# Trace distance

### Definition and Operational Meaning

The trace distance (also called the variational or Kolmogorov distance) δ : S(H) × S(H) → R is one of the most natural distance measures on S(H). It is intimately related to the problem of distinguishing two states in the following way: The value $\frac{1}{2}+\frac{1}{2}\delta(\rho,\sigma)$ is the average success probability when distinguishing (by a measurement) two states ρ and σ which are given with equal a priori probability.

Mathematically, it is defined as follows:

$$\delta(\rho,\sigma)=\frac{1}{2}\mathrm{tr}(|\rho-\sigma|)$$

The connection between the distinguishing problem and this definition in terms of the eigenvalues of the operator ρ − σ 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 Z, which is defined as

$$\delta(P,Q):=\frac{1}{2}\sum_{z\in\mathcal{Z}}|P(z)-Q(z)|=\max_{E\subseteq \mathcal{Z}} P(E)-Q(E).$$

The trace distance between two quantum states ρ, σ is then given by δ(ρ, σ) = maxEδ(E(ρ), E(σ)), where the maximum is taken over all POVMs E on H and the expression E(ρ) refers to the distribution of measurement outcomes obtained by measuring ρ using E. This explains why the trace distance has the mentioned operational meaning: The POVM E achieving the maximum in the above expression is the optimal POVM for distinguishing ρ and σ. Moreover, the trace distance is a natural generalization of the defined classical distance measure for probability distributions, as the definitions coincide if ρ and σ 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 S(H),  it is unitarily invariant and monotonous under local operations. The following list summarizes these and other properties.

1. 0 ≤ δ(ρ, σ) with equality if and only if ρ = σ
2. δ(ρ, σ) ≤ 1 with equality if and only if ρ is orthogonal to σ, i.e., if tr(ρσ) = 0
3. (symmetry): δ(ρ, σ) = δ(σ, ρ)
4. (triangle inequality) δ(ρ, σ) ≤ δ(ρ, θ) + δ(θ, σ)
5. (subadditivity w.r.t. tensor products) δ(ρ ⊗ ρʹ, σ ⊗ σʹ) ≤ δ(ρ, σ) + δ(ρʹ, σʹ)
6. (strong convexity)δ(∑zP(z)ρz, ∑zQ(z)σz) ≤ δ(P, Q) + ∑zP(z)δ(ρz, σz) for all probability distributions P, Q on Z and families of states {ρz}z and {σz}z.

Note that δ 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 ρ = ∣ψ⟩⟨ψ∣ and σ = ∣ϕ⟩⟨ϕ∣ are pure states, then $\delta(\rho,\sigma)=\sqrt{1-|\langle\psi|\phi\rangle|^2}$. However, it is not in general the case that $\delta(|\psi\rangle\langle\psi|,\sigma)=\sqrt{1-|\langle\psi|\sigma|\psi\rangle|}$. Also, if ρ, σ are states of a qubit, then the trace distance is equal to half the Euclidean distance between the corresponding vectors on the Bloch sphere.

• C. A. Fuchs, J. v.d. G., Cryptographic Distinguishability Measures for Quantum Mechanical States, quant-ph/9712042
• C. A. Fuchs, Distinguishability and Accessible Information in Quantum Theory, Ph.D. Thesis, University of New Mexico, 1996. quant-ph/9601020
• A. Gilchrist, N. K. Langford and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, quant-ph/0408063