TYPES OF ENTANGLEMENT (आरादुपकारक & संनिपत्योपकारक) – 5- Shri Basudeba Mishra
To explain quantum entanglement, physicists use the concept of quantum superposition – the idea that particles exist in multiple states at the same time. It is wrongly assumed that the particle collapses permanently to one of the states in the superposition on observation. For example, particles can exist in a superposition of spin up and spin down states simultaneously. Though there is a probability attached to each state, and it is possible to predict the average outcome from many measurements, and the likelihood of a single measurement being up or down depends on these probabilities, but is itself unpredictable. Let us examine this idea.
Standing before the beach in my home town, I used to watch the waves coming, breaking and going back repeatedly. Because of the topology, sometimes waves come from different angles, collide and merge and go back. At one moment, I could distinctly see two or three waves coming from different directions. After merger, when they go back, I could not identify which wave went where. They have gone to a superposition of states.
While water is physically existent and has a discreet form (स्थितिसिद्धः-सशरीरी), it doesn’t have a rigid three-fold internal structure (अहृदयम्). Hence, it is a boson (ऋतम्) as per the earlier definition. As such, it doesn’t follow the exclusion principle and can co-exist with each other (सामञ्जस्यम् – स+अञ्जूँ व्यक्तिमर्षणकान्तिग॒तिषु॑). For this reason, we can’t know the exact positions of the merged waves. Like God (अधिरीश्वरे – स्वस्वामिभावं) it exists (आसँ॒ उप॒वेश॑ने) everywhere (गत्याक्षेपे अथवा उपरिभागे) in an analog state without discreetness.
One may question how water and oil do not mix? The answer is density variation. Both are not similar, but two different objects. Oil coexists with similar oil in a superposition of states.
Thus, superposition (अध्यास) is our inability to know the exact state due to some hidden variables. This misleads us like mistaking a rope in darkness as a snake. Contrary to interpretation of most modern Vedantins, it is not ignorance. We may not have knowledge about the rope, but we must have knowledge about the snake (अज्ञात रज्जुकार्यस्य सर्पस्याज्ञातता कुतः). If we watch closely for some time, we can infer it is not a snake. The mistaken notion is Lie – Mithya (मेथृँ॑ मेधाहिंस॒नयोः॑ or मथेँ वि॒लोड॑ने) – that which appears in a twisted state to disguise information.
In the Schrodinger cat experience, the cat will be found either alive or dead at time t, based on whether the poison has already been released or not and whether the cat dies of natural causes or suffocation. Before measurement, the cat will not be in a ghostly state of half-alive-half-dead or undead like the Frankenstein’s monster. There is nothing like “collapse”. It is pure fantasy. We do not control all aspects of all operations. There may be aspects beyond our knowledge or control that influence the outcome of any operation (कर्मण्येवाधिकारस्ते मा फलेषु कदाचन). In the case of boson (ऋतम्), it is our inability to find the exact position of each before they mix up that is called superposition of state. That (nature of bosons) is the hidden variable. Since hidden variables are dealt in Bell’s experiment, let us discuss that.
BELL’S INEQUALITY.
Bell outlined three assumptions about the world, each with a corresponding mathematical statement:
1) Realism, which says objects have properties they maintain whether they are being observed or not (अस्तित्व);
2) Locality, which says nothing can influence something far enough away that a signal between them would need to travel faster than light (ज्ञेयत्व); and
3) Freedom of choice, which says physicists can make measurements freely and without influence from hidden variables (अभिधेयत्व).
Probing entanglement is thought as the key to testing these assumptions. If experiments show that nature obeys these assumptions, then we live in a world we can understand classically, and hidden variables are only creating the illusion of quantum entanglement. If experiments show that the world does not follow them, then quantum entanglement is real and the subatomic world is truly as strange as it seems. What Bell showed is that if the world obeys these assumptions, there’s an upper limit to how correlated entangled particle measurements can be.
Think of Polaroid filters in sunglasses. They absorb light polarized along one axis (say horizontal), but transmit light polarized along a perpendicular axis (say vertical). Suppose a light ray splits and is reflected in two directions – one along the horizontal and the other along vertical directions. It is said that there is a 50% probability of the light pulse going either way.
Let these rays be observed by two Polaroid filters – both aligned either horizontally or vertically. If the light pulses are perfectly aligned to the Polaroid filters, and if we could observe one light signal going through one lens, we could be sure that the other gets absorbed and vice versa. If we change the alignment of filters so that one is aligned horizontally, while the other vertically, then it is said that there is a 50% probability of both the pulses either going through or getting absorbed. What determines this probability? Is nature mathematically and linearly symmetrical? What is the proof?
Imagine the filters are aligned at a 45 degree angle? It is said that then the probability of the light pulse hitting the filters or not will be reduced by half, i.e., 25%. But why? We are probing whether it goes through or not. Not the angle through which it goes. If it is a massive particle of big enough size, the chances of it going through will be very much reduced with any change of angle. If we use entangled photons that go through, the chances will not be 50-50 – we can’t be sure – but if one photon made it through the 45° filter, then most likely the other also will go through. And if the former be absorbed, so will be the latter. There’s zero chance that one of the photons made it through while the other got absorbed.
One way to reconcile between realism and determinism was to add “hidden variables” to the wave-function to provide the most complete description of the system possible. These hidden variables might, for example, provide values for all components of the spin at all times, and thus dictate whether the result +hw/2 or -hw/2 was obtained in a measurement. However, Bohr and Heisenberg were convinced that one could not supplement quantum theory with hidden variables. Therefore they were overjoyed when, von Neumann claimed in 1932 to have proved that the application of hidden variables to quantum theory was indeed impossible.
WHAT IS WAVE-FUNCTION?
A wave is characterized by: amplitude, frequency, wavelength, time period and speed. The wave function is said to be a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. Probability distribution in three dimensions is established using the wave function. In a Wave Function, it is said that all measurable information about the particle is available. It should be continuous and single-valued.
It is said that a wave function must not be zero everywhere in space. It has to be continuous. It can’t tend to infinity. Its first derivative cannot be discontinuous for infinite number of points. Its first derivative may be discontinuous for a finite number of points. The wave equation is said to be a linear, homogeneous partial differential equation with constant coefficients. It has one dependent variable (q) and four independent variables (t, x, y, z). Interestingly, a wave function has no physical significance as it is a complex and non-observable quantity. Only its square has physical significance as a probability of finding the particle described by the wave function at a given point and time – that too proportionately.
(to be continued)