Transcript Classical World because of Quantum Physics

Faculty of Physics
University of Vienna, Austria
Institute for Quantum Optics and Quantum Information
Austrian Academy of Sciences
Quantum violation of macroscopic realism
and the transition to classical physics
Johannes Kofler
PhD Defense
University of Vienna, Austria
October 3rd, 2008
List of publications
Articles in refereed journals
Submitted
• J. Kofler and Č. Brukner
Conditions for quantum violation of macroscopic
realism
Phys. Rev. Lett. 101, 090403 (2008)
• T. Paterek, R. Prevedel, J. Kofler, P. Klimek, M.
Aspelmeyer, A. Zeilinger, and Č. Brukner
Mathemtical undecidability and quantum
randomness
• J. Kofler and Č. Brukner
Classical world arising out of quantum physics under
the restriction of coarse-grained measurements
Phys. Rev. Lett. 99, 180403 (2007)
• J. Kofler and Č. Brukner
Entanglement distribution revealed by macroscopic
observations
Phys. Rev. A 74, 050304(R) (2006)
• M. Lindenthal and J. Kofler
Measuring the absolute photo detection efficiency
using photon number correlations
Appl. Opt. 45, 6059 (2006)
• J. Kofler, V. Vedral, M. S. Kim, and Č. Brukner
Entanglement between collective operators in a
linear harmonic chain
Phys. Rev. A 73, 052107 (2006)
• J. Kofler, T. Paterek, and Č. Brukner
Experimenter’s freedom in Bell's theorem and
quantum cryptography
Phys. Rev. A 73, 022104 (2006)
Contributions in books
• J. Kofler and Č. Brukner
A coarse-grained Schrödinger cat
Quantum Communication and Security, ed. M.
Żukowski, S. Kilin, and J. Kowalik (IOS Press,
2007)
Proceedings
• R. Ursin et. al.
Space-QUEST: Experiments with quantum
entanglement in space
59th International Astronautical Congress (2008)
Articles in popular journals
• A. Zeilinger and J. Kofler
La dissolution du paradoxe
Sciences et Avenir Hors-Série, No. 148, p. 54
(Oct./Nov. 2006)
Classical versus Quantum
Phase space
Hilbert space
Continuity
Quantization, “clicks”
Newton’s laws
Schrödinger equation
Definite states
Superposition/Entanglement
Determinism
Randomness
- When and how do physical systems stop to behave quantum mechanically
and begin to behave classically?
- What is the origin of quantum randomness?
Isaac Newton
Ludwig Boltzmann
Albert Einstein
Niels Bohr
Erwin Schrödinger Werner Heisenberg
Double slit experiment
With electrons!
(or neutrons, molecules, photons, …)
With cats?
|cat left + |cat right ?
Why do we not see
macroscopic superpositions?
Two schools:
- Decoherence
uncontrollable interaction with environment; within quantum physics
- Collapse models
forcing superpositions to decay; altering quantum physics
Alternative answer:
- Coarse-grained measurements
measurement resolution is limited; within quantum physics
Macrorealism
Leggett and Garg (1985):
Macrorealism per se
“A macroscopic object, which has available to
it two or more macroscopically distinct states,
is at any given time in a definite one of those
states.”
Non-invasive measurability
“It is possible in principle to determine which
of these states the system is in without any
effect on the state itself or on the subsequent
system dynamics.”
Q(t1)
Q(t2)
t
t=0
t1
t2
The Leggett-Garg inequality
Dichotomic quantity:
t
t=0
t
Temporal correlations
t1
t2
t3
t4
All macrorealistic theories fulfill the
Leggett-Garg inequality
Violation  macrorealism per se or/and non-invasive measurability failes
When is the Leggett-Garg inequality violated?
Rotating spin-½
Rotating classical spin
precession around x
precession around x
measurement along z
sign of z component
Violation of the
Leggett-Garg inequality
Classical evolution
classical limit
½
Why don’t we see violations in everyday life?
Coarse-grained measurements
Model system: Spin j
macroscopic: j ~ 1020
Arbitrary state:
- Measure Jz, outcomes: m = – j, –j+1, ..., +j (2j+1 levels)
- Assume measurement resolution is much weaker than the intrinsic uncertainty such
that
neighbouring outcomes are bunched together into “slots” m.
m = –j
=
m = +j
1
2
3
4
Macrorealism per se
Probability for outcome m can be computed
from an ensemble of classical spins with
positive probability distribution:
Coarse-grained measurements:
any quantum state allows a
classical description
This is macrorealism per se.
J. K. and Č. Brukner, PRL 99, 180403 (2007)
Example: Rotation of spin j
j
Coarse-grained measurement
Sharp measurement
of spin z-component
–j
+j
1 3 5 7 ... Q = –1
–j
+j
2 4 6 8 ... Q = +1
classical limit
Fuzzy measurement
Violation of Leggett-Garg inequality
for arbitrarily large spins j
Classical physics of a rotating
classical spin vector
J. K. and Č. Brukner, PRL 99, 180403 (2007)
Coarse-graining  Coarse-graining
Sharp parity measurement
Neighbouring coarse-graining
(two slots)
(many slots)
1 3 5 7 ...
2 4 6 8 ...
Slot 1 (odd)
Slot 2 (even)
Violation of
Leggett-Garg inequality
Note:
Classical physics
Superposition versus Mixture
To see the quantumness of a spin j, you need to resolve j1/2 levels!
Albert Einstein and ...
Charlie Chaplin
Non-invasive measurability
Depending on the outcome, measurement reduces state
to
Fuzzy measurements only reduce previous
ignorance about the spin mixture:
For macrorealism we need more: Total ensemble without measurement should be the
weighted mixture of the evolved subensembles after a measurement:
Non-invasive measurability
t=0
ti
tj
t
t
J. K. and Č. Brukner, PRL 101, 090403 (2008)
The sufficient condition for macrorealism
The sufficient condition for macrorealism is
I.e. the statistical mixture has a classical time evolution, if no superpositions of
macroscopically distinct states are produced.
Given coarse-grained measurements, it depends on the Hamiltonian whether
macrorealism is satisfied.
“Classical” Hamiltonians
“Non-classical” Hamiltonians
eq. is fulfilled (e.g. rotation)
eq. not fulfilled (e.g. osc. Schrödinger cat, next slide)
J. K. and Č. Brukner, PRL 101, 090403 (2008)
Non-classical Hamiltonian
(no macrorealism despite of coarse-graining)
Hamiltonian:
Produces oscillating Schrödinger cat state:
Under fuzzy measurements it appears as a
statistical mixture at every instance of time:
But the time evolution of this mixture cannot
be understood classically:
time
Non-classical Hamiltonians are complex
Oscillating Schrödinger cat
Rotation in real space
“non-classical” rotation in Hilbert space
“classical”
Complexity is estimated by number of sequential local
operations and two-qubit manipulations
Simulate a small time interval t
O(N) sequential steps
1 single computation step
all N rotations can be done simultaneously
Relation quantum-classical
The origin of quantum randomness
Determinism (subjective randomness due to ignorance)?
Objective randomness (no causal reason)?
Mathematical undecidability
Axioms
Proposition:
true/false
if it can be proved/disproved from the axioms
“logically independent” or “mathematically undecidable”
if neither the proposition nor its negation leads to an inconsistency
(i) Euclid’s parallel postulate in neutral geometry
(ii) “axiom of choice” in Zermelo-Fraenkel set theory
intuitively: independent proposition contains new information
Information-theoretical formulation of undecidability (Chaitin 1982):
“If a theorem contains more information than a given set of axioms, then it is
impossible for the theorem to be derived from the axioms.”
Logical complementarity
Consider (Boolean) bit-to-bit function f(a) = b (with a = 0,1 and b = 0,1)
(A)
f(0) = 0
(B)
f(1) = 0
(C)
f(0) + f(1) = 0
logically complementary
Given any single 1-bit axiom, i.e. (A) or (B) or (C), the two other propositions are
undecidable.
Physical “black box” can encode
the Boolean function:
f(a) = 0
f(0) = 1
f(1) = 1
Example
qubit
a=1 a=0
f(1) = 1 f(0) = 0
Mathematical undecidability and quantum randomness
x
Preparation
Black box
Measurement
Information gain
f(0)
(A)
f(1)
(B)
f(0) + f(1)
(C)
z
x
y
However:
x
Random outcomes!
(B) is undecidable
within axiom (A)
Experimental test of mathematical undecidability:
1 qubit
“A qubit carries only one bit of information”
(Holevo 1973, Zeilinger 1999)
T. Paterek, R. Prevedel, J. K., P. Klimek, M. Aspelmeyer, A. Zeilinger, Č. Brukner, submitted (2008)
Generalization to many qubits
N qubits, N Boolean functions f1,…,fN
Black box:
New feature: Partial undecidability
T. Paterek, R. Prevedel, J. K., P. Klimek, M. Aspelmeyer, A. Zeilinger, Č. Brukner, submitted (2008)
Conclusions
Quantum-to-classical transition
under coarse-grained measurements
Quantum randomness: a manifestation
of mathematical undecidability
Thank you!
Appendix
Violation for arbitrary Hamiltonians
t
Initial state
t
t
State at later time t
t1 = 0
Measurement
t2
!
Survival probability
Leggett-Garg inequality
classical limit
Choose

can be violated for any E
t3
?
?
Continuous monitoring by an environment
Exponential decay of survival probability
- Leggett-Garg inequality is fulfilled (despite the non-classical Hamiltonian)
- However: Decoherence cannot account for a continuous spatiotemporal description
of the spin system in terms of classical laws of motion.
- Classical physics: differential equations for observable quantitites (real space)
- Quantum mechanics: differential equation for state vector (Hilbert space)
Experimental setups