Probability and the Classical/Quantum Divide

This paper considers the problem of distinguishing between classical and quantum domains in macroscopic phenomena using tests based on probability and it presents a condition on the ratios of the outcomes being the same (Ps) to being different (Pn). Given three events, Ps/Pn for the classical case, where there are no 3-way coincidences, is one-half whereas for the quantum state it is one-third. For non-maximally entangled objects ) 11 00 ( 1 1 2 + + = r r AA φ we find that so long as r < 5.83, we can separate them from classical objects using a probability test. For maximally entangled particles (r = 1), we propose that the value of 5/12 be used for Ps/Pn to separate classical and quantum states when no other information is available and measurements are noisy. INTRODUCTION That a signal be considered quantum is generally determined by characteristics such as superposition, entanglement, and complementarity. Information associated with either classical or quantum variables is measured in relation to the experimental arrangement and it is a function of the probabilities of the outcomes associated with the experiment. When calculating information, one should first determine whether the variable being measured is classical or quantum but this may be difficult if the measurement constraints preclude determination of, say, entanglement. If the variable is quantum, one must determine if it is pure or mixed and this would require prior testing and assumption of stationarity of the process with respect to time. One must also define the framework within which the question of information is being asked since, under certain conditions, an unknown pure state – associated with zero von Neumann entropy -can convey information [1]. There are many physical processes where the nature of the variables is well established both by theory and experiment. But what about new situations, outside of physics in macroscopic systems, where there isn’t consensus that the processes have a quantum mechanical basis? Quantum biology is one such area [2]-[5]. Reasons have also been advanced for considering quantum models of the brain [6]-[14] and if such functioning is true there ought to be evidence in favor of coincidences across space and time [15]. In many situations, such as determination of entanglement, the differentiation between classical and quantum effects is estimated by checking if the Bell inequality is violated [16]-[18]. In his original paper [16], Bell showed that under conditions of independence classical random variables A, B, C will satisfy | ) , ( B A Psame − ) , ( C A Psame | ≤ ) , ( C B Psame + 1 (1) 1 Department of Computer Science, Oklahoma State University, Stillwater, OK 74074. Probability and Classical/Quantum Divide 2 where ) , ( Y X Psame is the probability that the pair of random variables X, Y have some identical property. The Bell inequality is a consistency constraint on functions of two random variables and it can be described in many other ways (and we will use a slightly different form in our discussion). Bell showed that similar measurement of entangled quantum variables can lead to a violation of the inequality and, therefore, such violation can serve as a divide between classical and quantum variables. But Bell inequality is also violated in many classical situations where long-range correlations persist [19]-[22]. Statistics may also be used to distinguish between classical and quantum systems [23]. But this applicability will be limited to situations where the quantum process is in a state of thermal equilibrium. When a closed quantum system with a large number of eigenstates is opened to the environment, many of these states couple to the environmental states and undergo decoherence. This creates environment-induced superselection that leads to the survival of certain states [24] that remain correlated with the universe, obeying classical statistics, even though they are quantum mechanical. The classical/quantum divide is a central notion of the Copenhagen Interpretation (CI) of quantum mechanics [25],[26]. The Many Worlds Interpretation (MWI) takes the wavefunction to be the primary reality and assigns a wavefunction to the universe itself. In MWI there is no collapse of the wavefunction and the observation is a consequence of decoherence brought about by the environment. If CI is an inside-out view of the universe where the reality is constructed out of the perceptions of the experimenter, MWI is an outside-in view in which the mathematical function of the universe is the primary reality [27]. In our view CI is better than MWI in addressing the question of free-will which it does through the quantum Zeno effect if consciousness is seen to have a universal basis [28]. In application of quantum theory to beliefs (which sidesteps the question whether the brain must be described by a quantum model), questions A, B, A|B, and B|A are presented in sequence to each member of a group of people and the responses averaged [29],[30]. If it is shown that P(A|B) P(B) ≠ P(B|A) P(A), that would be evidence in support of a non-classical basis to such probability. In entangled states such as 11 3 . 0 10 1 . 0 01 3 . 0 00 9 . 0 − − − = φ , event order leads to different results for P(01) = 0.09 and P(10) = 0.01. Such an entangled state may be created by passing the state 00 through some appropriate unitary operator although it is not clear if it is easy to implement arbitrary operators [31]. Six reasons have been advanced for a quantum approach to mental events [29]: mental states are indefinite, judgments create rather than merely record, they disturb each other, and they do not always obey classical logic, they do not obey the principle of unicity, and cognitive phenomena may not be decomposable. To set up a proper quantum analogy for a questioninganswering system, we must assume that there is an ideal state function, in a suitable Hilbert space, that has been created by society and individuals are filters who change their state with a certain probability to that corresponding to the filter orientation. We take another look at what is possible to infer from experimental observations related to the nature of the system or the signals. Given pairwise data for variables, Boole showed what constraints had to be satisfied for the pairwise probabilities of a set of three evens to be consistent [32]. In this sense, Boole’s work prefigures Bell’s inequalities [16] although the focus of Bell was to find a setting that would clearly point to the differences between classical and quantum situations. We use the Bell-type inequality [33] on the probability of same outcomes for pairwise consideration of three pairs of bases. Probability and Classical/Quantum Divide 3 An entangled quantum state such as ) 11 00 ( 2 1 + = φ has the same form along all measurement bases. Thus for any random bases 1 0 , b b = φ ) ( 2 1 1 1 0 0 b b b b + and each of the qubits is in a mixed state. It is this feature that makes the measurements along different bases on the quantum state different from the measurements of classical states. The conditions for the probability constraints are well-defined both for classical objects and quantum states and we stress that the use of the ratios of “same” to “nonsame” events will be more reliable in the presence of noise than to consider just the absolute value of “same” events. For non-maximally entangled objects ) 11 00 ( 1 1 2 + + = r r AA φ we find that so long as r < 5.83, we can separate them from classical objects based on computation of “same” and “nonsame” events across appropriately chosen bases. PROBABILITY CONSTRAINTS It is quite clear that a constraint on n variables will be projected as several constraints on subsets of these variables. Therefore, if we are only given the constraints on the subsets of the variables, it cannot be said if they were derived from the same function on all the variables. Consider (with Boole) three events A, B, and C. Let P(AB) = r; P(BC) = s; and P(AC) = t. If a Venn diagram is drawn and we write Figure 1. Venn diagram for three events A, B, C η ν μ λ = = = = ) ( ; ) ( ; ) ( ; ) ( BC A P C B A P C AB P ABC P Then t s r = + = + = + ν λ η λ μ λ ; ; (2) A straightforward computation shows that the following constraints need to be satisfied for the data to be consistent: Probability and Classical/Quantum Divide