Purity (quantum mechanics)

In quantum mechanics, and especially quantum information theory, the purity of a quantum state is a scalar defined as

${\displaystyle \gamma \,\equiv \,{\mbox{tr}}(\rho ^{2})\,}$

where $\rho \,$ is the density matrix of the state. The purity defines a measure on quantum states, giving information on how much a state is mixed.

Mathematical properties

The purity of a quantum state satisfies ${\displaystyle {\frac {1}{d}}\leq \gamma \leq 1\,}$,^[1] where $d\,$ is the dimension of the Hilbert space upon which the state is defined. The upper bound is obtained by noting that ${\displaystyle tr(\rho )=1\,}$and ${\displaystyle tr(\rho ^{2})\leq tr(\rho )\,}$(see trace). If $\rho \,$ is a projection, which describe a pure state, then it holds that ${\displaystyle tr(\rho ^{2})=tr(\rho )=1\,}$ (see Projections). The lower bound is obtained by looking at the completely mixed state, represented by the matrix ${\displaystyle {\frac {1}{d}}I_{d}\,}$.

The purity of a quantum state is conserved under unitary transformations acting on the density matrix in the form ${\displaystyle \rho \mapsto U\rho U^{\dagger }\,}$, where ${\displaystyle U\,}$is an unitary matrix. Specifically, it is conserved under the time evolution operator ${\displaystyle U(t,t_{0})=e^{{\frac {-i}{\hbar }}H(t-t_{0})}\,}$, where $H\,$is the Hamiltonian operator.^[1][2]

Physical meaning

A pure quantum state can be represented as a single vector $| \psi \rangle$ in the Hilbert space. In the density matrix formulation, a pure state is represented by the matrix ${\displaystyle \rho _{pure}=|\psi \rangle \langle \psi |}$. However, a mixed state cannot be represented this way, and instead is represented by a linear combination of pure states ${\displaystyle \rho _{mixed}=\sum p_{i}|\psi _{i}\rangle \langle \psi _{i}|}$, while ${\displaystyle \sum p_{i}=1}$ for normalization. The purity parameter is related to the coefficients: If only one coefficient is equal to $1$, the state is pure; else the purity measures how much their values are similar. Indeed, the purity is $0$ when the state is completely mixed, i.e. ${\displaystyle \rho _{completley\ mixed}={\frac {1}{d}}\sum _{i=1}^{d}|\psi _{i}\rangle \langle \psi _{i}|={\frac {1}{d}}I_{d}}$, where ${\displaystyle |\psi _{i}\rangle }$ are the $d$ orthonormal vector that constitute the basis Hilbert space.^[3]

Geometrical representation

On Bloch sphere, pure states are represented by a point on the surface of the sphere, whereas mixed states are represented by an interior point. Thus, a purity of a state can be visualized as the degree in which it is close to the surface of the sphere. For example, the completely mixed state of a single qubit ${\displaystyle {\frac {1}{2}}I_{2}\,}$is represented by the center of the sphere, by symmetry.

A graphical intuition of purity can be gained by looking at the relation between the density matrix and Bloch sphere:

$\rho = \frac{1}{2}\left(I +\vec{a} \cdot \vec{\sigma} \right)$, where ${\vec {a}}$ is the vector representing the quantum state (on or inside the sphere), and ${\displaystyle {\vec {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}$ are Pauli matrices.

Since Pauli matrices are traceless, it still holds that ${\displaystyle tr(\rho )=1}$.

However, using $(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma}$:

${\displaystyle \rho ^{2}={\frac {1}{2}}[{\frac {1}{2}}(1+|a|^{2})I+{\vec {a}}\cdot {\vec {\sigma }}]}$, hence ${\displaystyle tr(\rho ^{2})={\frac {1}{2}}(1+|a|^{2})}$

Which agrees with the fact that only states on the sphere itself are pure (i.e. ${\displaystyle |a|=1}$).

Relation to other concepts

Linear entropy

Purity is trivially related to the Linear entropy $S_{L}\,$ of a state by

$\gamma =1-S_{L}\,.$

Projectivity of a measurement

For a quantum measurement, the projectivity^[4] is the purity of its pre-measurement state. This pre-measurement state is the main tool of the retrodictive approach of quantum physics, in which we make predictions about state preparations leading to a given measurement result. It allows us to determine in which kind of states the measured system was prepared for leading to such a result.

References

1. Jaeger, Gregg (2006-11-15). Quantum Information: An Overview. Springer Science & Business Media. ISBN 9780387357256.
2. Cappellaro, Paola (2012). "Lecture notes: Quantum Theory of Radiation Interactions, Chapter 7: Mixed states" (PDF). ocw.mit.edu. Retrieved 2016-11-26.
3. Nielsen, Michael A.; Chuang, Isaac L. (2011). Quantum Computation and Quantum Information: 10th Anniversary Edition. New York, NY, USA: Cambridge University Press.
4. Taoufik Amri, Quantum behavior of measurement apparatus, arXiv:1001.3032 (2010).