3.Classical vs. quantum
The very concept of the state of a system in quantum mechanics is different from its classical counterpart. In each, states are represented by different mathematical objects and have a different logical structure. In classical physics, the state of a system describes everything that is required to predict the future of that system. It is not so for a quantum system. The state of a quantum system is neither completely measurable nor completely predictable. Consequently, each system is interpreted differently. Each perspective demands a different mindset through which ideas, concepts, and interpretations flow. Even though certain formal terms, e.g., space, time, momentum, measurement, superposition, etc. are used in both systems, their mathematical underpinnings and mental interpretation are different. From either perspective, the other looks weird and apparently incompatible.
For example, in classical physics, Einstein’s general relativity describes gravity as the warping of space-time in the presence of mass, but to know what happens inside a massive black hole or what happened at the big bang requires a quantum mechanical explanation. The extremely weak gravitational waves predicted by Einstein were detected on 14 September 201511 and the Nobel Prize for physics (2017) promptly awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne “for decisive contributions to the LIGO detector and the observation of gravitational waves”12. It is likely that quantum entanglement (explained below) may provide an answer.13 For other differences pertinent to the present paper see Sections 3.1, 3.2, and 4.
The deeper we explore Nature, its laws become “more and more unreasonable and more and more intuitively far from obvious.”14 So it has been from Newton’s laws of motion to Maxwell’s equations of electromagnetism to Einstein’s general theory of relativity to the postulates of quantum mechanics. The last are so baffling that Richard Feynman said:
I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.15
Here are a few examples of the strangeness of quantum mechanics, the crown jewel of physics.
3.1 Some strange (non-classical) aspects of quantum mechanics
Classical physics presupposes that exact simultaneous values can be assigned to all physical quantities; quantum mechanics shows there are exceptions, e.g., the position and momentum of a particle. In such cases, the more precisely one quantity is measured, the less precisely can the other be measured. (See Heisenberg’s uncertainty principle in Section 4.4.)
Classical physics assumes that it is possible to measure the state of a system so delicately that it does not noticeably interfere with the system. In the quantum world it is simply impossible. Further, we intuitively assume that identical conditions always produce identical results. This too fails in quantum mechanics; it produces statistical results. When combined with the previous paragraph, it means that we cannot define the trajectory of a particle in quantum mechanics.
Unlike classical physics, in quantum mechanics there is no gradual transfer of energy from radiation field to matter. Planck’s assumption that energy transfer is a discontinuous process and is quantized has been amply confirmed by experiments.16
In Section 4.4, we present other quite crucial differences between classical and quantum physics. Taken together, it makes one wonder if every quantum bit (qubit) in a quantum computer is so different from a classical bit, how does a quantum computer work. The answer lies in making very clever use of superposition and entanglement of quantum states and in collapsing them.
3.2 Superposition, measurement, and entanglement
See Fig. 3.1.17 What did you see? Likely, a young woman (if you had gazed up; call it state |0 of the picture in quantum mechanical notation) or an old woman (if you had gazed down; call it state |1 of the picture)! You did not see some weird superposition (or summation) of the two. Although both coexist in the picture, each observer will see, at random, one or the other at a given time, independent of what others see then or at another time. It is in this metaphorical sense a quantum object is said to be in a state of superposition. Without any prior knowledge of the picture a viewer would randomly gaze up or down and accordingly see a young or an old woman. From a group of viewers, each of whom was shown an identical copy of this picture, you would get a statistical sample of the fraction of people who tend to look up or look down. Quantum systems when measured produce statistical results. Just as you did not see Fig. 3.1 as the sum of a young and an old woman, in quantum systems you can never see the system in its superposed state. You can only see one of its eigenstates, randomly chosen by Nature, and collect statistics. Where Fig. 3.1 differs from a quantum system is that no amount of observation will change the picture nor its effect on an observer. The state of the picture (a classical object) is immune to observation. A quantum object is not. A quantum object, when first observed, freezes (collapses) into the state it has revealed to the observer, and subsequent observations will reveal only the frozen state. Of course, there are means that can put the object into its original state (only if the original state is known) or some other state.
The response of two entangled quantum objects (metaphorically, as paired dance partners), to any change of state in one (e.g., due to its being observed) is an instant reactive change in the other independent of any distance separating them. The action implies perfect coordination without the benefit of anticipation. Perfect coordination in the classical world is possible only with preplanning, e.g., by following an agreed upon schedule of events.
3.3 Notations in quantum mechanics
Notations and the meanings we attach to them encode knowledge. In Table 3.1, we provide a few symbols and their frequently used combinations in quantum mechanics for easy reference. Familiarization requires only elementary pattern recognition skills. The notations were invented by Paul Dirac (1902-1984)18 in 1939 as a kind of dress code for quantum physicists to show off (just kidding!). He is one of the founders of quantum mechanics, and while founding, predicted the existence of antimatter, and won the Nobel Prize in Physics for it in 1933, at age 31.
In quantum computing, one talks about Hilbert space only to impress people. It is really about formatting of matrices—creating larger matrices by patching together smaller ones (the tensor product). It turns out that in the finite dimensional complex vector spaces that come up in quantum computing and quantum information, the Hilbert space is exactly the same thing as an inner product space. Also, beware that what appears as a variable in classical mechanics may appear as an operator in quantum mechanics. In natural languages, nouns becoming verbs and vice-versa are not uncommon. In classical physics, e.g., momentum is a variable but an operator in quantum physics.〉
3.4 Meet the qubit (the simplest quantum entity)
The simplest object that allows a state change is one with two states. We conventionally and arbitrarily label one as state 0 and the other as state 1 and provide rules for state change. While the classical binary bit at any given time can be either in state 0 or state 1, its counterpart the quantum bit, christened qubit by Ben Schumacher,19 can be and usually is in some superposition of the two quantum states |0〉 and|1〉. The general state representation of a qubit is the unit vector α|0〉 + |1〉 where and are complex numbers, constrained by |α|2 + |β|2 = 1. Unless |α| or |β| = 1, the qubit is in a state of superposition.
This qubit when observed (i.e., its state measured), will be seen in either state |0〉 or state |1〉 according to a statistical rule, i.e., given a very large number of identical copies of the qubit and each copy measured, the qubit will collapse to state |0〉 with probability |α|2 and to |1 with probability |β|2. This insightful, probabilistic aspect of Nature was uncovered by Max Born for which he received the 1954 Nobel Prize in Physics20. A quantum system, when appropriately observed or measured, randomly collapses to only one of its eigenstates |0〉 or |1〉 according to Postulate 3 in Section 4.1.
Two or more qubits can be entangled, and if handled with care, can be separated very far apart. Entanglement is a joint characteristic of two or more quantum particles. All interesting quantum algorithms make clever use of entanglement. As you gain experience with entanglement, you will see it as an enormous computing resource unknown in classical physics. As an aside, note that a qubit in the superposed state α|0〉 + β|1〉 may be viewed as being in a state of self-entanglement, because when α changes so does β instantly.
In a physical computer, qubits are physical objects (e.g., photons, electrons), which unless extremely well shielded, tend to randomly entangle with their environment. This decoheres their state of superposition and hence the information they encode. Thus, long computations are difficult to sustain. This is a hardware issue, but clever error correcting algorithms encoded in software for certain situations exist21. For an excellent tutorial on decoherence, see Marquardt and Püttmann.22
4.Quantum mechanics – postulates and consequences
A quantum system is described by its state vector, |Ψ〉. Its mathematical description and evolution is enshrined in the enigmatic postulates of quantum mechanics. Any system that fulfills the postulates is quantum mechanical. The postulates encompass all the observed weirdness of quantum systems described in Sections 3.1 and 3.2 and some more. The postulates neither tell us how a quantum system knows it is being observed nor the mechanism related to the collapse of | Ψ〉 when it is observed.
Note: The superscripts *, T, and † ( *T = T*) attached to various symbols, respectively, denote complex conjugate, transpose, and Hermitian conjugate (or adjoint) operations on the symbol.
 Abbott, et al (2016).
 RSAS (2017); Resnick (2017).
 Bose, et al (2017).
 Feynman (1965), Chapter 6.
 Feynman (1965), Chapter 6.
 This even includes data collected on the background radiation at the edge of the universe (remnants of the Big Bang) by the COBE satellite in 1990. The data fits perfectly Planck’s black-body radiation law. See, e.g., RSAS (2006).
 Picture credit: William Ely Hill (1887–1962), My wife and my mother-in-law, https://en.wikipedia.org/wiki/My_Wife_and_My_Mother-in-Law (in public domain)
 Dirac (1939). See also: Gidney (2016).
 Schumacher (1995). See also: Preskill (2015).
 Born (1954).
 Shor (1995).
 Marquardt & Püttmann (2008).