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# Decoherence, einselection, and the quantum origins of the classical

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### Abstract

Decoherence is caused by the interaction with the environment. Environment monitors certain observables of the system, destroying interference between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly non-local "Schr\"odinger cat" states. Classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit: Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation.

### Most cited references218

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### Mixed State Entanglement and Quantum Error Correction

(1996)
Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state $$|\xi\rangle$$ can be transmitted at some rate Q through a noisy channel $$\chi$$ without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state $$\hat{M}(\chi)$$ (obtained by sharing halves of EPR pairs through a channel $$\chi$$) yields a QECC on $$\chi$$ with rate $$Q=D$$, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts $$D_1(M)$$ and $$D_2(M)$$ that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for $$E(M)$$ when $$M$$ is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic $$Q$$ (or $$D$$) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.
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(1995)
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### Author and article information

###### Journal
2001-05-24
2003-06-19
###### Article
10.1103/RevModPhys.75.715
quant-ph/0105127
Rev. Mod. Phys. 75, 715 (2003)
Final version of the review, with brutally compressed figures. Apart from the changes introduced in the editorial process the text is identical with that in the Rev. Mod. Phys. July issue. Also available from http://www.vjquantuminfo.org
quant-ph

Quantum physics & Field theory