本文介绍了量子力学中的纯态和混合态。纯态可以用单个状态向量表示，完全确定系统状态；而混合态则表示我们对系统的状态只有部分或无知识，需要用概率组合来描述。混合态并非不同波函数的叠加，而是无相干混合。密度算子ρ^用于表示当系统初始状态不确定时的状态，其性质包括归一化和熵等。纯态的密度矩阵等于其投影算子，而混合态的密度矩阵由所有可能纯态的概率加权求和得到。
1. Pure States
A pure state is any state that can be represented by a single state vector ∣ ψ ⟩ |\psi \rangle ∣ψ⟩. What this means in practice is that at any time we can say for certain (with 100% probability) that our system is in state ∣ ψ ⟩ |\psi \rangle ∣ψ⟩. In other words, for a pure state we have complete knowledge of the system and therefore know exactly what state it is in. ∣ 0 ⟩ |0\rangle ∣0⟩, ∣ 1 ⟩ |1\rangle ∣1⟩, ∣ 0 ⟩ + ∣ 1 ⟩ 2 \frac{|0\rangle+|1\rangle}{\sqrt{2}} 2∣0⟩+∣1⟩ are all examples of pure states.
2. Mixed States
A system is in a mixed state if we have limited or no knowledge of the state that it is prepared in. Thus we cannot be 100% sure if it is in pure state ∣ ψ 1 ⟩ |\psi_1\rangle ∣ψ1⟩ or ∣ ψ 2 ⟩ |\psi_2\rangle ∣ψ2⟩ or any other possible states. We therefore must describe the state of the system as a probabilistic combination of all the pure states it could possibly have been prepared in. There are several reasons this may be the case:
- inconsistencies with lab equipment;
- entanglement with an external system that is out of our control. (See Sec.4 below.)
Mixed states are not superpositions of different wavefunctions, but incoherent mixtures. For example, if ∣ V ⟩ |V\rangle ∣V⟩ is a vertically polarized photon, and ∣ H ⟩ |H\rangle ∣H⟩ is a horizontally polarized photon, then the superposition ( 1 / 2 ) ( ∣ V ⟩ + ∣ H ⟩ (1/\sqrt{2}) (|V\rangle+|H\rangle (1/2)(∣V⟩+∣H⟩) is a diagonally polarized photon, while the unpolarized photon is a mixture of half ∣ V ⟩ |V\rangle ∣V⟩ and half ∣ H ⟩ |H\rangle ∣H⟩, described by the density matrix 1 / 2 ( ∣ V ⟩ ⟨ V ∣ + ∣ V ⟩ ⟨ H ∣ ) 1/\sqrt{2}(|V\rangle \langle V | + |V\rangle\langle H|) 1/2(∣V⟩⟨V∣+∣V⟩⟨H∣). The superposition is in both states, the mixture is in perhaps one or perhaps the other.
Suppose we want to compute the ensemble expectation of an operator A A A. In a pure state ∣ ψ n ⟩ |\psi_n\rangle ∣ψn⟩, the quantum expectation is < A > p u r e = ∫ ψ n ∗ A ψ n = ⟨ ψ n ∣ A ∣ ψ n ⟩ = Tr ( ∣ ψ n ⟩ ⟨ ψ n ∣ A ) \left<A\right>_{pure}=\int\psi_n^*A\psi_n=\langle\psi_n|A|\psi_n\rangle=\text{Tr}(|\psi_n\rangle\langle\psi_n|A) ⟨A⟩pure=∫ψn∗Aψn=⟨ψn∣A∣ψn⟩=Tr(∣ψn⟩⟨ψn∣A) but in the ensemble of pure states { ∣ ψ n ⟩ } \{|\psi_n\rangle\} { ∣ψn⟩} the expectation is the probabilistic weighted averages of the expectation values of the pure states: < A > m i x e d = ∑ n p n ( ∫ ψ n ∗ A ψ n ) = ∑ n p n ⟨ ψ n ∣ A ∣ ψ n ⟩ = ∑ n p n Tr ( ∣ ψ n ⟩ ⟨ ψ n ∣ A ) = Tr ( ρ ^ A ) . \left<A\right>_{mixed}=\sum_n p_n\left(\int\psi_n^*A\psi_n\right)=\sum_n p_n\langle\psi_n|A|\psi_n\rangle=\sum_n p_n\text{Tr}(|\psi_n\rangle\langle\psi_n|A)=\text{Tr}(\hat{\rho}A). ⟨A⟩mixed=n∑pn(∫ψn∗Aψn)=n∑pn⟨ψn∣A∣ψn⟩=n∑pnTr(∣ψn⟩⟨ψn∣A)=Tr(ρ^A). The Traces in the above equalities will be discussed in the following.
3. The Density Operator, ρ ^ \hat{\rho} ρ^
We want to avoid carrying around the particular states ∣ ψ n ⟩ |\psi_n\rangle ∣ψn⟩. Pick any complete orthonormal basis ∣ ϕ i ⟩ |\phi_i\rangle ∣ϕi⟩, then the identity operator is I ^ = ∑ i ∣ ϕ i ⟩ ⟨ ϕ i ∣ \hat{I}=\sum_i |\phi_i\rangle \langle\phi_i| I^=∑i∣ϕi⟩⟨ϕi∣. Substitute the identity into < A > m i x e d \left<A\right>_{mixed} ⟨A⟩mixed, we have < A > m i x e d = ∑ i ⟨ ϕ i ∣ A ( ∑ n p n ∣ ψ n ⟩ ⟨ ψ n ∣ ) ∣ ϕ i ⟩ = Tr ( A ρ ^ ) \left<A\right>_{mixed}=\sum_i \langle\phi_i|A\left(\sum_n p_n |\psi_n\rangle \langle\psi_n|\right)|\phi_i\rangle=\text{Tr}(A\hat{\rho}) ⟨A⟩mixed=i∑⟨ϕi∣A(n∑pn∣ψn⟩⟨ψn∣)∣ϕi⟩=Tr(Aρ^)
The density operator can be used to represent the state of the system when we don’t know the starting state with certainty. It is a more general form of the state vector (used to describe pure states). (Eq.1) ρ ^ = ∑ n = 1 N p n ∣ ψ n ⟩ ⟨ ψ n ∣ \hat{\rho}=\sum_{n=1}^N p_n |\psi_n\rangle \langle\psi_n|\tag{Eq.1} ρ^=n=1∑Npn∣ψn⟩⟨ψn∣(Eq.1) where p i p_i pi is the probability of the system being in pure state ∣ ψ i ⟩ |\psi_i\rangle ∣ψi⟩ at the start. The set of pure states { ∣ ψ i ⟩ } \{|\psi_i\rangle\} { ∣ψi⟩} in a general mixture need not be eigenstates or even orthogonal. The number N could be anything, and is not limited by the dimension of the Hilbert space(比如，可以与其他系统发生各种量子纠缠).
We could interpret the weights p i p_i pi as probabilities, but we have to be careful: we should not think of p i p_i pi as the probability to find the particle in the state ∣ ψ i ⟩ |\psi_i\rangle ∣ψi⟩!
Properties
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Reduction to the pure case.
If the starting state is a pure one ∣ ψ ⟩ |\psi\rangle ∣ψ⟩ (i.e. there is only one possible state the system could be in at the start), then ρ ^ p u r e = ∣ ψ ⟩ ⟨ ψ ∣ \hat{\rho}_{pure}=|\psi\rangle\langle\psi| ρ^pure=∣ψ⟩⟨ψ∣. This is another definition for a pure state – any state that has a density operator ρ ^ = ∣ ψ ⟩ ⟨ ψ ∣ \hat{\rho}=|\psi\rangle\langle\psi| ρ^=∣ψ⟩⟨ψ∣ is a pure one (i.e. its density matrix consists of a single projector).在坐标表象下从纯态的密度矩阵 ρ ^ p u r e \hat{\rho}_{pure} ρ^pure重构出纯态的波函数 ψ ( x ) \psi(x) ψ(x)
[*Refer to Sethna《Entropy Order Parameters Complexity》7.1.1]
In the position basis ∣ x ⟩ |x\rangle ∣x⟩, this pure-state density matrix has matrix elements ρ p u r e ( x , x ′ ) = ⟨ x ∣ ρ ^ p u r e ∣ x ′ ⟩ = ψ ∗ ( x ′ ) ψ ( x ) \rho_{pure}(x,x')=\langle x|\hat{\rho}_{pure}|x'\rangle=\psi^*(x')\psi(x) ρpure(x,x′)=⟨x∣ρ^pure∣x′⟩=ψ∗(x′)ψ(x). We can reconstruct the wavefunction from a pure-state density matrix, up to an overall physically unmeasurable phase. In particular, since ψ ( x ) \psi(x) ψ(x) is normalized, ∫ d x ∣ ρ ( x , x ′ ) ∣ 2 = ∫ d x ∣ ψ ∗ ( x ′ ) ∣ 2 ∣ ψ ( x ) ∣ 2 = ∣ ψ ∗ ( x ′ ) ∣ 2 \int dx|\rho(x,x')|^2=\int dx |\psi^*(x')|^2|\psi(x)|^2=|\psi^*(x')|^2 ∫dx∣ρ(x,x′)∣2=∫dx∣ψ∗(x′)∣2∣ψ(x)∣2=∣ψ∗(x′)∣2 and thus ψ ( x ) = ρ ( x , x ′ ) d x ~ ∣ ρ ( x ~ , x ′ ) ∣ 2 . \psi(x)=\frac{\rho(x,x')}{\sqrt{d\tilde{x}|\rho(\tilde{x},x')|^2}}. ψ(x)=dx
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