Entanglement entropy and the simulation of Quantum Mechanics

The relation between entanglement entropy and the computational difficulty of classically simulating Quantum Mechanics is briefly reviewed. Matrix product states are proven to provide an efficient representation of one-dimensional quantum systems. Further applications of the techniques based on matrix product states, some of their spin-off and their recent generalizations to scale invariant theories and higher dimensions systems are also discussed. ‡ 1. Entanglement entropy as a measure of quantum correlations A common misconception states that, in general, large quantum mechanical system can not be efficiently described by classical means. This prejudice can be illustrated with the simple example of a system composed of n two-level systems or qubits. The Hilbert space of this system corresponds to the direct product C2⊗n and an arbitrary state can be expressed in the natural (also called computational) basis |ψ〉 = ∑ i1,i2,...,in=0,1 c12n |i1, i2, . . . , in〉. (1) In order to fully specify an arbitrary state, it seems necessary to provide all the c1n coefficients, that is, 2 complex numbers (minus a global phase and a normalization constraint that we can ignore for the counting of the scaling of needed resources). As n grows, the classical representation of a quantum state requires exponential resources. Furthermore, the processing of the state, e.g. the computation of its time evolution, and the computation of observables also requires exponentially many operations. The exponential effort needed to deal with Quantum Mechanics can also be advocated using an argument based on entropy. The precise statement says that an average random state in the Hilbert space is known to carry maximal von Neumann entropy. Let us describe in more detail this point. Consider a partition of the original state into two parties, A and B. If party A ignores party B, the description of its subsystem is based on the reduced density matrix ρA = trB|ψ〉〈ψ|. (2) ‡ Contribution to the Proceedings of the IRGAC conference held at Barcelona, July 2006. Entanglement entropy and the simulation of Quantum Mechanics 2 The description that party A is making of the system ignores quantum correlations between A and B. If A would suddenly discover that it was correlated to B a surprise would take place. The amount of that surprise is quantified by the von Neumann entropy S(ρA) = −tr (ρA log ρA) . (3) It is well-known that the entropy attached to party A ignoring party B equals the reciprocal one, that is, the entropy attached to party B when ignoring party A. This is a consequence of the Schmidt decomposition |ψ〉 = χ=min(dimHA,dimHB)