Trace distance

== Definition and Operational Meaning == The trace distance (also called the variational or Kolmogorov distance) $\backslash delta:\backslash mathcal\{S\}(\backslash mathcal\{H\})\backslash times\backslash mathcal\{S\}(\backslash mathcal\{H\})\backslash rightarrow\; \backslash mathbb\{R\}$ is one of the most natural distance measures on $\backslash mathcal\{S\}(\backslash mathcal\{H\})$. It is intimately related to the problem of distinguishing two states in the following way: The value $\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{2\}\backslash delta(\backslash rho,\backslash sigma)$ is the average success probability when distinguishing (by a measurement) two states $\backslash rho$ and $\backslash sigma$ which are given with equal a priori probability. Mathematically, it is defined as follows: :$\backslash delta(\backslash rho,\backslash sigma)=\backslash frac\{1\}\{2\}\backslash mathrm\{tr\}(|\backslash rho-\backslash sigma|)$ The connection between the distinguishing problem and this definition in terms of the eigenvalues of the operator $\backslash rho-\backslash sigma$ is related to a result by Helstrom, who solved a more general problem known as the binary decision problem in the theory of quantum hypothesis testing. An alternative expression for the trace distance can be given using the classical variational distance between two probability distributions $P,Q$ on an alphabet $\backslash mathcal\{Z\}$, which is defined as :$\backslash delta(P,Q):=\backslash frac\{1\}\{2\}\backslash sum\_\{z\backslash in\backslash mathcal\{Z\}\}|P(z)-Q(z)|=\backslash max\_\{E\backslash subseteq\; \backslash mathcal\{Z\}\}\; P(E)-Q(E).$ The trace distance between two quantum states $\backslash rho,\backslash sigma$ is then given by $\backslash delta(\backslash rho,\backslash sigma)=\backslash max\_\{\backslash mathcal\{E\}\}\backslash delta(\backslash mathcal\{E\}(\backslash rho),\backslash mathcal\{E\}(\backslash sigma))$, where the maximum is taken over all POVMs $\backslash mathcal\{E\}$ on $\backslash mathcal\{H\}$ and the expression $\backslash mathcal\{E\}(\backslash rho)$ refers to the distribution of measurement outcomes obtained by measuring $\backslash rho$ using $\backslash mathcal\{E\}$. This explains why the trace distance has the mentioned operational meaning: The POVM $\backslash mathcal\{E\}$ achieving the maximum in the above expression is the optimal POVM for distinguishing $\backslash rho$ and $\backslash sigma$. Moreover, the trace distance is a natural generalization of the defined classical distance measure for probability distributions, as the definitions coincide if $\backslash rho$ and $\backslash sigma$ are diagonal in the same basis. == Properties == The trace distance satisfies many useful properties that are desirable for quantum distance measures. Most importantly, it is a metric on $\backslash mathcal\{S\}(\backslash mathcal\{H\}),$ it is unitarily invariant and monotonous under local operations. The following list summarizes these and other properties. #$0\backslash leq\; \backslash delta(\backslash rho,\backslash sigma)$ with equality if and only if $\backslash rho=\backslash sigma$ #$\backslash delta(\backslash rho,\backslash sigma)\backslash leq\; 1$ with equality if and only if $\backslash rho$ is orthogonal to $\backslash sigma$, i.e., if $\backslash mathrm\{tr\}(\backslash rho\backslash sigma)=0$ #(symmetry): $\backslash delta(\backslash rho,\backslash sigma)=\backslash delta(\backslash sigma,\backslash rho)$ #(triangle inequality) $\backslash delta(\backslash rho,\backslash sigma)\backslash leq\backslash delta(\backslash rho,\backslash theta)+\backslash delta(\backslash theta,\backslash sigma)$ #(subadditivity w.r.t. tensor products) $\backslash delta(\backslash rho\backslash otimes\backslash rho\text{'},\backslash sigma\backslash otimes\backslash sigma\text{'})\backslash leq\; \backslash delta(\backslash rho,\backslash sigma)+\backslash delta(\backslash rho\text{'},\backslash sigma\text{'})$ #(strong convexity):$\backslash delta(\backslash sum\_z\; P(z)\backslash rho\_z,\backslash sum\_z\; Q(z)\backslash sigma\_z)\backslash leq\; \backslash delta(P,Q)+\backslash sum\_z\; P(z)\backslash delta(\backslash rho\_z,\backslash sigma\_z)$ for all probability distributions $P,Q$ on $\backslash mathcal\{Z\}$ and families of states $\backslash \{\backslash rho\_z\backslash \}\_z$ and $\backslash \{\backslash sigma\_z\backslash \}\_z$. Note that $\backslash delta$ is robust under addition of subsystems as a consequence of the subadditivity with respect to tensor products. == Useful identities == In certain special cases, the trace distance can be evaluated easily. As an example, if both $\backslash rho=|\backslash psi\backslash rangle\backslash langle\backslash psi|$ and $\backslash sigma=|\backslash phi\backslash rangle\backslash langle\backslash phi|$ are pure states, then $\backslash delta(\backslash rho,\backslash sigma)=\backslash sqrt\{1-|\backslash langle\backslash psi|\backslash phi\backslash rangle|^2\}$. However, it is '''not''' in general the case that $\backslash delta(|\backslash psi\backslash rangle\backslash langle\backslash psi|,\backslash sigma)=\backslash sqrt\{1-|\backslash langle\backslash psi|\backslash sigma|\backslash psi\backslash rangle|\}$. Also, if $\backslash rho,\backslash sigma$ are states of a qubit, then the trace distance is equal to half the Euclidean distance between the corresponding vectors on the Bloch sphere. == Related papers == * C. A. Fuchs, J. v.d. G., ''Cryptographic Distinguishability Measures for Quantum Mechanical States'', [http://arxiv.org/abs/quant-ph/9712042 quant-ph/9712042] * C. A. Fuchs, ''Distinguishability and Accessible Information in Quantum Theory'', Ph.D. Thesis, University of New Mexico, 1996. [http://arxiv.org/abs/quant-ph/9601020 quant-ph/9601020] * A. Gilchrist, N. K. Langford and M. A. Nielsen, ''Distance measures to compare real and ideal quantum processes'', [http://arxiv.org/abs/quant-ph/0408063 quant-ph/0408063] == See also == * [[Index of notational and terminological conventions]] * [[Distance measures]] * [[Fidelity]] [[Category:Handbook of Quantum Information]]