Transcript Exponential complexity and ontological theories of quantum

Exponential complexity and ontological
theories of quantum mechanics
Alberto Montina
Dip. Fis., Univ. di Firenze
[email protected]
Sestri Levante 4-6 Giugno 2008
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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
Exponential complexity of quantum mechanics
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.
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)
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
space
Quantum Monte Carlo methods and sign problem
W1
W2
W3
x(t1)
x(t0)
W4
time
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.
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)
Example: QMC method with BCS states
  exp( ijbˆibˆ j ) 0
overcomplete base
Quartic Hamiltonian → Stochastic equation for 
   t F 
  
ij
__________
ij
 (t )
  (t )
ij
A. Montina and Y. Castin,
Phys. Rev. A 73, 013618 (2006)
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 the classical
rules of the probability theory. Examples: Bohm mechanics and
Nelson stochastic mechanics.
General framework:
   ( X | , )
 (t )  (1  itHˆ )  (t )

 ( X , t  t )   dX P( X | X , t )  ( X , t )
 Pˆ    dX PM ( X )  ( X | , )
P, PM and  are POSITIVE distributions!
Is the evaluation of the classical trajectory in the
ontological space a polynomially complex problem?
Ontological space
The system is in the ontic state X0 at time t. At time t+t is in X1 with
probability P(X1|X0,t).
ˆ
X(t1)→Generate event P
with probability PM[X(t1)]
X(t0)
time
A. Montina, Phys. Rev. Lett. 97, 180401 (2006)
Theorem on the ontological space dimension
A. Montina, Phys. Rev. A 77, 022104 (2008)
Given an ontological Markovian theory of a Ndimensional 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
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.
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
Conclusion and perspectives
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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].
Theorem
I[]→ Union of the supports of P(X| ,) for any .
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.