Necessary and sufficient conditions for macroscopic realism from quantum mechanics
Johannes Kofler
Max Planck Institute of Quantum Optics (MPQ) Garching/Munich, Germany
Quantum Theory: from foundations to technologies – QTFT Linnaeus University, Växjö, Sweden
9 June 2015
Introduction
• How does our macroscopic & classical world arise out of quantum mechanics? − Decoherence (within quantum mechanics) − Spontaneous collapse models (altering quantum mechanics) − Complementary to decoherence: coarse-grained measurements
• Macroscopic realism: “every object is in a definite macrostate at all times and can be measured non-invasively” → Leggett-Garg inequality (LGI)
• Quantum mechanics: superpositions of macroscopically distinct states (“Schrödinger cats”) → violation of LGI
• Alternative to the Leggett-Garg inequalities: “no-signaling in time” (NSIT)
? ?
With photons, electrons, neutrons, molecules etc.
With cats?
|cat left⟩ + |cat right⟩ ?
Measurement problem Quantum-to-classical transition
Macroscopic superpositions
Candidates
Heavy molecules1
(position)
Nanomechanics4
(position, momentum)
Superconducting devices2
(current)
Atomic gases3
(spin)
1 S. Gerlich et al., Nature Comm. 2, 263 (2011) 3 B. Julsgaard et al., Nature 413, 400 (2001) 2 M. W. Johnson et al., Nature 473, 194 (2011) 4 G. Cole et al., Nature Comm. 2, 231 (2011)
Macrorealism
• Macrorealism per se:1 given a set of macroscopically distinct states, a macroscopic object is at any given time in a definite one of these states
• Non-invasive measurability:1 measurements reveal the state without any effect on the state itself or on the subsequent dynamics
• Leggett-Garg inequality (LGI):
1 A. J. Leggett and A. Garg, PRL 54, 857 (1985)
• Quantum mechanics:
t1 t2 t3 t4 t0
Q Q Q Q ±1
S := ⟨A1B1⟩ + ⟨A2B1⟩ + ⟨A1B2⟩ – ⟨A2B2⟩ ≤ 2
K := ⟨Q1Q2⟩ + ⟨Q2Q3⟩ + ⟨Q3Q4⟩ – ⟨Q1Q4⟩ ≤ 2
Bell:
KQM = 2√2 ≈ 2.83
locality
non-invasiveness =
=
time
J. Dressel et al., PRL 106, 040402 (2011) M. E. Goggin et al., PNAS 108, 1256 (2011) A. Fedrizzi et al., PRL 106, 200402 (2011)
Exp. LGI violations for microscopic systems: G. Waldherr et al., PRL 107, 090401 (2011) G. C. Knee et al., Nature Comm. 3, 606 (2012) A. Asadian et al., PRL 112, 190402 (2013)
a
B = ±1 A = ±1
b
Quantum mechanics says “yes” (if you manage to defy decoherence)
Are macroscopic superpositions possible?
Local realism vs. macrorealism
Quantum mechanics says “yes” (use entanglement)
Are non-classical correlations possible?
Local realism (e.g. classical physics) says “no” (only classical correlations)
Bell test has given experimental answer in favor of quantum mechanics
Macrorealism (e.g. classical physics, objective collapse models) says “no” (only classical temporal correlations)
Leggett-Garg test can/will give experimental answer, community still split
Practical relevance qu. computation, qu. cryptography
Practical relevance witnessing temporal qu. coherence
Ideal negative measurements Taking only those results where no interaction with the object took place
How to enforce non-invasiveness?
Locality vs. non-invasiveness
Space-like separation Special relativity guarantees impossibility of physical influence
How to enforce locality?
Bohmian mechanics Space-like separation is of no help: non-local influence on hidden variable level Realistic, non-local
Bohmian mechanics Ideal negative measurements are of no help: wavefunction “collapse” changes subsequent evolution Macrorealistic per se, invasive
? ?
–1 +1
–1 +1
Sharp measurements (resolution of individual quantum levels)
Decoherence or coarse-grained meas. (smeared phase space observables)
Sharp vs. coarse-grained measurements
To see quantumness: need to resolve j1/2 levels & protect system from environment
J.K. and Č. Brukner, PRL 99, 180403 (2007)
Macroscopic spin j
Oscillating Schrödinger cat “non-classical” rotation in Hilbert space
Rotation in real space “classical”
N sequential steps per ∆t 1 single computation step per ∆t all N rotations can be done simultaneously
Non-classical evolutions are complex
J.K. and Č. Brukner, PRL 101, 090403 (2008)
N elemen- tary spins ½
time time
“+” “+”
∆t ∆t ∆t ∆t
The quantum-to-classical transition
decoherence or
J.K. and Č. Brukner, PRL 101, 090403 (2008) J.K. and Č. Brukner, PRL 99, 180403 (2007)
Alternative to LGI
No-signaling (NS): “A measurement on one side does not change the outcome statistics on the other side.”
tA tB t0
A B
1 J.K. and Č. Brukner, PRA 87, 052115 (2013)
BI NS
LR
QM
LGI NSIT
MR
QM
BI necessary for LR tests NS “useless”
LGI not essential for MR tests alternative: NSIT (interference) more physical, simpler, stronger, more robust to noise
a
B A
b
No-signaling in time (NSIT): “A measurement does not change the outcome statistics of a later measurement.”1
Double slit experiment
t1
Picture: N. Bohr, in Quantum Theory and Measurement, eds. J. A. Wheeler and W. H. Zurek, Princeton University Press (1983)
t2
II Block lower slit at x = –d/2:
III Block upper slit at x = +d/2:
t0
x = ±d/2 x
fringes
no fringes
II,III: ideal negative measurements
≠
NSIT is violated due to interference terms
LGI impossible to construct
I Both slits open:
t
x
Mach-Zehnder interferometer
LGI NSIT
violated in specific parameter regimes violated up to measure 0
J. K. and Č. Brukner, PRA 87, 052115 (2013)
Necessary conditions for MR
1 L. Clemente and J.K., PRA 91, 062103 (2015)
Variety of necessary conditions for macrorealism1
Necessary and sufficient for MR
2 L. Clemente and J.K., PRA 91, 062103 (2015)
Necessary and sufficient2 for MR012 Sufficient1 for LGI012
1 O. J. E. Maroney and C. G Timpson, arXiv:1412.6139
NSIT for quantum measurements
L. Clemente and J.K., PRA 91, 062103 (2015)
Violation requires non-classical resource1,2,3,4
non-classical states / sharp measurements or non-classical Hamiltonians
1 J.K. and Č. Brukner, PRL 101, 090403 (2008) 2 T. Wang, R. Ghobadi, S. Raeisi, C. Simon, PRA 88, 062114 (2013) 3 H. Jeong, Y. Lim, M. S. Kim, PRL 112, 010402 (2014). 4 P. Sekatski, N. Gisin, N. Sangouard, PRL 113, 090403 (2014)
coarse-grained observables
“classical” Hamiltonians
proj.
proj.
proj.
Conclusion & Outlook
• Quantum-to-classical transition: − classical Hamiltonians & coarse-grained measurements
• No-signaling in time (NSIT): − alternative to the Leggett-Garg inequalities (simpler, stronger) − combination of NSIT clauses: necessary and sufficient for macrorealism
• Open problems: − general trade-off “measurement precision vs. Hamiltonian complexity” − coarse-graining requires notion of “neighborhood” of states
Acknowledgments
Časlav Brukner Lucas Clemente J.K. and Č. Brukner, PRA 87, 052115 (2013) L. Clemente and J.K., PRA 91, 062103 (2015)
