Download “What is quantum theory about?” Jos Uffink March 26, 2010, Utrecht

“What is quantum theory about?”
Jos Uffink
Institute of History and Foundations of Science, Utrecht University,
[email protected]
March 26, 2010, Utrecht
1. Introduction: Is quantum theory about objects or information?
2. A look at Quantum Teleportation
3. Preliminary conclusions
4. How could quantum states represent personal beliefs?
5. The Quantum de Finetti Theorem
6. Looking back at some 19th century debates
7. Conclusions
1. Introduction
◮
Quantum theory is our best candidate today for a
fundamental theory of physics.
◮
but how to conceive of what this theory decribes?
◮
is QT about the behaviour of microconstiuents of a system
(whether they be particles, fields, strings, etc)?
◮
Or is it about the information that humans have about the
system?
The question is old (cf Bohr-Einstein debate)
But it has taken on a new dimension with the advent of quantum
information theory.
I want to . . . argue for this thesis: the interpretation of
quantum mechanics as a theory about the representation
and manipulation of information in our world, not a
theory about the mechanics of nonclassical waves or
particles. J. Bub, Found. Phys. (2005).
A quantum state is viewed here as an expression of an
agents personalist Bayesian degrees of belief, or
probabilities, concerning the results of measurements.
C.A. Fuchs and R Schack http://arxiv.org/abs/0912.4252.
2. Quantum teleportation
Alice
|ψi1 = |φi
Bob
2. Quantum teleportation
Alice
Bob
|ψi1 = |φi
Source
|Ψi23 =
√1 (|
2
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
|ψi1 = |φi
2
|Ψi23 =
Source
√1 (|
2
3
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
Measure on 1&2
|ψi1 = |φi
in Bell basis
2
|Ψi23 =
Source
√1 (|
2
3
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
Measure on 1&2
|ψi1 = |φi
in Bell basis
send outcome to Bob
2
|Ψi23 =
Source
√1 (|
2
3
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
Measure on 1&2
|ψi1 = |φi
in Bell basis
send outcome to Bob
2
|Ψi23 =
Source
√1 (|
2
After manipulation:
3
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
Measure on 1&2
|ψi1 = |φi
in Bell basis
send outcome to Bob
2
|Ψi23 =
Source
√1 (|
2
After manipulation:
|ψi3 = |φi
3
↑i| ↓i − | ↓i| ↑i)
2. Quantum teleportation
Alice
Bob
Measure on 1&2
|ψi1 = |φi
in Bell basis
send outcome to Bob
2
|Ψi23 =
Source
√1 (|
2
After manipulation:
|ψi3 = |φi
3
↑i| ↓i − | ↓i| ↑i)
Figure: Quantum teleportation.
What is surprising about this protocol?
◮
◮
◮
A quantum state transferred, from particle 1 to 3, which never
had interaction with 1.
Only two bits of information are needed; whereas a
specication of |φi requires two real paprameters.
The association with Star Trek’s “Mr. Scotty, beam me up”
What is surprising about this protocol?
◮
◮
◮
A quantum state transferred, from particle 1 to 3, which never
had interaction with 1.
Only two bits of information are needed; whereas a
specication of |φi requires two real paprameters.
The association with Star Trek’s “Mr. Scotty, beam me up”
However:
◮
Particle 3 might be quite different from particle 1! (Say: 1 is
a neutron, or a squid, and 3 a photon). It is only the state
that is transferred; not the system!
But then again:
◮
What makes up the identity of a system? Is it the matter out
of which it is composed or rather its state?
◮
If we take humans as example, it is not the matter out of
which we are made that determines who we are.
◮
This is an old lesson: Already in the mid-nineteenth century
chemists figured out that molecular structure can be more
important than composition.
On the other hand:
◮
If my physical state could be perfectly copied to a computer
memory, which would last indefinitely, would that mean that
“I” am immortal? (cf. F.Tipler’s Physics of immortality
3. Preliminary conclusions
◮
the quantum state is often more important to what a system
is, than its material composition.
◮
In as far as quantum theory is a theory about quantum states,
it might be a theory about information.
◮
But the theory also needs a Hamiltonian, a dimension of the
state space, and non-dynamic quantities like mass, charge, etc.
4. Could quantum states represent personal beliefs?
◮
Major obstacle against this view is that almost every working
physicist takes a quantum state to be a (possibly unknown)
objective attribute of a physical system.
◮
The task for those who claim it is rather a representation of
personal beliefs is to show how this (in their eyes mistaken)
view can be so fruitful. =⇒ The quantum de Finetti
Theorem.
The (classical) de Finetti theorem (1937) shows that, even if you
deny the existence of objective probabilities, if your personal beliefs
satisfy a certain condition (exchangeability), they can be
represented as if there was an unknown objective probability and
you were uncertain about its value.
5. The Quantum de Finetti Theorem: Fuchs & Schack
(2004)
Let ρN be any (possibly mixed) state of a quantum system of N
similar subsystems (i.e.: each subsystem has a state space of
dimension d).
Suppose that state is exchangeable in the sense that for any
sequence of measurements E 1 (α1 ), . . . E N (αN ) performed on the
system the probability for the outcomes are permutation invariant,
i.e.
(1)
Probρ (α1 , . . . , αN ) = ProbρN (απ(1) , . . . , απ(N) )
where π denotes a permutation of {1, . . . , N} and
ProbρN (α1 , . . . , αN ) = Tr ρN E 1 (α1 ) ⊗ · · · ⊗ E N (αN )
(2)
Quantum De Finetti Theorem:
If ρN is exchangeable for all N = 1, 2, . . . then
Z
N
P(ρ)ρ ⊗ · · · ⊗ ρ dρ
ρ =
(3)
Sd
where Sd is the single-particle state space, and P(ρ) some
probability density over Sd .
◮
◮
◮
In other words: if ρN , representing your personal beliefs, is
exchangeable, it is as if each particle has an unknown,
objective state ρ of its own.
This theorem goes a long way in explaining why we can
operate with assuming particles have a quantum state, and
how we can learn about it by making measurements
But of course, it does not explain why measurements have to
be represented by projectors (or positive operators) E (α)
acting on a Hilbert space.
6. Looking back on some 19th century debates
◮
Mechanicism: There are only particles and forces. The goal of
physics is to provide a detailed mechanical account of them.
◮
Energeticsm: There are only various kinds of energy. The goal
of physics is to describe transformations between these kinds
of energy.
Closely related is the distinction principle theories vs constructive
theories (Einstein 1920):
◮
Constructive theories start by postulating (simple) hypotheses
about basic entities. They aim to reconstruct the observable
phenomena from these.
◮
Principle theories start from “empirical principles”, i.e. bold
generalizations of empirical facts. They proceed by deducing
their consequences using only logic.
According to Einstein each approach had its own merits, in terms
of understanding and security.
“ Suppose that we have before us a machine; the initial wheel-work and
the final wheel-work alone are visible, but the transmission, by which the
movement is communicated from one to the other are hidden in the
interior; we do not know whether the communication is made by gearing
or by belts, or by connecting-rods etc.
Is it impossible to understand anything about this machine so long as we
do not take it to pieces? Surely not, and the principle of the conservation
of energy suffices to deduce the most interesting point. Suppose we learn
that the final wheel turns ten times less quickly than the initial wheel; we
are able thence to conclude that a couple applied to the one will be
balanced by a couple ten times greater applied to the other. For that
there is no need to penetrate the mechanism . . .
Well, in regard to the universe, the principle of the conservation of energy
renders the same service. This is also a machine, and almost all its parts
are profoundly hidden from us; but in observing the movement of those
that we can see, we can, by aid of this principle, draw conclusions which
remain true whatever may be the details of the invisible mechanism
which animates them. ”
Poincar´e The principles of mathematical physics (1904)
7. Conclusions
◮
Today, we do not regard energy and matter as mutually
conflicting ontologies underlying physics.
◮
Rather: each provides a valuable method from which theories
can be approached.
◮
Similarly, I would argue there is no need for opposition
between a viewpoint in which quantum states describe the
mechanics of physical systems, or human manipulation of
information.
◮
The claim that quantum theory could be understood as a pure
theory about information has (not yet) been substantiated. (If
only because quantum theory is not just about quantum
states).