nLab density matrix

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,

$${|\psi_{\alpha}\rangle} = \left( \begin{aligned} \vdots \\ x_{n-1} \\ x_{n} \\ x_{n+1} \\ \vdots \end{aligned} \right) .$$

The Hermitian adjoint, ${\langle\psi_{\alpha}|} = ({|\psi_{\alpha}\rangle})^{\dagger}$, 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 $Q$ that arises from some random process such that the state ${|\psi_{\alpha}\rangle}$ has a probability $p_{\alpha} \in [0,1]$ (we often speak of having ‘prepared’ the state with the associated probability). The possible states ${|\psi_{\alpha}\rangle}$ 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 $\mathbf{A}$ on the system as a whole, i.e. on the ensemble. The expectation value of $\mathbf{A}$ over the state ${|\psi_{\alpha}\rangle}$ is $\langle{A}\rangle_{\alpha} = \langle\psi_{\alpha} | \mathbf{A} |\psi_{\alpha}\rangle$. Over the entire ensemble, this becomes

$$\langle{A}\rangle = \sum_{\alpha} p_{\alpha} \langle{A}\rangle_{\alpha} = \sum_{\alpha} p_{\alpha} tr {|\psi_{\alpha}\rangle} {\langle\psi_{\alpha}|} \mathbf{A} .$$

Given the above, we define the density operator to be

$$\mathbf{\rho} = \sum_{\alpha} \rho_\alpha {|\psi_{\alpha}\rangle}{\langle\psi_{\alpha}|} .$$

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

Characterisation

An operator $\rho$ 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 $V\otimes W$, the partial trace $\tr_W$ specifies a mixed state on $V$. 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 $|\psi\rangle\langle\psi|$ under partial trace.

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

$$\rho=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |.$$

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

$$\tilde{\rho}=\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\in V\otimes V^*.$$

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

$$\begin{aligned} \tr_{V^*}(\tilde{\rho} \tilde{\rho}^{\dagger})&=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right)^{\dagger}\\ &=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}\langle\psi_\alpha |\otimes |\psi_\alpha \rangle\right)\\ &=\sum_{\alpha,\beta}\sqrt{\rho_{\alpha}}\sqrt{\rho_{\beta}}|\psi_\alpha \rangle \langle \psi_\beta | \otimes \langle \psi_\alpha | \psi_\beta \rangle\\ &=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |=\rho. \end{aligned}$$

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.

Related concepts

• quantum operation

• quantum computation

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

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

• Michael A. Nielsen, Isaac L. Chuang, §2.4 in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

• Ingemar Bengtsson, Karol Życzkowski, Chapters 8 and 12 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

• Benjamin Schumacher, Michael Westmoreland, §8 of: Quantum Processes, Systems, and Information, Cambridge University Press (2010) [doi:10.1017/CBO9780511814006]

• Greg Kuperberg, § 1.4 of: A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf]

On quantum circuits with density matrices:

• Dorit Aharonov, Alexei Kitaev, Noam Nisan, Quantum Circuits with Mixed States, Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC) (1998) 20-30 [arXiv:quant-ph/9806029, doi:10.1145/276698.276708]