A first look at quantum Darwinism

The last section settled the accounts of decoherence: the environment photographs nonstop, and coherence outside the pointer states evaporates in an instant. But the final piece of the story is still off the board — objectivity.

The phenomenon first: you have never actually “seen” the dust grain

Ten people stand around a dust grain on a table. Everyone sees it in the same position; the tenth person’s looking does nothing to the first nine’s conclusions; and nobody has “disturbed” the grain. Think this utterly ordinary scene through with chapter 3’s measurement postulates and it is riddled with problems: measurement changes the state, so why do ten “measurements” not interfere with one another? An unknown quantum state cannot be copied, by chapter 9’s no-cloning theorem — so how can ten people each hold a consistent piece of information?

The key that cracks the case is admitting a humble fact: not one of those people ever touched the grain. What each person intercepted is a small handful of photons scattered off the grain into their own pupils — the very environment particles that were “photographing” the system in the last section. Observers never measure the system itself; observers only read the environment’s records.

And so the objectivity question is re-posed: this photo album the environment keeps — why does every page say the same thing?

Branching structure: one piece of information, billions of copies

Decoherence pushes the joint system-environment state towards a very special shape. Let the grain’s pointer states be $\ket{x_1},\ket{x_2}$ (last section: selected by the interaction), and let the environment consist of $N$ sub-blocks (each handful of photons counts as one block). Scattering turns every block into a photograph:

$$\ket{\Psi}=a\,\ket{x_1}\ket{e_1^{(1)}}\ket{e_1^{(2)}}\cdots\ket{e_1^{(N)}} +b\,\ket{x_2}\ket{e_2^{(1)}}\ket{e_2^{(2)}}\cdots\ket{e_2^{(N)}}\tag{10.6.1}$$

where $\braket{e_2^{(k)}}{e_1^{(k)}}\approx0$ (every block’s photograph suffices to resolve the position). This is called branching structure: one and the same piece of information — “where the grain is” — is stored in the environment in $N$ independent copies. For a dust grain in air, every second adds molecular-collision records of order $10^{18}$; $N$ is astronomical.

What a fragment can read out, and what it cannotadvanced~8 minexpand

Reading the position: any small fragment is enough. Some observer intercepts only the $k$-th block of the environment. Trace out everything else (the system + the other blocks); the cross terms carry factors $\braket{x_2}{x_1}\braket{e_2^{(1)}}{e_1^{(1)}}\cdots\approx0$, leaving

$$\hat\rho^{(k)}=|a|^2\ket{e_1^{(k)}}\bra{e_1^{(k)}}+|b|^2\ket{e_2^{(k)}}\bra{e_2^{(k)}}\tag{10.6.2}$$

Since $\ket{e_1^{(k)}},\ket{e_2^{(k)}}$ are nearly orthogonal, a measurement distinguishing them reads out “which branch the grain favours” with probability 1 — and does so without touching the grain or any other fragment. Ten observers read ten different fragments and their conclusions agree automatically (they are all copies of the same branching structure), with no mutual disturbance. That is the mechanism of objectivity: not ten people measuring the same system, but ten people reading ten printed copies of the same book.

Reading the phase: no number of fragments suffices (short of all of them). Where did the relative phase between $a$ and $b$ go? Try to find it on the system itself: trace out the whole environment, and $\hat\rho_S=|a|^2\ket{x_1}\bra{x_1}+|b|^2\ket{x_2}\bra{x_2}$ — the phase is not there. Try any combination of fragments: as long as even one block is left out, tracing over that one block kills every cross term — the phase is not there either. The phase is a global property, written into the joint correlations of all $N$ blocks with the system, and it can be retrieved only by collecting every last record and performing an interference measurement on the whole. With $N$ astronomical, that is impossible in any operational sense.

Squaring the books with no-cloning. What got copied $10^{30}$ times is only one classical bit (“branch one or branch two”) — not the quantum state itself: the amplitudes $a,b$ and the phase were never copied. The no-cloning theorem forbids copying unknown superpositions; it has never forbidden transcribing the labels of a set of orthogonal states many times over — a CNOT gate has always been able to do that (chapter 9). The only information that can be massively copied is information in the pointer basis — which explains, in reverse, why our entire world-picture is written in pointer quantities (position and functions of it).

∎Key formulas

• Branching structure

  $$\ket{\Psi}=a\ket{x_1}\ket{e_1^{(1)}}\cdots\ket{e_1^{(N)}}+b\ket{x_2}\ket{e_2^{(1)}}\cdots\ket{e_2^{(N)}}$$

  The pointer label is stored in N independent copies across the environment

• Reduced state of one fragment

  $$\hat\rho^{(k)}=|a|^2\ket{e_1^{(k)}}\bra{e_1^{(k)}}+|b|^2\ket{e_2^{(k)}}\bra{e_2^{(k)}}$$

  Any fragment reads out the pointer information without disturbing the system or the other fragments

• Objectivity criterion

  $$R_\delta\gg1\ \text{(classical quantities)};\qquad R_\delta\approx1\ \text{(quantum coherent information)}$$

  Redundancy = number of environment fragments that independently supply the system’s information

?Self-check3 questions

1. 1.

   Quantum Darwinism explains "ten people see the same dust grain, agree, and disturb nothing" by:

   • Each person intercepts a different record copy in the environment — they read different copies of the same branching structure
   • The grain’s wavefunction is repeatedly collapsed to the same position by ten measurements
   • Macroscopic objects do not obey quantum mechanics, so measurement does not disturb them
   • The ten people just happened to tacitly perform the same kind of measurement

   Submit
2. 2.

   Which statements about how information is distributed in a branching-structure state are correct? (Select all that apply.)

   Select all that apply

   • Any small environment fragment suffices to read out the pointer (which-branch) information
   • The relative phase is not on the system, nor on any combination of fragments, unless every record is collected
   • What is redundantly copied is the pointer label — one classical bit — not the quantum state itself
   • Collecting half the environment fragments is enough to read out the phase

   Submit
3. 3.

   Why does "copying the pointer label 10³⁰ times" not violate the no-cloning theorem?

   • The theorem forbids copying unknown superpositions; transcribing the labels of orthogonal states many times is allowed — a CNOT can do it
   • The no-cloning theorem only holds for photons
   • Because the macroscopic environment is not a quantum system
   • It actually is violated, which makes quantum Darwinism doubtful

   Submit

What comes next

Chapter 10 closes here. Looking back over it: density matrices, the Bloch sphere, channels, Lindblad, decoherence, Darwinism — we stayed inside one language throughout (states and operators, Hilbert space, tracing out), pulling the camera further and further back, until the classical world developed into view at the far end of the lens.

The next chapter does something wholly different: it replaces the language altogether. No state vectors, no operators — back to Feynman in 1948: a particle goes from A to B, and you add up every path it could possibly take, each path carrying a phase $\ee^{\ii S/\hbar}$. You will see that all of quantum mechanics can be rebuilt from that one sentence, that double-slit interference becomes its most trivial special case, and that classical mechanics’ principle of least action floats up out of the sea of phases by itself, under the name of “stationary phase”.