Download 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, classical &
coarse-grained measurements
Johannes Kofler and Časlav Brukner
Young Researchers Conference
Perimeter Institute for Theoretical Physics
Waterloo, Canada, Dec. 3–7, 2007
Classical versus Quantum
Phase space
Hilbert space
Continuity
Quantization, “Clicks”
Newton’s laws
Schrödinger + Projection
Local Realism
Violation of Local Realism
Macrorealism
Violation of Macrorealism
Determinism
Randomness
- Does this mean that the classical world is substantially different from the
quantum world?
- When and how do physical systems stop to behave quantumly and begin to
behave classically?
- Quantum-to-classical transition without environment (i.e. no decoherence)
and within quantum physics (i.e. no collapse models)
A. Peres, Quantum Theory: Concepts and Methods (Kluwer 1995)
What are the key ingredients for a
non-classical time evolution?
The candidates:
The initial state of the system
The Hamiltonian
The measurement observables
Answer:
At the end of the talk
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
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
The Leggett-Garg inequality
Dichotomic quantity: Q(t)
t
t=0
t
Temporal correlations
t1
t2
t3
t4
All macrorealistic theories fulfill the
Leggett–Garg inequality
Violation  at least one of the two postulates fails
(macrorealism per se or/and non-invasive measurability).
Tool for showing quantumness in the macroscopic domain.
When is the Leggett-Garg inequality violated?
Rotating spin-1/2
Evolution
Observable
1/2
for
Violation of the Leggett-Garg inequality
Rotating classical spin
precession around x
+1
Classical evolution
–1
classical
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
?
?
Why don’t we see violations in everyday life?
- (Pre-measurement) Decoherence
- Coarse-grained measurements
Model system: Spin j, i.e. a qu(2j+1)it
Arbitrary state:
Assume measurement resolution is much weaker than the intrinsic uncertainty such
that
neighbouring outcomes in a Jz measurement are bunched together
into “slots” m.
–j
+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:
Fuzzy measurements: any
quantum state allows a classical
description (i.e. hidden variable
model).
This is macrorealism per se.
J. Kofler and Č. Brukner, PRL 99, 180403 (2007)
Example: Rotation of spin j
sharp parity measurement
classical limit
fuzzy measurement
Classical physics of a rotated
classical spin vector
Violation of Leggett-Garg inequality
for arbitrarily large spins j
J. Kofler 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:
But for macrorealism we need more than that:
Non-invasive measurability
t=0
ti
tj
t
t
J. Kofler and Č. Brukner, quant-ph/0706.0668
The sufficient condition for macrorealism
The sufficient condition for macrorealism is
I.e. the statistical mixture has a classical time evolution, if measurement and time
evolution commute “on the coarse-grained level”.
Given fuzzy measurements (or pre-measurement decoherence), 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)
J. Kofler and Č. Brukner, quant-ph/0706.0668
Non-classical Hamiltonians
(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
- „Cosine-law“ between macroscopically distinct states
- Coarse-graining (even to northern and southern hemisphere) does not “help” as j and –j are well separated
is not fulfilled
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
What are the key ingredients for a
non-classical time evolution?
The candidates:
The initial state of the system
The Hamiltonian
The measurement observables
Answer:
Sharp measurements
Any (non-trivial)
Hamiltonian produces
a non-classical time
evolution
Coarse-grained measurements
(or decoherence)
“Classical” Hamiltonians:
classical time evolution
“Non-classical” Hamiltonians:
violation of macrorealism
Relation Quantum-Classical
fuzzy measurements
Quantum Physics
macroscopic objects &
classical Hamiltonians
macroscopic objects & nonclassical Hamiltonians or
sharp measurements
Macro Quantum Physics
(no macrorealism)
Discrete Classical Physics
(macrorealism)
limit of infinite
dimensionality
Classical Physics
(macrorealism)
Conclusions and Outlook
1.
Under sharp measurements every Hamiltonian leads to a non-classical
time evolution.
2.
Under coarse-grained measurements macroscopic realism (classical
physics) emerges from quantum laws under classical Hamiltonians.
3.
Under non-classical Hamiltonians and fuzzy measurements a quantum
state can be described by a classical mixture at any instant of time but
the time evolution of this mixture cannot be understood classically.
4.
Non-classical Hamiltonians seem to be computationally complex.
5.
Different coarse-grainings imply different macro-physics.
6.
As resources are fundamentally limited in the universe and practically
limited in any laboratory, does this imply a fundamental limit for
observing quantum phenomena?