Mixed state

In physics, in particular in quantum mechanics, mixed-state is a concept introduced for describing a quantum mechanical system whose state is not known precisely, but can possibly be in a collection of pure states (i.e., elements of the Hilbert space of the quantum mechanical system with unity norm) with a certain probability of being in one of the pure states in this collection. For example, consider a finite-dimensional quantum system that has been prepared in a pure state $\scriptstyle \phi$ and measurement of an observable X on this system with, say, eigenvalue-eigenvector pairs $\scriptstyle (\lambda _{i},\psi _{i})$ for i = 1, ..., n, where $\scriptstyle \lambda _{i}$ and $\scriptstyle \psi _{i}$ is an eigenvalue and eigenvector of X, respectively. For a simple illustration, consider the situation where an (external) observer is told that a projective measurement of X has been carried out, but the particular outcome of the measurement is not revealed to this observer. Now, when a projective measurement is performed and the measurement outcome recorded is $\scriptstyle \lambda _{i}$, according to quantum mechanics the probability $\scriptstyle p_{i}$ of obtaining this measurement result is $\scriptstyle p_{i}\,=\,|\langle \phi ,\psi _{i}\rangle |^{2}$ (here $\scriptstyle \langle \cdot ,\cdot \rangle$ denotes the inner product between two vectors in the associated Hilbert space), and immediately after measurement the state of the system becomes (or, in popular terminology, “collapses” to) $\scriptstyle \psi _{i}$. However, to our observer, from whom the measurement outcome has been kept secret, based on the information that measurement has been performed alone the best description that he or she has of the current (post-measurement) state of the system is that it can be any of the states in the collection $\scriptstyle \{\psi _{i};\,i\,=\,1,\ldots ,n\}$ with probability $\scriptstyle p_{i}$ of being in state $\scriptstyle \psi _{i}$. Thus, to the observer, the system after the measurement would be in a mixed state.

The mixed state is an extremely useful extension of the original (but limited) notion of “state” in quantum mechanics (that is, pure states) that allows an effective description of various scenarios that are encountered in theory and experiments that cannot be described with the pure state formalism, such as selective measurements (on ensembles). Although an intuitively obvious way of representing a mixed state would be to write it as (in this instance, for the case of the finite dimensional quantum system in the above paragraph) $\scriptstyle \{(p_{i},\psi _{i});\;i\,=\,1,\ldots ,n)\}$, physicists have found that a more effective operational way of representing a mixed state is via a so-called density operator. In the density operator formalism, a pure state $\scriptstyle \phi$ would represented by a projection operator $\scriptstyle P\,=\,\phi \phi ^{*}$ (here $\scriptstyle \phi ^{*}$ denotes the linear functional $\scriptstyle \phi ^{*}(\cdot )\,=\,\langle \phi ,\cdot \rangle$, which for quantum mechanical systems described on $\scriptstyle \mathbb {C} ^{n}$ can be identified with the Hermitian transpose of $\scriptstyle \phi$), while a mixed state $\scriptstyle \{(p_{i},\psi _{i});\;i\,=\,1,\ldots ,n)\}$ in the notation above is described by the density operator $\scriptstyle \rho \,=\,\sum _{i=1}^{n}p_{i}P_{i}$ with $\scriptstyle P_{i}\,=\,\psi _{i}\psi _{i}^{*}$ (note that each $\scriptstyle P_{i}$ is also a density operator). Density operators are trace class operators on the Hilbert space of the system with unity trace. The density operator corresponding to a pure state satisfies $\scriptstyle {\rm {tr(\rho ^{2})\,=\,1}}$, while the density operator of a mixed state satisfies $\scriptstyle {\rm {tr(\rho ^{2})\,<\,1}}$, where $\scriptstyle {\rm {tr}}(\cdot )$ denotes the trace of the operator.

It is worth emphasizing that the correspondence of a mixed state to a density operator is not one to one, if fact it is many to one. That is, different mixed states can have the same density operator representation. Conversely, given a density operator it is not possible to say that it corresponds to a particular mixed state.