backslash rho-\backslash sigma))|=\backslash sum\_i\; |\backslash operatorname(\backslash Pi\_i\; (P-Q))|\backslash le\; \backslash sum\_i\; \backslash operatorname(\backslash Pi\_i(P+Q))=\; \backslash operatorname|\backslash rho-\backslash sigma|.$$This shows that$$\backslash max\_\backslash Pi\; \backslash delta(P\_,P\_)\; \backslash le\; T(\backslash rho,\backslash sigma),$$where P _{\Pi,\rho} denotes the classical probability distribution resulting from measuring \rho with the POVM \Pi , (P _{\Pi,\rho} ) _{i} \equiv \operatorname{Tr}(\Pi _{i} \rho) , and the maximum is performed over all POVMs \Pi\equiv\{\Pi _{i\}} _{i} .

To conclude that the inequality is saturated by some POVM, we need only consider the projective measurement with elements corresponding to the eigenvectors of

\rho - \sigma . With this choice,$$\backslash delta(P\_,P\_)\; =\backslash frac12\backslash sum\_i\; |\backslash operatorname(\backslash Pi\_i(\backslash rho-\backslash sigma))|=\; \backslash frac12\; \backslash sum\_i\; |\backslash lambda\_i|\; =\; T(\backslash rho,\backslash sigma),$$where λ _{i} are the eigenvalues of \rho - \sigma .

Physical interpretation

By using the Hölder duality for Schatten norms, the trace distance can be written in variational form as ^[1]

T(\rho,\sigma) = (1)/(2) \sup _{-I\leq} Tr [U(\rho - \sigma)] = \sup _{0\leq} 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 (12)/() 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 byp _{guess} = (12)/(p( 0 |\rho)) + (12)/(= p( 1 |\sigma)) (12)/(Tr (P) _{0\rho)+} (12)/(Tr (P) ^{1 \sigma) = (12)/(\left( 1+)} Tr \left(P _{0(\rho-\sigma)\right)\right).}

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

p ^{max} _{guess} = \sup _{P _{0}} (12)/(\left( 1+) Tr \left(P _{0 (\rho - \sigma)\right)\right) = (12)/(( 1)} + T(\rho,\sigma)) of correctly identifying in which state Alice prepared the system.^[2]

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)}

• On pure states, it can be expressed uniquely in term of the inner product of the states:

T(|\psi\rangle\langle\psi|,|\phi\rangle\langle\phi|) = \sqrt{ 1- |\langle\psi | \phi\rangle| ^{2}} ^[3]

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 inequalities1- \sqrt{F(\rho,\sigma)} \le T(\rho,\sigma) \le\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.

Notes and References

1. Book: Nielsen. Michael Nielsen. Chuang. Isaac L.. Isaac Chuang. Quantum Computation and Quantum Information. Cambridge University Press. Cambridge. 2010. 2nd. 844974180. 978-1-107-00217-3. 9. Distance measures for quantum information.
2. S. M. Barnett, "Quantum Information", Oxford University Press, 2009, Chapter 4
3. Book: Wilde . Mark . Quantum Information Theory . 2017 . 10.1017/9781316809976 . 1106.1445. 9781107176164 . 2515538 .