Advanced ERC Grant: QUAGATUA AvH Senior Research Grant + Feodor Lynen Hamburg Theory Prize Chist-Era DIQIP Maciej Lewenstein Detecting Non-Locality in Many Body Syst ems Enrico Fermi School Course 191 EU IP SIQS EU STREP EQuaM Advanced ERC Grant: OSYRIS John Templeton Foundation ICFO-Cellex- Severo Ochoa Polish Science Foundatio
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Advanced ERC Grant: QUAGATUA AvH Senior Research Grant + Feodor Lynen Hamburg Theory Prize Chist-Era DIQIP Maciej Lewenstein Detecting Non-Locality in.
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Detecting non-locality in many body systems - Outline
2. Non-locality in many body systems•2.1 Correlations – DIQIP approach•2.2 Non-locality in many body systems•2.3 Physical realizations with ultracold ions
1. Entanglement in many body systems•1.1 Computational complexity•1.2 Entanglement of pure states (generic, and not…)•1.3 Area laws•1.4 Tensor network states
Many body physics from a quantum information perspective R. Augusiak, F. M. Cucchietti, M. Lewenstein Lect. Notes Phys. 843, 245-294 (2010).
Ultracold atoms in optical lattices: Simulating quantum many-body systems M. Lewenstein, A. Sanpera, V. Ahufinger Oxford University Press (2012)
What can be simulated classically? What is computationally hard (examples)?
Ultracold atoms in optical lattices: Simulating quantum many-body physics, M. Lewenstein, A. Sanpera, V. Ahufinger, in print Oxford University Press (2012)
“Good” entanglement measure for pure states Take reduced density matrix: ρA = TrB(ρAB) = TrB(|ΨAB›‹ΨAB|), and then take von Neumann entropy E(|ΨAB›‹ΨAB|) = S(ρA) = S(ρB), where S(ρ) = -Tr(ρ log ρ). Note that maximally entangled states have E(|ΨAB›‹ΨAB|) = log dANote: For mixed states a super hard
problem…
1.2 Why computations may be hard? Entanglement of a generic state
1.2 Why computations may be hard? Entanglement of a generic state
1.3 Why there are some hopes? - Area laws
Classical area laws
Thermal area laws
Quantum area laws in 2D?
Quantum area laws in 1D
1.3 Area laws
Area law: Averaged values of correlations, between the regions A and B, scale as the size of the boundary of A. For instance for quantum pure (ground states): S(ρA) ~ ∂A (Jacob Beckenstein, Mark Srednicki…)
A
B
1.3 Area laws for thermal states
AB
1.3 Quantum area laws in 1D
1.3 Quantum area laws in 1D
1.3 Quantum area laws in 2D, 3D …
?
One can prove generally S(ρA) ≤ |∂A| log(|∂A|)
1.4 TNS and quantum many-body systems
1... 1| | ,...,Ni i Nc i i
We need coefficients to represent a state.2N
To determine physical quantitites (expectation values) an exponential number of computations is required.
Many-body quantum systems are difficult to describe.
J. Tura, R. Augusiak, A.B. Sainz, T. Vértesi, M. Lewenstein, and A. Acín, Detecting the non-locality of quantum many body states, arXiv:1306.6860, Science 344, 1256 (2014).
J. Tura, A.B. Sainz, T. Vértesi, A. Acín, M. Lewenstein, R. Augusiak, Translationally invariant Bell inequalities with two-body correlators, arXiv:1312.0265, in print to special issue of J. Phys. A on “50 years of Bell’s Theorem”.
•2.1 Correlations – DIQIP approach•2.2 Non-locality in many body systems
Analytic example: Family of many body Bell inequalities
2.3 Physical realizations with ultracold
ions
2.3 Realizations with ultracold ions/atoms
2.3 Realizations with ultracold ions
2.3 Realizations with ultracold ions
2.3 Realizations with ultracold ions
2.3 Realizations with ultracold ions
Detecting non-locality in many body systems - Conclusions
2. Non-locality in many body systems
•“Weak” entanglement ≈ Locality with respect to “simple” Bell inequalities.•“Strong” non-locality and symmetry ≈ Classical computability?
1. Entanglement in many body systems
• “Weak” entanglement ≈ Area laws ≈ Classical computability!
Many body physics from a quantum information perspective R. Augusiak, F. M. Cucchietti, M. Lewenstein Lect. Notes Phys. 843, 245-294 (2010).
Ultracold atoms in optical lattices: Simulating quantum many-body systems M. Lewenstein, A. Sanpera, V. Ahufinger Oxford University Press (2012)
Quantum Optics Theory ICFO
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Ultracold atoms in optical lattices: Simulating quantum many-body physicsM. Lewenstein, A. Sanpera, and V. Ahufinger, Oxford University Press (2012)Atomic Physics: Precise measurements & ultracold matterM. Inguscio and L. Fallani, Oxford University Press (2013)
Quantum simulators, precise measurements and ultracold matter