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quantum information theory Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
, quantum state purification refers to the process of representing a mixed state as a pure quantum state of higher-dimensional
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. The purification allows the original mixed state to be recovered by taking the
partial trace In linear algebra and functional analysis, the partial trace is a generalization of the trace (linear algebra), trace. Whereas the trace is a scalar (mathematics), scalar-valued function on operators, the partial trace is an operator (mathemati ...
over the additional degrees of freedom. The purification is not unique, the different purifications that can lead to the same mixed states are limited by the Schrödinger–HJW theorem. Purification is used in algorithms such as
entanglement distillation Entanglement distillation (also called entanglement purification) is the transformation of ''N'' copies of an arbitrary Quantum entanglement, entangled state \rho into some number of approximately pure Bell pairs, using only LOCC, local operation ...
, magic state distillation and algorithmic cooling.


Description

Let \mathcal H_S be a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and consider a generic (possibly mixed)
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
\rho defined on \mathcal H_S and admitting a decomposition of the form \rho = \sum_i p_i, \phi_i\rangle\langle\phi_i, for a collection of (not necessarily mutually orthogonal) states , \phi_i\rangle \in \mathcal H_S and coefficients p_i \ge 0 such that \sum_i p_i = 1. Note that any quantum state can be written in such a way for some \_i and \_i. Any such \rho can be ''purified'', that is, represented as the
partial trace In linear algebra and functional analysis, the partial trace is a generalization of the trace (linear algebra), trace. Whereas the trace is a scalar (mathematics), scalar-valued function on operators, the partial trace is an operator (mathemati ...
of a
pure state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space \mathcal H_A and a pure state , \Psi_\rangle \in \mathcal H_S \otimes \mathcal H_A such that \rho = \operatorname_A\big(, \Psi_\rangle\langle\Psi_, \big). Furthermore, the states , \Psi_\rangle satisfying this are all and only those of the form , \Psi_\rangle = \sum_i \sqrt , \phi_i\rangle \otimes , a_i\rangle for some orthonormal basis \_i \subset \mathcal H_A. The state , \Psi_\rangle is then referred to as the "purification of \rho". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state. Because all of them admit a decomposition in the form given above, given any pair of purifications , \Psi\rangle, , \Psi'\rangle \in \mathcal H_S \otimes \mathcal H_A, there is always some unitary operation U : \mathcal H_A \to \mathcal H_A such that , \Psi'\rangle = (I\otimes U) , \Psi\rangle.


Schrödinger–HJW theorem

The Schrödinger–HJW theorem is a result about the realization of a mixed state of a
quantum system Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger ( ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was an Austrian-Irish theoretical physicist who developed fundamental results in quantum field theory, quantum theory. In particul ...
who proved it in 1936, and after Lane P. Hughston,
Richard Jozsa Richard Jozsa is an Australian mathematician who holds the Leigh Trapnell Chair in Quantum Physics at the University of Cambridge. He is a fellow of King's College, Cambridge, where his research investigates quantum information science. A pion ...
and
William Wootters William "Bill" Kent Wootters is an American theoretical physicist, and one of the founders of the field of quantum information theory. In a 1982 joint paper with Wojciech H. Zurek, Wootters proved the no-cloning theorem, at the same time as De ...
who rediscovered in 1993. The result was also found independently (albeit partially) by
Nicolas Gisin Nicolas Gisin (born 1952) is a Swiss physicist and professor at the University of Geneva, working on the foundations of quantum mechanics, quantum information, and communication. His work includes both experimental and theoretical physics. He has ...
in 1989, and by Nicolas Hadjisavvas building upon work by E. T. Jaynes of 1957, while a significant part of it was likewise independently discovered by N. David Mermin in 1999 who discovered the link with Schrödinger's work. Thanks to its complicated history, it is also known by various other names such as the GHJW theorem, the HJW theorem, and the purification theorem. Consider a mixed quantum state \rho with two different realizations as ensemble of pure states as \rho = \sum_i p_i , \phi_i\rangle\langle\phi_i, and \rho = \sum_j q_j , \varphi_j\rangle\langle\varphi_j, . Here both , \phi_i\rangleand , \varphi_j\rangle are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state \rho reading as follows: : Purification 1: , \Psi_^1\rangle=\sum_i\sqrt, \phi_i\rangle \otimes , a_i\rangle; : Purification 2: , \Psi_^2\rangle=\sum_j\sqrt, \varphi_j\rangle \otimes , b_j\rangle. The sets \and \ are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix U_A such that , \Psi^1_\rangle = (I\otimes U_A), \Psi^2_\rangle. Therefore, , \Psi_^1\rangle = \sum_j \sqrt, \varphi_j\rangle\otimes U_A, b_j\rangle, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.


References

{{DEFAULTSORT:Schrodinger-HJW theorem Quantum information theory