Pure and mixed state

The terms pure and mixed state (better: mixture of states ) denote certain quantum mechanical states in quantum statistics , the sub-area of quantum mechanics for many-body systems .

Pure state

A pure state exists when the system under consideration is in a firmly defined state, which is described by a state vector from the Hilbert space . Then there is a likelihood of finding it in that state . Thus is the density operator ${\displaystyle |\psi \rangle }$ ${\displaystyle p=1}$ ${\displaystyle |\psi \rangle }$

${\displaystyle \rho _{\psi }=P_{\psi }=|\psi \rangle \langle \psi |}$

just the projection on the state . This is idempotent , i.e. i.e., it applies . ${\displaystyle |\psi \rangle }$ ${\displaystyle \rho _{\psi }^{2}=\rho _{\psi }}$

An alternative definition of a pure state can also be extended to the more general concept of state for C * -algebras of operators. A state on a C * algebra is a positive linear functional with norm 1, i.e. a mapping with and . The set of states forms a convex set. A pure state is a state that is extremal in . This means that a pure state cannot be described as a convex combination (a linear combination with positive coefficients, the sum of which is 1) of two other states. ${\displaystyle A}$ ${\displaystyle \varphi \colon A\to \mathbb {K} }$ ${\displaystyle \varphi (A_{+})\subseteq \mathbb {R} _{+}}$ ${\displaystyle \|\varphi \|=1}$ ${\displaystyle S(A)}$ ${\displaystyle S(A)}$

Mixed states

The counterpart to a pure state is a mixture of states . This is sometimes also referred to as a mixed state , which is unsuitable because then a distinction must be made between coherent and incoherent mixture, and the coherent mixture does not lead to a mixed state but to a pure state. Usually the "coherent mix" is referred to as an overlay. Here, wave functions are linearly superimposed with complex factors that express phase information, among other things. , Incoherent mixture 'describes first ensemble in which objects in pure states randomly with probabilities , are mixed. The phases of the do not play a role. Such a 'state' is then ascribed to the individual objects in the ensemble. It should be noted, however, that the object will generally not show the states with the specified probability during a measurement . This only happens when the merged states are orthogonal. ${\displaystyle |\psi \rangle =a_{1}|\psi _{1}\rangle +a_{2}|\psi _{2}\rangle +...,}$ ${\displaystyle a_{i}}$ ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle p_{i}>0}$ ${\displaystyle \textstyle \sum _{i}p_{i}=1}$ ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle |\psi _{i}\rangle }$

In optics, there is an analogous difference between the coherent addition of amplitudes ( wave optics ) and the incoherent addition of intensities ( ray optics ). In the case of incoherent superposition, i.e. a mixture of states, the projection operator is replaced by the density operator , which is formed as a linear combination of the individual projection operators:

${\displaystyle {\hat {\rho }}=\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|}$.

Mixtures of states of different composition can only be distinguished by measurements (of suitable observables) if their density operators differ. A pure state is the special case of a mixture with only one component.

If the mixed pure states are orthogonal, then they are the eigen-states of the density operator and they are the associated eigen-values. ${\displaystyle p_{i}}$

Mixtures of states often arise, e.g. B. if different pure states are generated with relative frequencies when preparing the system . The expectation value of any operator (with eigenvalues and eigenstates ) is the weighted sum of the expectation values ​​of in each of the incoherently mixed states : ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle p_{i}}$ ${\displaystyle {\hat {O}}}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle |\lambda _{i}\rangle }$ ${\displaystyle p_{i}}$ ${\displaystyle {\hat {O}}}$ ${\displaystyle |\psi _{i}\rangle }$

${\displaystyle \langle {\hat {O}}\rangle _{\hat {\rho }}=\mathrm {Sp} ({\hat {\rho }}\cdot {\hat {O}})=\sum _{i}p_{i}\mathrm {Sp} \left(|\psi _{i}\rangle \langle \psi _{i}|\cdot {\hat {O}}\right)=\sum _{i}p_{i}\sum _{j}\langle \lambda _{j}||\psi _{i}\rangle \langle \psi _{i}|\cdot {\hat {O}}|\lambda _{j}\rangle }$ ${\displaystyle =\sum _{i}p_{i}\langle \psi _{i}|\underbrace {\sum _{j}\lambda _{j}|\lambda _{j}\rangle \langle \lambda _{j}|} _{{\text{Spektraldarstellung von }}{\hat {O}}}|\psi _{i}\rangle =\sum _{i}p_{i}\;\langle \psi _{i}|{\hat {O}}|\psi _{i}\rangle }$

If the pure states are orthogonal to one another, then the weight indicates the probability of finding the system in the pure state . If they are not orthogonal, this does not apply. Rather, the following applies to the probability of finding the mixture in a certain state : ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle p_{i}}$ ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle |\psi _{m}\rangle }$

${\displaystyle \mathrm {Sp} ({\hat {\rho }}\cdot {\hat {\rho }}_{m})=\mathrm {Sp} \left(\left\{\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|\right\}\cdot |\psi _{m}\rangle \langle \psi _{m}|\right)=\sum _{i}p_{i}\;|\langle \psi _{m}|\psi _{i}\rangle |^{2}\neq p_{m}}$

Different compositions can produce the same density matrix :

${\displaystyle \rho _{1}={\frac {1}{2}}(|z+\rangle \langle \!z\!+\!|+|z-\!\rangle \langle \!z\!-\!|)={\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix}}}$

${\displaystyle \rho _{2}={\frac {1}{2}}(|x+\rangle \langle x+|+|x-\rangle \langle x-|)={\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix}}}$

Here are the states and the eigenstates of the spin 1/2 system to the z or x axis. The two formulas with different projection operators appear to mark the z-axis on the one hand and the x-axis on the other. However, they are identical, as can be seen if one uses the usual identifications and (see e.g. Bell state ). ${\displaystyle |z\pm \rangle }$ ${\displaystyle |x\pm \rangle }$ ${\displaystyle |z-\rangle \,=\,{\frac {1}{\sqrt {2}}}(|\uparrow \rangle \ -|\downarrow \rangle )=\ {\frac {1}{\sqrt {2}}}({\begin{pmatrix}1\\0\end{pmatrix}}-...)}$ ${\displaystyle |x+\rangle \,=\,{\frac {1}{\sqrt {2}}}(|\rightarrow \rangle +|\leftarrow \rangle )\ ={\frac {1}{\sqrt {2}}}(\,{\begin{pmatrix}1\\1\end{pmatrix}}+...)}$

The density operator of an incoherent mixture of states (of at least two states) can be recognized by the fact that the following applies to it: while the equals sign always applies to a coherent mixture, ${\displaystyle \rho ^{2}<\rho ,}$ ${\displaystyle \rho ^{2}=\rho .}$

Examples

The most prominent example of incoherent superposition are thermodynamics or statistical physics ( quantum statistics ). Here is . It is the reciprocal of the Fermi-temperature T, more precisely, with the Boltzmann constant is the Hamiltonian (energy operator) of the system; so it is  Z (T) finally what is called the partition function , which corresponds. The following applies to the thermodynamic entropy of the system: with the Von Neumann entropy statistically defined in the article Bell State. Both entropies are therefore essentially identical. ${\displaystyle \rho ={\frac {\exp(-\beta {\hat {H}})}{Z(T)}}}$ ${\displaystyle \beta }$ ${\displaystyle \beta =1/(k_{\mathrm {B} }\cdot T),}$ ${\displaystyle k_{\mathrm {B} };}$   ${\displaystyle {\hat {H}}}$ ${\displaystyle p_{i}={\frac {\exp(-\beta E_{i})}{Z(T)}};}$ ${\displaystyle Z(T)=\sum _{i}\,\exp(-\beta E_{i}),}$ ${\displaystyle \sum _{i}p_{i}=1}$ ${\displaystyle S(T)}$ ${\displaystyle S(T)=k_{\mathrm {B} }S_{N}\equiv -k_{\mathrm {B} }T\cdot \ln Z(T),}$ ${\displaystyle S_{N}=\sum _{i}-p_{i}\ln p_{i}\ (=-\ \mathrm {Sp} \,\rho \cdot \ln \rho ).}$

The most prominent example of coherent superpositions is laser radiation . Here the laser atoms shine in phase, i.e. in time. Transitions between different “pure” energy states of the irradiated system are induced, whereby the transition rate, that is the number of transitions divided by time, is not constant as with incoherent radiation, but z. B. grows very rapidly over a certain period of time. The generated radiation intensity ( ) is much greater than with incoherent excitation ( compared with incoherent light. The number of atoms involved is extremely large, so that it is many powers of ten greater than ) ${\displaystyle \propto |{\vec {E}}|^{2}}$ ${\displaystyle \propto N^{2},}$ ${\displaystyle \propto N}$ ${\displaystyle N}$ ${\displaystyle N^{2}}$ ${\displaystyle N.}$

In the case of incoherent excitation, the energy transitions do not take place in time, but z. B. with random phases. A so-called “ golden rule ” by E. Fermi now applies to the mean transition rate .

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• H. Lin, An Introduction to the Classification of Amenable C * -algebras , World Scientific (2001)