Factorization and Entanglement in Quantum Systems

We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of this issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type (p,q) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space. The phenomenon of entanglement is of central importance in the interpretation of quantum mechanics. Historically, entanglement was the focus of the famous EinsteinPodolsky-Rosen (EPR) paper [1], which suggested that standard quantum mechanics is an incomplete theory of physical reality. The central argument of the EPR paper was that more information about incompatible variables such as momentum and position could in principle be deduced about an entangled two particle quantum state than quantum mechanics permits, effectively giving information about each particle separately, and therefore supporting a classical perspective. The resolution of this “paradox” is the observation that information extraction in quantum mechanics always comes at a cost: it is not possible to actually extract information about incompatible variables from a given state without destroying the state being looked at before the information extraction process is completed, and this invalidates the argument used by EPR [2]. An apparently unrelated issue is the following. Throughout the history of quantum mechanics, a constant topic of debate has been where the boundary between the classical and quantum worlds should be. We believe that there is now sufficient evidence to support the notion that there is no such boundary, and that the classical world view is no more than an emergent, i.e. effective, view of a universe which is entirely quantum mechanical in origin [3]. The evidence we cite is the near universal validity of the quantized-field approach to elementary particles, numerous experimentally observed violations of Bell inequalities and galactic lensing. In the latter process, we can imagine observing (say) one photon per day over several years to build up patterns analogous to the interference bands seen in double slit experiments, the difference being that the scale of the process is cosmological rather than local. In addition to these more exotic applications, the validity of quantum principles is supported by the overwhelming success of quantum mechanics in applied physics, biology and chemistry, on both terrestrial and astrophysical scales. With a recognition that the semi-classical observers of standard quantum mechanics should ideally be regarded as quantum systems themselves, it has become more fashionable to extend the quantum description to include them with the systems under observation. This can be done whilst maintaining a semi-classical perspective by writing a quantum state vector Ψ for an OS (combined observer plus system under observation) as a direct product Ψ ≡ θ⊗φ, where θ represents a state of the observer O and φ represents a state of the system S under observation. Such a state will be called separable. In general, we shall use the word separable when we talk about states constructed from direct products of vectors, and factorizable when we refer to Hilbert spaces constructed from direct (tensor) products of (factor) Hilbert spaces. When applied to Hilbert spaces, the term separable traditionally refers to the possibility of finding a countable basis for it, regardless of any issue of factorizability. When the dimensions of realistic Hilbert spaces which might model the universe are considered, then the a priori probability that a state chosen at random in such a space be separable is zero, as will be seen from our discussion of concurrency below. The observed separability of the universe into vast numbers of identifiably distinct subsystems is on the face of it surprising from this point of view. However, this does not take into account the crucial role of dynamics, which imposes very specific constraints on which states are physically accessible in the course of time. For example, suppose all the possible outcomes of some quantum process are separable states. Then there will be zero probability of getting an entangled state outcome in that process. In this article, classicity (or classicality) is regarded as synonymous with the possibility of making distinctions between different objects, such as different spatial positions, or physical subsystems. In quantum mechanics, entanglement may be regarded as a breakdown of such a possibility. When physicists discuss isolated systems within a wider universe, they invariably model the totality by separable states, with some of the factors representing states of the isolated systems and other factors representing the rest of the universe. The conventional procedure is then to ignore these other factors (the environment), and discuss only those factors representing the isolated systems. Certainly, it seems impossible to discuss experiments in physics without assuming that the states of interest are factored out from the rest of the environment. The development of decoherence theory has not altered this in the least. Separability is therefore as fundamental to quantum physics as entanglement. This leads to the following question: given a finite dimensional Hilbert space H of dimension d ≡ dimH, when is it possible to think of a state Ψ in H as a separable state? By this we mean we would like to know the circumstances which guarantee that Ψ is a tensor product of the form Ψ = ψ ⊗ φ, where ψ is some vector in some factor space H1 of H and φ is another vector in another factor H2 of H. Let H be a finite dimensional Hilbert space of dimension dimH. If H can be expressed in the bi-partite form H = H (d1) 1 ⊗H (d2) 2 , (1) where H (di) i , i = 1, 2 is a Hilbert space with dimension di, then we shall say that H is factorizable. Clearly dimH must itself be factorizable and given by the rule dimH = d1d2. This elementary result may have important cosmological implications. According to a number of authors [3-6] the universe is described by a time dependent pure quantum state Ψ, an element in a Hilbert space HU of enormous but finite dimension. We note that the notion that the universe is a quantum system has been criticised principally on the grounds that there is no evidence that all physical systems must possess quantum states [7], and also because it appears inconsistent to discuss probabilities when there is only one universe. These arguments can be met with three counter arguments: first, the absence of any boundary between the quantum and classical worlds and the empirical validity of quantum mechanics actually strongly supports the notion that all systems must run on quantum principles, and so by extension does the universe; second, a pure state formalism eliminates the need for a density matrix approach to quantum cosmology; third, quantum probabilities make sense if they are interpreted correctly in terms of predictions about the possible future state of the universe made by physicists who are themselves part of the quantum universe. This is not inconsistent with the notion that the universe is in a definite state in the present. Given this quantum perspective about the universe, the apparently overwhelmingly classical appearance of the universe, with a classical looking spatial structure which permits the separation of vast numbers of subsystems of the universe spatially, is interpreted by us as evidence that the current state of the universe Ψ has separated into a vast number of factors. If this is true then HU must be factorizable into a vast number of factor spaces, and therefore dimHU itself must be highly factorizable. In particular, the Hilbert space of the universe cannot have prime dimension according to this scenario. Hilbert spaces with a high degree of factorizability are readily constructed. Recent approaches to fundamental physics inspired by spin networks and quantum computation [5,8] considers HU to be the direct product of a (usually vast) number N of qubit Hilbert spaces, viz HU = H (2) 1 ⊗H (2) 2 ⊗ . . .⊗H (2) N , (2) and then dimHU = 2 N . None of the individual qubit factor spaces H (2) i are factorizable, so that (2) represents a complete, or maximal, factorization of HU . We shall call such a qubit factorization a primordial factorization. In the most general case, a primordial factorization will be of the form H = H (p1) 1 ⊗H (p2) 2 ⊗ . . .⊗H (pN ) N , (3) where the pi are prime numbers and dimH = p1p2 . . . pN . If the factorizability ζ of H is defined as the ratio N/ dimH then qubits provide the maximum factorizability for a given N , i.e., ζ = N/2 . Qubits are favoured by various authors because they represent the most elementary attributes of logic, that is, “yes” and “no” (or equivalently, “true” and “false”) can be identified with the two elements of a qubit “spin-up”, “spin-down” basis. Given an N− qubit system with N > 2 then it is possible to consider partial factorizations (or splits) of H of the form H = H ) ⊗H N−n), (4)