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A general way to describe a system $S$ that is entangled with an environment $E$ is

$\rho_{S}=Tr(\rho_{SE})=\sum\limits_{m,n}c_mc^*_n |s_m\rangle \langle s_n| \langle e_n|e_m\rangle$

with $\psi_S=\sum\limits_n c_n|s_n\rangle$ the state of the system and $|e_n\rangle $ the corresponding environment states.

In general the different environment states can not be considered as orthogonal, thus my question more neatly formulated is:

Is there a general mechanism, such that the environment states become orthonormal, or is this actually just a hypothesis essential for the emergence of decoherence (that has been supported by many model calculations and experiments).

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  • $\begingroup$ This is (unfortunately) not the most general way to describe decoherence. Otherwise the much used Born approximation $\rho_{SE}(t)\approx \rho_S(t) \otimes \rho_E(t)$ (with $\rho_E$, the density matrix of the environment, and $\rho_{SE}$, the density matrix of the combined System) would not make sense. $\endgroup$
    – meigge
    Oct 9, 2013 at 16:15

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The situation depends on the specification of "system" and "environment", and the detailed form of the interaction Hamiltonian. Clearly, if your environment has only a few degrees of freedom then the Poincare recurrence time is finite, and the assertion that the coupling results in decoherence is false at certain points in time, when revivals are seen in the system coherences. Usually, one is interested in models where the environment has many more degrees of freedom than the system, in which case the hypothesis holds for many physically reasonable system-environment interaction models (see the answer by Trimok). If the interaction couples each state of the system differently to many-body states of the environment Hilbert space, over time one expects these environment states will evolve to become orthogonal due to the interaction, implying complete decoherence of the system.

However, it is not difficult to find reasonable models where this is not true, for example two qubits interacting linearly with a common bosonic bath but not with each other. This situation arises quite naturally when considering, for example quantum dots interacting with lattice phonons or trapped atomic impurities immersed in a BEC. The bath cutoff frequency $\omega_c$ and signal speed $c$ define a length scale $c/\omega_c$. If the distance between qubits is much smaller than this length scale, the decoherence between certain states of the two-qubit system can be almost perfectly suppressed. This allows one to preserve entanglement in such decoherence-free subspaces. See Palma et al., Proc. Roy. Soc. Lond. A452 (1996) 567-584.

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A baby model : Suppose your system state is $|\psi \rangle = \sum c_n |s_n \rangle$, with the $s_n$ normed, but not orthogonal : $\langle s_m|s_n \rangle = \cos \theta_{mn}$, with $\cos \theta_{nn} =1$, and $\cos \theta_{mn} < 1$ if $m \neq n$ . Suppose that the system entangles with an environment reduced to one particle, we would have :

$\psi_E = |\psi \rangle = \sum c_n |s_n \rangle (|s_n \rangle)_E$

With a real environment made of a big number $N$ of particles, we would have :

$\psi_E = |\psi \rangle = \sum c_n |s_n \rangle (\overbrace {|s_n \rangle \otimes |s_n \rangle \otimes ... |s_n \rangle}^{N})_E = \sum c_n |s_n \rangle (\otimes^N |s_n \rangle)_E$

But, now, we have :

$(\otimes^N \langle s_n |)_E (\otimes^N |s_m \rangle)_E = (\cos \theta_{mn})^N$

So, we see, that when $N$ is big enough, the environment states entangled with the system states become orthogonal, and then the reduced density matrix corresponds to a mixed state.

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  • $\begingroup$ Your model is based on the assumption that the environment can be written in a tensor product form, but is this always true? E.g. also for an environment consisting of coupled harmonic oscillators? $\endgroup$
    – meigge
    Oct 8, 2013 at 12:59
  • $\begingroup$ I'd say you can always write the environment in tensor product form since there is little large-scale entanglement in our universe. You can always find subspaces of the universe Hilbert space which are not entangled. Problem is, the fact that there is no (or little) persistent entanglement is due to decoherence. It's the chicken and the egg. $\endgroup$
    – Jonas Greitemann
    Oct 8, 2013 at 13:23
  • $\begingroup$ Surely I can always find subspaces of the universe Hilbert spaces, which are not entangled, but for nearly all of them $\theta_{mn}=0$ (as a particle on alpha centauri is presumably ignorant of a measurement on the earth). The environment subsystems for which $\theta_{mn}\neq 0$ are typically spatially near the measured system and thus "close" to each other. I don't find it obvious that sufficiently many of them can be considered as separated. $\endgroup$
    – meigge
    Oct 8, 2013 at 14:31
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    $\begingroup$ @meigge Although I agree with your point, many of the environmental states/systems that seem most natural can be written as product states. Going beyond these kinds of simplistic models is part of modern research into open systems. To use your example, an environment of coupled harmonic oscillators can be written in the normal mode basis. If the interaction between the system and environment is weak, then this coupling will be linear in the normal mode operators, so an initially thermal environmental state should remain in a product state over the normal modes. $\endgroup$
    – Mark Mitchison
    Oct 8, 2013 at 15:07

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