1. Classical World because of Quantum Physics
Institute for Experimental Physics
University of Vienna
Institute for Quantum Optics and Quantum Information
Austrian Academy of Sciences
Classical World because of Quantum Physics
Johannes Kofler and Časlav Brukner
University of Leeds, United Kingdom
September 2006

2. Classical versus Quantum
Phase space
Continuity
Newton’s laws
Local Realism
Macrorealism
Determinism
Hilbert space
Events, ”Clicks”
Schrödinger + Projection
Violation of Local Realism
Violation of Macrorealism
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?

3. Macrorealism [Leggett–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

4. Leggett–Garg inequality
Quantity: Q
Temporal correlations
All macrorealistic theories fulfill the
Leggett–Garg inequality
t = 0 t t1 t2 t3 t4
Violation  no objective properties prior to and independent of measurements

5. When is Macrorealism violated?
Spin-1/2
Evolution Observable
1/2
for
Violation of macrorealism
Classical Spin
precession around x +1
classical
Macrorealism –1

6. Violation of Macrorealism for macroscopically large spins?
Spin-j precession in magnetic field
(totally mixed state!) j
Parity of eigenvalue m of Jz measurement
classical limit
Violation of macrorealism for arbitrarily large spins j
Shown for local realism [Mermin, Peres]

7. The quantum-to-classical transition
Coherent spin state (t = 0):
exact measurement
fuzzy measurement
fuzzy measurement & limit of large spins
This is (continuous and non-invasive) classical physics of a rotated classical spin vector!

8. Transition to Classicality: General state
Quantum
Classical
General density matrix:
Probability to detect in a slot: 
f can be negative!
Probability for result m:
h is non-negative!
Hamilton operator:
Hamilton function:
Classical limit:
Ensemble of classical spins with probability distribution h

9. Relation Quantum-Classical Discrete Classical Physics
inaccurate measurements
Discrete Classical Physics
(macrorealism)
Quantum Physics
macroscopic objects
macroscopic objects
limit of large spins
limit of large spins
Macro Quantum Physics
(no macrorealism)
Classical Physics
(macrorealism) ?

10. Is there a fundamental limit for observability of quantum phenomena?
Algorithmic Complexity of Quantum States [Mora and Briegel]:
The length of the shortest string that encodes the preparation procedure for a given state with a given precision ε.
Most of the states of N qubits are of complexity
2N random amplitudes
Number of bits needed to realize the state:
Number of bits in the Universe [Lloyd]:
Limit [Davis]: 
Every state is effectively a mixture over all pure states that are not distinguishable within the precision ε

11. Conclusions
Classical physics emerges from quantum laws under the restriction of coarse-grained measurements, not alone through the limit of large quantum numbers.
Conceptually different from decoherence. Not dynamical, puts the stress on observability and works also for fully isolated systems.
As the resources in the world are limited, there is a fundamental limit for observability of quantum phenomena (even if there is no such limit for the validity of quantum theory itself).
quant-ph/