nLab density matrix

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Context

Measure and probability theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

In quantum mechanics a density matrix is a linear endomorphism of a Hilbert space of pure quantum states that represents a statistical ensemble of quantum states, hence states which are not necessarily pure states but mixed states.

The space of density matrices inside all suitable endomorphisms is called the Bloch region.

Notation

In the following definition, we use Dirac’s “bra-ket” notation for vectors where a state vector, describing a pure state of a quantum system, is represented by a “ket” which is a column vector,

|ψ α⟩=(⋮ x n−1 x n x n+1 ⋮). {|\psi_{\alpha}\rangle} = \left( \begin{aligned} \vdots \\ x_{n-1} \\ x_{n} \\ x_{n+1} \\ \vdots \end{aligned} \right) .

The Hermitian adjoint, ⟨ψ α|=(|ψ α⟩) †{\langle\psi_{\alpha}|} = ({|\psi_{\alpha}\rangle})^{\dagger}, is called a “bra” (hence “bra(c)ket”) and is a row vector.

The density operator of an ensemble

Suppose we have a quantum state QQ that arises from some random process such that the state |ψ α⟩{|\psi_{\alpha}\rangle} has a probability p α∈[0,1]p_{\alpha} \in [0,1] (we often speak of having ‘prepared’ the state with the associated probability). The possible states |ψ α⟩{|\psi_{\alpha}\rangle} need not be orthogonal and we thus call such a collection of states a mixed state. More specifically, a mixed state is often described as an ensemble of quantum systems.

Suppose we now measure some observable A\mathbf{A} on the system as a whole, i.e. on the ensemble. The expectation value of A\mathbf{A} over the state |ψ α⟩{|\psi_{\alpha}\rangle} is ⟨A⟩ α=⟨ψ α|A|ψ α⟩\langle{A}\rangle_{\alpha} = \langle\psi_{\alpha} | \mathbf{A} |\psi_{\alpha}\rangle. Over the entire ensemble, this becomes

⟨A⟩=∑ αp α⟨A⟩ α=∑ αp αtr|ψ α⟩⟨ψ α|A. \langle{A}\rangle = \sum_{\alpha} p_{\alpha} \langle{A}\rangle_{\alpha} = \sum_{\alpha} p_{\alpha} tr {|\psi_{\alpha}\rangle} {\langle\psi_{\alpha}|} \mathbf{A} .

Given the above, we define the density operator to be

ρ=∑ αρ α|ψ α⟩⟨ψ α|. \mathbf{\rho} = \sum_{\alpha} \rho_\alpha {|\psi_{\alpha}\rangle}{\langle\psi_{\alpha}|} .

We call the matrix representation of the density operator, relative to a given basis, the density matrix.

Characterisation

An operator ρ\rho is the density operator associated to some ensemble if and only if it is a positive operator with trace 1. (Nielsen and Chuang Theorem 2.5, p. 101)

Coherence

Diagonal density matrices with at least two non-zero terms on the diagonal represent mixed states. Density matrices that possess non-zero off-diagonal terms represent superposition states. Such states are referred to as coherent and the off-diagonal entries are called the coherences. Any physical process that has the effect of suppressing the coherences is known as decoherence.

Quantum doubles

Given a pure state on the tensor product of Hilbert spaces V⊗WV\otimes W, the partial trace tr W\tr_W specifies a mixed state on VV. Moreover, every mixed state comes from the partial trace of a pure state. Hence, this gives an alternate characterization of what a density matrix is: The image of the “pure state” outer products |ψ⟩⟨ψ||\psi\rangle\langle\psi| under partial trace.

Even better, given a density matrix ρ∈V\rho\in V one can explicitly construct a pure state which traces to it. Expand ρ\rho as

ρ=∑ αρ α|ψ α⟩⟨ψ α|.\rho=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |.

The states |ψ α⟩| \psi_{\alpha}\rangle serve as a basis for VV, and the states ⟨ψ α|\langle\psi_\alpha | serve as a basis for V *V^*. Hence, we can consider the state

ρ˜=∑ αρ α|ψ α⟩⊗⟨ψ α|∈V⊗V *.\tilde{\rho}=\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\in V\otimes V^*.

Tracing V *V^* out of ρ˜\tilde{\rho} allows us to recover ρ\rho. This can be seen in the following computation:

tr V *(ρ˜ρ˜ †) =(∑ αρ α|ψ α⟩⊗⟨ψ α|)⊗(∑ αρ α|ψ α⟩⊗⟨ψ α|) † =(∑ αρ α|ψ α⟩⊗⟨ψ α|)⊗(∑ αρ α⟨ψ α|⊗|ψ α⟩) =∑ α,βρ αρ β|ψ α⟩⟨ψ β|⊗⟨ψ α|ψ β⟩ =∑ αρ α|ψ α⟩⟨ψ α|=ρ. \begin{aligned} \tr_{V^*}(\tilde{\rho} \tilde{\rho}^{\dagger})&=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right)^{\dagger}\\ &=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}\langle\psi_\alpha |\otimes |\psi_\alpha \rangle\right)\\ &=\sum_{\alpha,\beta}\sqrt{\rho_{\alpha}}\sqrt{\rho_{\beta}}|\psi_\alpha \rangle \langle \psi_\beta | \otimes \langle \psi_\alpha | \psi_\beta \rangle\\ &=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |=\rho. \end{aligned}

Limitations

Note that a density operator, as the representation of the state of a quantum system, is less restrictive than a state vector which specifies the wavefunction. On the other hand, two different state vectors can give rise to the same density operator. However, in that case, the two vectors are the same up to a phase, so arguably the density operator still describes the physical state unambiguously.

More controversially, two entirely different probabilisitic combinations of state vectors can give rise to the same density operator. Roger Penrose, for one, has argued that this means that that the density operator does not describe mixed states unambiguously. But one can also argue the reverse: that mixed states with the same operator really are the same physical state, since they are observationally indistinguishable.

quantum probability theory – observables and states

References

The notion of density matrices originates with

(in precursor discussion of what later came to be called quantum measurement channels).

Most accounts of quantum physics discuss density matrices in some form. See for instance:

In the context of quantum computation:

In the context of entanglement:

In the context of decoherence:

In the context of open quantum systems:

Further:

On quantum circuits with density matrices:

Last revised on October 8, 2024 at 12:35:36. See the history of this page for a list of all contributions to it.

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