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arXiv:quant-ph/0509100 · Physical Review A

Physical purification of quantum states

Matthias Kleinmann, Hermann Kampermann, Tim Meyer, Dagmar Bruß

Published Jun 8, 2006• 29 citations
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Abstract

We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two noncommuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.

DOI
10.1103/physreva.73.062309

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We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two noncommuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.
We prove that no physical map can provide a perfect purification of an unknown quantum state. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, for which the reduced density matrix is identical to the original state. We investigate the possibility of performing an imperfect purification, by either admitting a non-pure output state, or a non-zero distance of its reduced state to the original. Surprisingly, the restriction of the input set to two states leads to non-trivial purifying maps.
Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.
Purification is a well-known tool to represent mixed quantum states as pure states on an enlarged Hilbert space. A purification of a given state is not unique and its spatial entanglement strongly depends on the particular choice. This entanglement is linked to how efficiently the purified state can be represented using matrix-product states (MPS). Here, the authors introduce an MPS-based method that allows the user to efficiently find the minimally entangled purification of a given mixed state and demonstrate that it can significantly reduce the entanglement in certain cases.
Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions and classical communication.
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of nnn copies of a two-qubit state.
Pure-state quantum phases are defined through finite-depth local unitary connections. This concept of phases has recently been generalized to mixed states in terms of locally reversible channels. In this work, we establish a fundamental relation between mixed-state phases and pure-state phases. We show that a locally reversible channel between mixed states can be lifted to a local isometry between their canonical purifications; thus, if two mixed states are in the same phase, then their canonical purifications are in the same pure-state phase. Furthermore, an explicit construction of the lifting isometry is given by ``half'' of the Petz recovery map, suggesting that the Petz recovery map \emph{without rotation} is enough to connect mixed state phases. We illustrate our results with two examples: toric code under bit flip noise and Chern insulator under fermion number noise.