Density matrix

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This article is about a concept in physics. For the psychiatric condition, see Mixed state (psychology).

A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. The formalism was introduced by John von Neumann (according to other sources independently by Lev Landau and Felix Bloch) in 1927.

It is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises when one considers either an ensemble of systems, or one system when its preparation history is uncertain.

Situations in which a density matrix is used include: a quantum system in thermal equilibrium (at finite temperatures); nonequilibrium time-evolution that starts out of a mixed equilibrium state; and entanglement between two subsystems, where each individual system must be described, via the partial trace operation, by a density matrix even though the complete system may be in a pure state; and in analysis of quantum decoherence. See also quantum statistical mechanics.

A density operator is an operator corresponding to a density matrix under some orthonormal basis. Thus it is a non-negative, self-adjoint, trace class operator of trace one.

1 The need for a statistical description
2 Formulation
3 Measurement
- 3.1 Entropy
4 C*-algebraic formulation of states
5 Notes and references
6 See also

[edit] The need for a statistical description

By quantum mechanics, the state vector ψ^[1] of a system completely determines the statistical behavior of an observable O. This means that if O is represented by an operator A on the Hilbert space H of the system, then for any real-valued function F^[2] defined on the real numbers, the expectation value of F(O) ^[3] is the quantity

$\langle F(A) \psi \mid \psi \rangle.$

or written as

$\langle \psi \mid F(A) \mid \psi \rangle.$

in Dirac notation.

Now consider the example of a "mixed quantum system" prepared by statistically combining two different pure states φ, ψ each with probability 1/2. The preparation process for such a system consists in tossing an unbiased coin and using the preparation process for φ or for ψ depending on whether the toss outcome is heads or tails.

It is not hard to show that the statistical properties of the observable O for the system prepared in such a mixed state are completely determined. However, there is no vector ξ which determines this statistical behavior in the sense that the expectation value of F(O) is

$\langle \xi \mid F(A) \mid \xi \rangle.$

Nevertheless: there is a unique operator ρ such that the expectation value can be written as

$\operatorname{Tr}(F(A) \rho).$

The operator ρ is the density operator of the mixed system. A simple calculation shows that for the example mentioned above:

$\rho = \frac{1}{2} \mid \phi\rangle \langle \phi \mid + \frac{1}{2} \mid \psi\rangle \langle \psi \mid.$

[edit] Formulation

For a finite dimensional Hilbert space, the most general density matrix is of the form

$\rho = \sum_j p_j |\psi_j \rang \lang \psi_j|,$

where the coefficients p_j are non-negative and add up to one. This represents a statistical mixture of pure states. If the given system is closed, then one can think of a mixed state as representing a single system with an uncertain preparation history, as explicitly detailed above; or we can regard the mixed state as representing an ensemble of systems, i.e. large number of copies of the system in question, where p_j is the proportion of the ensemble being in the state $|\psi_j \rang$ . An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, i.e. it is a pure ensemble.

It must be noted, however, that if the system is not closed, then it is simply not correct to claim that it has some definite but unknown state vector, as the density operator may record physical entanglements to other systems.

Example Consider a quantum ensemble of size N with occupancy numbers n₁, n₂,...,n_k corresponding to the orthonormal states |1>,...,|k>, respectively, where n₁+...+n_k = N, and, thus, the coefficients p_j = n_j /N. For a pure ensemble, where all N particles are in state |i>, we have n_j = 0, for all j ≠ i, from which we recover the corresponding density matrix ρ = |i >< i|.

However the density matrix of a mixed state does not capture all the information about a mixture; in particular, the coefficients p_j and the kets ψ_j are not recoverable from the matrix ρ without additional information. This non-uniqueness implies that different ensembles or mixtures may correspond to the same density matrix. Such equivalent ensembles or mixtures cannot be distinguished by measurement of observables alone. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator if and only if there is a matrix U with

$\sum_k u^*_{ik} u_{kj} = \delta_{ij}$

i.e, U is unitary and such that

$\mid \psi_i'\rangle \sqrt p_i' = \sum_{j} u_{ij} \mid \psi_j\rangle \sqrt {p_j}$

This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M^* = N N^* if and only if M = NU for some unitary U. (See square root of a matrix for more details.) Thus there is an unitary freedom in the ket mixture or ensemble that gives the same density operator. However if the kets in the mixture are orthonormal then the original probabilities p_j are recoverable as the eigenvalues of the density matrix.

In operator language, a density operator is a positive semidefinite, hermitian operator acting on the state space of trace 1. A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ is a pure state if and only if

$\; \rho = \rho^2$ ,

i.e. the state is idempotent. This is true regardless of whether H is finite dimensional or not.

Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set.

It follows from the spectral theorem for compact self-adjoint operators that every mixed state is an infinite convex combination of pure states. This representation is not unique. Furthermore, a theorem of Andrew Gleason states that certain functions defined on the family of projections and taking values in [0,1] (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See quantum logic for more details.

[edit] Measurement

Let A be an observable of the system, and suppose the ensemble (copies of the system in question, or one system in an uncertain state) is prepared in mixed state. The corresponding density operator is:

$\rho = \sum_j p_j |\psi_j \rang \lang \psi_j| .$

The average value of the measurement can be calculated by extending from the case of vectorial pure states:

$\lang A \rang = \sum_j p_j \lang \psi_j|A|\psi_j \rang = \operatorname{tr}[\rho A].$

Similarly, if A has spectral resolution

$A = \sum_i a_i |a_i \rang \lang a_i| = \sum _i a_i P_i,$

where $P_i = |a_i \rang \lang a_i|$ , the corresponding density operator after the measurement is given by:

$\; \rho ^' = \sum_i P_i \rho P_i.$

[edit] Entropy

The von Neumann entropy $S$ of a mixture can be expressed in terms of the probabilities $p i$ or in terms of the trace and logarithm of the density matrix $ρ$ :

$S = -\sum_i p_i \ln \,p_i = -\operatorname{tr}[\rho \ln \rho]$ :

This entropy can increase but never decrease^[4]^[5] with a measurement. The entropy of a pure state is zero, while that of a proper mixture always greater than zero. Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state. Thus the act of measurement induces a fundamental irreversible change on the density matrix; this is analogous to the "collapse" of the state vector, or wavefunction collapse.

[edit] C*-algebraic formulation of states

It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable^[6]^[7]. For this reason, observables are identified to elements of an abstract C*-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. Note that by using the GNS construction, we can recover Hilbert spaces which realize A as an algebra of operators.

Geometrically, a pure state on a C*-algebra A is a state which is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A.

The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics.

The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures, as noted in the introduction.

[edit] Notes and references

^ We assume ψ is a pure state in this example.
^ Technically, F must be a Borel function
^ F(O) is defined to be the result of measuring O and then applying F to the outcome.
^ Michael A Nielsen, Issac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000), ISBN 0-521-63503-9, Chapter 11: Entropy and information, Theorem 11.9, "Projective measurements cannot decrease entropy"
^ Hugh Everett, The Theory of the Universal Wavefunction (1956), Appendix I. "Monotone decrease of information for stochastic processes" pp 128-129 in The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 0-691-08131-X
^ See appendix, G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963
^ G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972.