Origin of the exponential complexity: ensemble description.

1. Exponential complexity and ontological theories of quantum mechanicsAlberto MontinaDip. Fis., Univ. di Firenzealberto.montina@inoa.itSestri Levante 4-6 Giugno 2008

2. Origin of the exponential complexity: ensemble description. • Classical analogy: ensemble vs single system description -> Monte Carlo methods with a finite number of realizations • Quantum Monte Carlo methods: sign problem for Fermions and real time dynamics-> exponential growth of the statistical errors • Are the ontological theories of quantum mechanics a possible solution of the sign problem? • Exponential growth of the ontological space dimension with the physical size [A. Montina, PRL (2006); Phys. Rev. A 77, 022104 (2008)]. • Future expectations for foundational quantum physics and quantum Monte Carlo methods

3. single particle: (x) N particles: (x1, x2,…,xN) lattice-> (i1, i2,…,iN) i1, i2,…,iN=1..M Number of lattice points=MN Exponential increase of the Hilbert space dimension-> Born's argument against the realistic Schroedinger's interpretation of the wave-function. In general, the state space of two systems is the tensorial product of the Hilbert spaces of each single system. Exponential complexity of quantum mechanics

4. Born’s statistical interpretation and classical analogue of the exponential complexity The wave function is not a real field, but a mathematical object which enables one to evaluate the probabilities of events. |(x1, x2,…,xN)|2=(x1, x2,…,xN)  probability distribution The wave function contains the complete statistical information of an ensemble of infinite number of systems. Classical analogy: BROWNIAN PARTICLES description of a single realization Ensemble description: Exponential complexity problem (x1, x2,…,xN;p1, p2,…,pN)

5. Classical and Quantum Monte Carlo methods. CLASSICAL SYSTEMS: In a Monte Carlo method (MC), one does not evaluate the evolution of the multi-particle probability distribution, but the averages over a finite number of realizations. The evaluation of the stochastic trajectories is a polynomially complex problem. QUANTUM SYSTEM: Quantum MC: one does not evaluate the evolution of the multi-particle wave-function, but the averages over a finite number of realizations in a suitable “small” sampling space. Necessary condition for a good QMC method: the dimensionality of the sampling space must grow polynomially with the physical size, i.e. with the number of particles.

6. W2 W3 W1 space x(t1) x(t0) W4 time Quantum Monte Carlo methods and sign problem The sampling space is the configuration space. Its dimension grows as the number of particles! Wi trajectory weights. In classical mechanics they are POSITIVE probabilities. Feynman path integral: the weights Wi are not positive real numbers destructive interference among different paths. One needs to consider a very large number of realizations. Bad QMC method.

7. Alternative sampling spaces Feynman path integral in the configuration space  Alternative sampling space: Coherent states |=Exp(-||2+  â†) |0 Fock states Phi(x1) Phi(x2)…Phi(xN) I. Carusotto, Y. Castin J. Dalibard PRA (2001). BCS states A. Montina, Y. Castin, PRA (2006). With a suitable choice of the sampling space, the sign problem can be mitigated. However all the known quantum Monte Carlo methods for real time dynamics are affected by an exponential growth of the statistical errorsA~exp( N t)

8. Example: QMC method with BCS states overcomplete base Quartic Hamiltonian → Stochastic equation for  A. Montina and Y. Castin, Phys. Rev. A 73, 013618 (2006)

9. Can an ontological theory be as a solution of the sign problem? An ontological theory provides a description of single quantum system by means of well-defined variables and using theclassical rules of the probability theory. Examples: Bohm mechanics and Nelson stochastic mechanics. General framework: P, PM and  are POSITIVE distributions!

10. Is the evaluation of the classical trajectory in the ontological space a polynomially complex problem? The system is in the ontic state X0 at time t. At time t+t is in X1 with probability P(X1|X0,t). X(t1)→Generate event with probability PM[X(t1)] Ontological space X(t0) time A. Montina, Phys. Rev. Lett. 97, 180401 (2006)

11. Theorem on the ontological space dimension A. Montina, Phys. Rev. A 77, 022104 (2008) Given an ontological Markovian theory of a N-dimensional quantum system, the corresponding ontological space dimension can not be smaller than 2N-2. Consequence: the ontic space dimension grows exponential with the physical size. The prize paid for the solution of the sign problem is the exponential growth of the sampling space→ Bad consequences for a quantum Monte Carlo application

12. Foundational consequences Born’s argument against Schroedinger’s realistic interpretation of the wave-function applies to ANY Markovian realistic theory of quantum mechanics. The hypotheses and their consequences can be exchanged→ Constructive result: Hypotheses: quantum mechanics is reducible to a realistic theory in a “polynomially growing” ontic space. Consequence: the ontological theory is not Markovian, i.e., the dynamics is a LONG MEMORY PROCESS or, more drastically, the theory is not CAUSAL.

13. Bell theorem and non-causality Bell theorem→ any ontological theory equivalent to quantum mechanics MUST be non-local + Lorentz invariance of the OT ↓ non-causality of the ontological theory

14. Conclusion and perspectives • Solution of the sign problem with OT→exponential growth of the sampling space dimensionality. • Generalization to a larger class of Monte Carlo methods (e.g. not normalized distributions) • Foundational perspectives: searching for a non-Markovian (long memory or non-causal) ontological theory of quantum phenomena with polynomially growing sampling space. [L. Hardy J.Phys. A40, 3081 (2007)→Quantum Gravity theory with non-causal structure].

15. Theorem I[]→ Union of the supports of P(X| ,) for any .

16. Definition of S(X): S(X) contains any vector |> whose I() contains X. -There exists a X such as S(X) is not null. -S(X) does not contain every vector of the Hilbert space.