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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.
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,
The Hermitian adjoint, ${\langle\psi_{\alpha}|} = ({|\psi_{\alpha}\rangle})^{\dagger}$, is called a “bra” (hence “bra(c)ket”) and is a row vector.
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
Given the above, we define the density operator to be
We call the matrix representation of the density operator, relative to a given basis, the density matrix.
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)
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.
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
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
Tracing $V^*$ out of $\tilde{\rho}$ allows us to recover $\rho$. This can be seen in the following computation:
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.
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:
Last revised on May 8, 2023 at 16:24:52. See the history of this page for a list of all contributions to it.