{{Short description|Metric in quantum mechanics}} 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.
== Definition == The trace distance is defined as half of the trace norm of the difference of the matrices:<math display="block">T(\rho,\sigma) := \frac{1}{2}\|\rho - \sigma\|_{1} = \frac{1}{2} \mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)} \right],</math>where <math>\|A\|_1\equiv \operatorname{Tr}[\sqrt{A^\dagger A}]</math> is the trace norm of <math>A</math>, and <math>\sqrt A</math> is the unique positive semidefinite <math>B</math> such that <math>B^2=A</math> (which is always defined for positive semidefinite <math>A</math>). This can be thought of as the matrix obtained from <math>A</math> taking the algebraic square roots of its eigenvalues. For the trace distance, we more specifically have an expression of the form <math>|C|\equiv \sqrt{C^\dagger C}=\sqrt{C^2}</math> where <math>C=\rho-\sigma</math> is Hermitian. This quantity equals the sum of the singular values of <math>C</math>, which being <math>C</math> Hermitian, equals the sum of the absolute values of its eigenvalues. More explicitly, <math display="block">T(\rho,\sigma) = \frac12 \operatorname{Tr}|\rho-\sigma| = \frac12\sum_{i=1}^{r}|\lambda_i|,</math> where <math>\lambda_i\in\mathbb R</math> is the <math>i</math>-th eigenvalue of <math>\rho-\sigma</math>, and <math>r</math> is its rank.
The factor of two ensures that the trace distance between normalized density matrices takes values in the range <math>[0,1]</math>.
== Connection with the total variation distance == The trace distance serves as a direct quantum generalization of the total variation distance between probability distributions. Given two probability distributions <math>P</math> and <math>Q</math>, their total variation distance is defined as<math display="block">\delta(P,Q) = \frac12\|P-Q\|_1 = \frac12 \sum_k |P_k-Q_k|.</math>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. More explicitly, this is the quantity<math display="block">\max_\Pi \frac12\sum_i |\operatorname{Tr}(\Pi_i \rho) - \operatorname{Tr}(\Pi_i\sigma)|,</math>with the maximization performed with respect to all possible POVMs <math>\{\Pi_i\}_i</math>.
To understand why this maximum equals the trace distance between the states, note that there is a unique decomposition <math>\rho-\sigma=P-Q</math> with <math>P,Q \ge 0</math> positive semidefinite matrices with orthogonal support. With these operators we can write concisely <math>|\rho-\sigma|=P+Q</math>. Furthermore <math>\operatorname{Tr}(\Pi_i P),\operatorname{Tr}(\Pi_i Q)\ge0</math>, and thus <math>|\operatorname{Tr}(\Pi_iP)-\operatorname{Tr}(\Pi_i Q))| \le \operatorname{Tr}(\Pi_iP)+\operatorname{Tr}(\Pi_i Q)</math>. We thus have<math display="block">\sum_i |\operatorname{Tr}(\Pi_i (\rho-\sigma))| =\sum_i |\operatorname{Tr}(\Pi_i (P-Q))| \le \sum_i \operatorname{Tr}(\Pi_i(P+Q)) = \operatorname{Tr}|\rho-\sigma|.</math>This shows that<math display="block">\max_\Pi \delta(P_{\Pi,\rho},P_{\Pi,\sigma}) \le T(\rho,\sigma), </math>where <math>P_{\Pi,\rho}</math> denotes the classical probability distribution resulting from measuring <math>\rho</math> with the POVM <math>\Pi</math>, <math>(P_{\Pi,\rho})_i \equiv \operatorname{Tr}(\Pi_i \rho)</math>, and the maximum is performed over all POVMs <math>\Pi\equiv\{\Pi_i\}_i</math>.
To conclude that the inequality is saturated by some POVM, we need only consider the projective measurement with elements corresponding to the eigenvectors of <math>\rho-\sigma</math>. With this choice,<math display="block">\delta(P_{\Pi,\rho},P_{\Pi,\sigma}) = \frac12\sum_i |\operatorname{Tr}(\Pi_i(\rho-\sigma))| = \frac12 \sum_i |\lambda_i| = T(\rho,\sigma), </math>where <math>\lambda_i</math> are the eigenvalues of <math>\rho-\sigma</math>.
== Physical interpretation == By using the Hölder duality for Schatten norms, the trace distance can be written in variational form as <ref name="nielsen">{{Cite book|last1=Nielsen|first=Michael A.|authorlink1=Michael Nielsen|last2=Chuang|first2=Isaac L.|authorlink2=Isaac Chuang|title=Quantum Computation and Quantum Information|publisher=Cambridge University Press|location=Cambridge|year=2010|edition=2nd|oclc=844974180|isbn=978-1-107-00217-3|chapter=9. Distance measures for quantum information}}</ref> :<math> 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)]. </math>
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 <math>\rho</math> or <math>\sigma</math>, each with probability <math>\frac 12</math> and sends it to Bob who has to discriminate between the two states using a binary measurement. Let Bob assign the measurement outcome <math>0</math> and a POVM element <math>P_0</math> such as the outcome <math>1</math> and a POVM element <math>P_1=1-P_0</math> to identify the state <math>\rho</math> or <math>\sigma</math>, respectively. His expected probability of correctly identifying the incoming state is then given by :<math> p_{\text{guess}} = \frac 12 p(0|\rho) + \frac 12 p(1|\sigma) = \frac 12 \mathrm{Tr}(P_0\rho)+ \frac 12 \mathrm{Tr}(P_1\sigma)=\frac 12 \left(1+ \mathrm{Tr}\left(P_0(\rho-\sigma)\right)\right). </math>
Therefore, when applying an optimal measurement, Bob has the maximal probability
:<math> p^{\text{max}}_{\text{guess}} = \sup_{P_0} \frac 12 \left(1+ \mathrm{Tr}\left(P_0(\rho-\sigma)\right)\right) =\frac 12 (1 + T(\rho,\sigma)) </math> of correctly identifying in which state Alice prepared the system.<ref>S. M. Barnett, "Quantum Information", Oxford University Press, 2009, Chapter 4</ref>
== Properties == The trace distance has the following properties<ref name="nielsen" /> * It is a metric on the space of density matrices, i.e. it is non-negative, symmetric, and satisfies the triangle inequality, and <math>T(\rho,\sigma) = 0 \Leftrightarrow \rho=\sigma</math> * <math>0 \leq T(\rho,\sigma) \leq 1</math> and <math>T(\rho,\sigma)=1 </math> if and only if <math>\rho</math> and <math>\sigma</math> have orthogonal supports * It is preserved under unitary transformations: <math>T(U\rho U^\dagger,U\sigma U^\dagger) = T(\rho,\sigma) </math> * It is contractive under trace-preserving CP maps, i.e. if <math>\Phi</math> is a CPTP map, then <math>T(\Phi(\rho),\Phi(\sigma))\leq T(\rho,\sigma)</math> * It is convex in each of its inputs. E.g. <math>T(\sum_i p_i \rho_i,\sigma) \leq \sum_i p_i T(\rho_i,\sigma)</math> * On pure states, it can be expressed uniquely in term of the inner product of the states: <math>T(|\psi\rangle\langle\psi|,|\phi\rangle\langle\phi|) = \sqrt{1-|\langle\psi | \phi\rangle|^2} </math> <ref>{{cite book|last1=Wilde |first1=Mark |title=Quantum Information Theory |date=2017 |doi=10.1017/9781316809976 |arxiv=1106.1445|isbn=9781107176164 |s2cid=2515538 }}</ref>
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 <math>F(\rho,\sigma)</math> is related to the trace distance <math>T(\rho,\sigma)</math> by the inequalities
:<math> 1-\sqrt{F(\rho,\sigma)} \le T(\rho,\sigma) \le\sqrt{1-F(\rho,\sigma)} \, . </math>
The upper bound inequality becomes an equality when <math>\rho</math> and <math>\sigma</math> 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.
== References == {{reflist}}
Category:Quantum information science