# Decoherence: Does the environment must be a tensor product of two sub-systems to remove any ambiguity?

In decoherence theory, we describe the decoherence of a system $$S$$ by the mean of an interaction with an environment. In short, if I consider: $$|\psi_S \rangle=a|0_S\rangle+b|1_S\rangle$$, the system will interact with the environment and I will have:

$$\left( a|0_S\rangle + b|1_S\rangle \right) |0_E\rangle \rightarrow a|0_S0_E(t)\rangle +b|1_S1_E(t)\rangle$$

Tracing out the environment, we have, with $$r(t)=\langle 1_E(t) | 0_E(t) \rangle$$

$$\rho_{S}=|a|^2 |0\rangle \langle 0| + |b|^2 |1 \rangle \langle 1 | +ab^* r(t) |0\rangle \langle 1| + a^*b r^*(t) |1\rangle \langle 0 |$$

On a short timescale, we will have $$r(t) \approx 0$$, the final state of the system will be a classical mixture. This is usually how we describe decoherence process.

When we talk about a measurement process, replacing $$E$$ by an apparatus $$A$$ leaves an un-satisfactory explanation. Indeed if my environment played the role of an apparatus, I would have for the global state, assuming the process completly killed the coherence of the system, and modelling for simplicity the apparatus as a two level system:

$$|\psi_{SA}\rangle= a|0_S0_A\rangle + b|1_S1_A\rangle$$

In the particular case $$a=b=1$$, this state is also $$|++\rangle +|--\rangle$$. The measurement is then ill-defined (did we measure $$\sigma_x$$, or $$\sigma_z$$: it looks the same in those two basis !). I call it the "what did we measure" problem in what follows.

A way to solve this issue is then to use a third system (the environment) that will remove such ambiguity. Basically at the end we will have a $$SAE$$ state: $$a|0_S0_A0_E\rangle + b|1_S1_A1_E\rangle$$ which because of the third system removes all ambiguity (I don't go in the details but this is a result from tri-orthogonal decomposition theorem).

## My question is:

I totally understand the "what did we measure" problem with the apparatus. But then it should also be a conceptual problem for a "natural" decoherence (i.e system interacting with environment). However in all environmental decoherence study I looked we always assume system+environment as the total system of study. We do not include some extra part to remove this issue.

Is this ambiguity implicitly removed by the fact the environment $$E$$ is actually composed of many sub-systems. Basically: $$H_E = H_{E_1} \otimes H_{E_2}$$, and the state is actually something like:

$$|\psi_{SE_1E_2}\rangle= a|0_S0_{E_1}0_{E_2}\rangle + b|1_S 1_{E_1}1_{E_2}\rangle$$

And I can then interpret the decoherence as a measurement process as well (the part $$E_1$$ of the environment measured my system).

Would you agree that it is precisely the fact we can "factorise" the environment into two sub-environment that explains the fact we don't have a "what did we measure" problem with the environment ?

• We can prove the uniqueness of such a form for the state if we assume a fixed tripartite factorization of the Hilbert space, but if we don't assume that, then uniqueness is lost (and the proof is easy). So if you're asking whether the ability to factorize the environment at all solves the what-did-we-measure problem, the answer would be no; but if you're asking whether fixing a factorization of the environment solves the what-did-we-measure problem, the answer would be yes. Are you asking either of these questions? Or something else? Jun 14, 2020 at 17:09
• @ChiralAnomaly thank you for your answer. If I understand you correctly: I must assume to have a tensor product of three (or more right ?) terms to remove this ambiguity. If on the other hand I do not assume this, I could have a particular case in which only two hilbert space are considered: $S$ and $E$ as a whole. And in this case there is still the ambiguity. But the thing is that the physics shouldn't depend on the way I want to write down the vector (which is implicit somehow in our discussion). The basis ambiguity should either be removed or not. Jun 14, 2020 at 19:44
• Then my question is: in practice in environmental decoherence, when people usually explain it with two Hilbert space (system+environment), should it actually be understood as the fact environment is composed of many sub-systems and the basis ambiguity is removed because of them ? How to make sense of basis ambiguity with environmental decoherence ? Jun 14, 2020 at 19:45
• Maybe I will formulate my question differently: to have a well defined environmental decoherence model, do we need to have a well defined physics to have an environment composed of at least two hilbert space so that the ambiguity can be removed ? Is it implicit in all decoherence model in which there are only system and environmental hilbert space ? Because there is a great emphasis about this problem when talking about measuring apparatus but the problem seems like forgotten when we study environmental decoherence which I find weird. Jun 14, 2020 at 19:48
• Having at least two factors in the environment is necessary (so the answer to your comment is yes), but my point was that it's not sufficient. If the factorization of the environment's Hilbert space is not fixed, then the number of factors is irrelevant: the basis ambiguity remains no matter how many factors we allow. This is related to the fact that a $N^K$-dimensional Hilbert space can be factorized into a number $K$ of $N$-dimensional Hilbert spaces in uncountably-infinitely many different ways. Not only is $K\geq 2$ environment-factors required, but those factors must also be fixed. Jun 14, 2020 at 22:33