Relative States and the Environment: Einselection, Envariance, Quantum Darwinism, and the Existential Interpretation (notes)
Relative State Quantum interpretation by Hugh Everett III:
“explains “collapse of the wavepacket” by postulating that observer perceives the state of the “rest of the Universe” relative to his own state, or – to be more precise – relative to the state of his records”
This has a problem:
“One is now forced one to seek sets of preferred, eﬀectively classical but ultimately quantum states that can deﬁne branches of the universal state vector, and allow observers to keep reliable records.”
(What about this, is it relevant? Original paper here: “The Invariant Set Hypothesis: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity“. If the reality of the universe inhabits a tiny edge, a fractal coastline on the edge of chaos, then this would be a candidate for preferred states. If the mechanism of quantum Darwinism could show a tendency to produce a set of preferred states which lay on a fractal then this would link the ideas of Zurek and Palmer in these two papers).
Solving this requires a “quantum origin of objective existence”. Zurek shows “how [the] mathematical structure of quantum theory supplemented by the only uncontroversial measurement axiom (that demands immediate repeatability – and, hence, predictability – of idealized measurements) leads to preferred sets of states”.
“Only states that can survive decoherence can produce information theoretic progeny…This quantum Darwinism allows observers to use [the] environment as a witness – to acquire information about pointer states indirectly, leaving system of interest untouched and its state unperturbed.”
Intro: The clash of quantum determinism (as defined by the Schrodinger equation) and quantum randomness (random collapse of wavepacket) causes interpretational controversies.
Quantum theory based on 2 axioms which establish the formal structure, but do not say anything about collapse or probabilities:
(i) State of a quantum system is represented by a vector in its Hilbert space HsubS implies =>quantum superposition.
(ii) Evolutions are unitary (i.e., generated by Schrodinger equation) implies => unitarity of quantum evolutions
A third and fourth axiom tie these abstract state vectors in Hilbert space to experimental data:
(iii) Immediate repetition of a measurement yields the same outcome (this is idealized and difficult in practice).
(iv) Measurement outcome is one of the orthonormal states – eigenstates of the measured observable (the collapse postulate, where controversy lies).
[1.1] Starting from a general state |ψS ⟩ in a Hilbert space of the system (axiom (i)),
an initial state |A0 ⟩ of the apparatus, and assuming unitary evolution (axiom (ii))
one is led to a superposition of outcomes, which contradicts axiom (iv)
In other words, the math predicts that the universe contains all possible outcomes but experimental measurement only ever yields one.
(The Copenhagen Interpretation says that the measurer is classical and that the collapse occurs on the boundary between the quantum superimposed axiom i and ii and the classical.)
Everett’s relative interpretation says that a measurement (including appearance of the collapse) is al-
ready described by (1.1). One just need to include observer in the wavefunction. At this point the observers universe is in accordance with one state (the problem is that you end up with infinite states for the bits of the apparatus, or whatever become entangled with another quantum system), so: “there is something that (in spite of the egalitarian superposition principle enshrined in axiom (i)) picks out certain preferred quantum states, and makes them eﬀectively classical.”
In other words, Everett merely says that the universe is in one state for an observer, but many branching states for other observers or parts of an experiment. “Everett decomposes the global entangled tensor state into a superposition of branches – Cartesian products – labeled by observer’s records.”
Also - what is the probability that a particular outcome will be observed, This ‘Preferred basis’ problem was settled by environment induced superselection (einselection).
However, they come at a price that might have been unacceptable to Everett: Decoherence and einselection usually employ reduced density matrices. Their physical signiﬁcance derives from averaging, and is thus based on probabilities – on Born’s rule:
axiom (v) Probability pk of ﬁnding an outcome |sk ⟩ in a measurement of a quantum system that was previously prepared in the state |ψ⟩ is given by |⟨s|ψ⟩|^2.
(unlike axiom (iv), axiom (v) doesn’t contradict the first 3 and can be used to justify preferred basis and symptoms of collapse via decoherence and einselection).
axiom (o) The Universe consists of systems (often omitted as obvious) in absence of systems (a universe divisible into parts) the measurement problem disappears.
To provide a quantum account of classical reality, Zurek derives axioms (iv) and (v) from the non-controversial (0)-(iii) and demonstrates objective existence of ‘pointer states’.
For the first part, Zurek shows that:
“any set of states will do providing they are orthogonal…states are (ein)selected by the dynamics of the process of information acquisition”.
For the second task, the problem is: “How to account for objective existence of “reality” using only “unreal” quantum ingredients?”
(Universe is divided into System S and Environment E) “Continuous monitoring of S by its environment results in redundant records of in E. Thus, many observers can ﬁnd out the state of the system indirectly, from small fragments of the same E that caused decoherence. Recent and still ongoing studies discussed in Section IV show how this replication selects “ﬁttest” states that can survive monitoring, and yield copious information-theoretic oﬀspring: Quantum Darwinism favors pointer observables at the expense of their complements. Objectivity of preferred is quantiﬁed by their redundancy – by the number of copies of the state of the system deposited in E . Stability in spite of the environment is clearly a prerequisite for large redundancy. Hence, pointer states do best in this information - theoretic “survival of the ﬁttest”.”
[half way through page 3]
Objective existence can be acquired (via quantum Darwinism) only by a relatively small fraction of all degrees of freedom within the quantum Universe: The rest is needed to “keep records”. Clearly, there is only a limited (if large) memory space available for this at any time. This limitation on the total memory available means that not all quantum states that exist or quantum events that happen now “really happen” – only a small fraction of what occurs will be still in the records in the future. So the ﬁnite memory capacity of the Universe implies indeﬁniteness of the present and impermanence of the past.