In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as
Contents
- Mathematical properties
- Physical meaning
- Geometrical representation
- Linear entropy
- Entanglement
- Inverse Participation Ratio IPR
- Projectivity of a measurement
- References
where
Mathematical properties
The purity of a normalized quantum state satisfies
The purity of a quantum state is conserved under unitary transformations acting on the density matrix in the form
Physical meaning
A pure quantum state can be represented as a single vector
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
A graphical intuition of purity can be gained by looking at the relation between the density matrix and Bloch sphere:
Since Pauli matrices are traceless, it still holds that
However, using
Which agrees with the fact that only states on the sphere itself are pure (i.e.
Linear entropy
Purity is trivially related to the Linear entropy
Entanglement
A 2-qudits pure state
For 2-qubits (pure or mixed) states, the Schmidt number (number of Schmidt coefficients) is at most 2. Using this and Peres–Horodecki criterion (for 2-qubits), a state is entangled if its partial transpose has at least one negative eigenvalue. Using the Schmidt coefficients from above, the negative eigenvalue is
And the purity is
One can see that the more entangled the composite state is (i.e. more negative), the less pure the subsystem state.
Inverse Participation Ratio (IPR)
In the context of localization, a quantity closely related to the purity, the so called inverse participation ratio (IPR) turns out to be useful. Instead of the trace over the square of the density matrix, it is defined as the integral (or sum for finite system size) over the square of the density in some space, e.g., real space, momentum space, or even phase space, where the densities would be the square of the real space wave function
A small value of the IPR then corresponds to a delocalized state (or a strongly mixed state to keep the analogy to the purity), as can be seen by calculating the IPR for a totally delocalized state
In the context of localization, it is often not necessary to know the wave function itself; it often suffices to know the localization properties. This is why the IPR is useful in this context. The IPR basically takes the full information about a quantum system (the wave function; for a
Projectivity of a measurement
For a quantum measurement, the projectivity 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.