# Distance measures between states

### Introduction

There are various ways of introducing a notion of *distance* between two quantum states. In general, a distance measure quantifies the extent to which two quantum states behave in the same way. While these distance measures are usually given by certain mathematical expressions, they often possess a simple operational meaning, i.e., they are related to the problem of distinguishing two systems. This is why the distance measures in quantum information are sometimes refered to as *distinguishability measures*.

Even more generally, one can ask how *close* to quantum channels or systems are to each other. Typically, distance measures are used to express how much the system under consideration (e.g., in an experiment) deviates from an ideal system (e.g., the theoretical model that is used). Basically, if the real system is close to the ideal system in the sense the value of a certain distance measure is small, then any statement about the ideal system approximately also holds for the real system.

### Basic properties of distance measures for quantum states

In mathematical terms, a distance measure is represented by a two-argument function *d* : S(H) × S(H) → R. Usually, the basic properties associated with a distance *d* are that it is a *metric*, i.e., it satisfies

- (positivity):
*d*(*ρ*,*σ*) ≥ 0 with equality if and only if*ρ*=*σ* - (symmetry):
*d*(*ρ*,*σ*) =*d*(*σ*,*ρ*) - (triangle inequality):
*d*(*ρ*,*σ*) ≤*d*(*ρ*,*θ*) +*d*(*θ*,*σ*)

These axioms capture the most intuitive properties that we expect to hold for a good distance measure. In the case of quantum states, certain additional properties are also desirable. For example, the distance between two states should not change when a unitary operation is applied to both of them.

- (unitary invariance):
*d*(*ρ*,*σ*) =*d*(*U**ρ**U*† ,*U**σ**U*† ) for every unitary*U*

Another useful properties that can be satisfied by a distinguishability measure is that states obtained by applying the same quantum operation are not more distinguishable than the original states, that is

- (montonicity under quantum operations):
*d*(*T*(*ρ*),*T*(*σ*)) ≤*d*(*ρ*,*σ*) for every trace-preserving quantum operation*T*

A special case of this is the operation of taking the partial trace: If a distance measure *d* is monotonous under quantum operations, then *d*(*ρ**A*, *σ**A*) ≤ *d*(*ρ**A**B*, *σ**A**B*). Intuitively, this says that two systems are harder to distinguish when only one part of the two systems is accessible.

It is also natural to have the following property, which expresses the fact that adding an independent systems does not increase distinguishability:

- (stability under addition of systems):
*d*(*ρ*⊗*θ*,*σ*⊗*θ*) =*d*(*ρ*,*σ*)

Examples of such distance measures include the so-called fidelity, the variational distance and the Shannon distinguishability. Unfortunately, they do not satisfy all the properties stated here and not all of them have a clear operational interpretation (or it is unknown). However, they are still useful to quantify the extent to which two systems or quantum states behave in the same way.

### Related papers

- 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, Departement IRO, Universite de Montreal, 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