*5.2.1 "*Qubit"

The term for a **classical physical system** that exist in two unambiguously distinguishable states, representing 0 and 1, is often called a 'bit.' It is commonly acknowledged that the elementary quantity of information is the bit, which can take on one of two values, usually "0" and "1". If we consider any physical realization of a bit, it requires a system with two well defined states, For example in a switch *off* represents "0" and *on* represents "1". On the other hand a bit can also be represented by a certain voltage level in a logical circuit or a pit in a compact disc or a pulse of light in a glass fiber or the magnetization on a magnetic tape. For classical systems it is helpful to have the two states separated by a large energy barrier so that the value of the bit cannot change spontaneously.

On the other hand, in quantum information science, the basic variable is the **"qubit"***:* a quantum variant of the bit. In order to encode information in a two-state quantum system, it is customary to designate the two quantum states |0i and |1i. The term "Qubit" seems to have been used first by Benjamin Schumacher [21], in his "Quantum coding."

Electrons possess a quantum feature called **spin**, a type of intrinsic angular momentum. In the presence of a magnetic field, the electron may exist in two possible spin states, usually referred to as 'spin up' and 'spin down'.

One of the innovative and unusual features of Quantum Information Science is the idea of "superposition" (explained below) of different states. A quantum system can be in a "superposition" of different states at the same time. Consequently, a quantum bit can be in both the |0i state and the |1i state simultaneously. This new feature has no parallel in classical information theory. Schumacher in [21], coined the word "qubit" to describe a quantum bit.

The job of the weird symbols "|" and "⟩" (the so-called the "bra-ket" notation, was introduced by Paul Dirac in [22]. It is essentially to remind us that *mathematically* we are talking about **vectors** that represent qubit states labeled 0 and 1 and *physically* they represent states of some quantum system. This helps us to distinguish them from things like the bit values 0 and 1 or the numbers 0 and 1. One way to represent this with the help of mathematics is to use two orthogonal vectors:

$$|\mathbf{0}\rangle = \begin{bmatrix} \mathbf{1} \\ \mathbf{0} \end{bmatrix} |\mathbf{1}\rangle = \begin{bmatrix} \mathbf{0} \\ \mathbf{1} \end{bmatrix} \tag{2}$$

Thus one of the novel features of Quantum Information Science is that a quantum system can be in a "superposition" of different states. In a sense, the quantum bit (or "qubit") can be in both the |0i state and the |1i state *at the same time.* This is one of the reasons why in 1995 Schuhmacher coined the word "qubit" to describe a quantum bit.

It is also claimed that in Quantum Computing a "qubit" carries information. The question is where is the extra information kept. The usual answer is that the extra information lies embedded in a *superposition* of *entangled states* (the two terms will be explained below). The peculiar feature of this is that any accessing of the information destroys the superposition and with it the extra information itself.

Suppose that the two vectors |0i and |1i are **orthonormal**. This means they are both orthogonal and normalized (a normalized vector is a vector in the same direction but with a norm (length) 1) and 'orthogonal' (**Figure 1**) means the vectors are at right angles):

Consider a situation where the two vectors |0i and |1i are linearly independent. This means that we cannot describe |0i in terms of |1i and vice versa. Nevertheless, it is feasible to describe all possible vectors in 2D space using both the vectors |0i and |1i and our rules of addition and multiplication by scalars,

It is maintained that the vectors |0i and |1i form a "basis" because of the fact that (i) the vectors |0i and |1i are linearly independent, and (ii) can be used to describe any vector in 2D space (**Figure 2**) using vector addition and scalar multiplication. When the vectors are both orthogonal and normalized, they construe an "orthonormal" basis.

**Figure 2.** *Vectors in 2D space.*

**Figure 1.**

#### *5.2.2 ψ function, Schrödinger equation and the dynamics of quantum mechanics*

In Quantum Mechanics, the **wave function**, *ψ*, plays the central role and represents a variable quantity that mathematically describes the wave characteristics of a particle. At a given point of space and time the value of the wave function of a particle represents the probability of the position of the particle at the time. *ψ* function may be thought of as an expression for the **amplitude** of the wave of a particle. However, the spatial probability density is given by the squared modulus of the wave function, *ψ* 2 . The Schrödinger equation is as follows:

$$i\hbar \frac{\Psi}{t} = \frac{-h^2}{2m} \frac{{}^2 \Psi}{\varkappa^2} + V(\varkappa)\Psi(\varkappa, t) \equiv \hat{\mathcal{H}}\,\Psi(\varkappa, t) \tag{3}$$

where *i* is the imaginary unit, is the time-dependent wave function, is h-bar, *V* (*x*) is the potential, and is the Hamiltonian operator.

The Schrödinger equation is supposed to answer the question as to how the states of a system change with time. It is in the form of a differential equation and it captures the 'dynamics' of quantum mechanics: it describes how the wave function of a physical system evolves over time.

Schrödinger's equation gives an answer to the question: what happens to the de Broglie wave associated with an electron if a force (gravitational or electromagnetic) acts on it. The equation gives the possible waves associated with the particle a number associated with any position in space at an arbitrary time (i.e. functions of position and time). The general form of this wave function is:

$$
\psi(\mathbf{x}, \mathbf{y}, \mathbf{z}; \mathbf{t}) \tag{4}
$$

The essence of the Schrödinger's equation is that, given a particle and given the force system that acts (say, gravitational or electromagnetic), it yields the wave function solutions for all possible energies. Thus a particle can be described by a state vector or wave function whose evolution is provided by the Schrödinger equation. Hence, the Schrödinger equation, being a time-evolution equation, will make *ψ* vary with time.

## *5.2.3 "*Qubit" *and Schrödinger equation*

The time-dependent **Schrödinger equation** gives the time evolution of Ψ. The entangled states are created by distributing the qubits **between** the particles so that each particle carries one **qubit**. By assuming that **a freely moving particle is the qubit carrier, it is found that both the particle position in physical space and the qubit state, change in time in accordance with the Schrödinger equation**.

#### *5.2.4 What is quantum computing?*

Basically, Quantum computing is concerned with processing information by harnessing and exploiting the amazing laws of quantum mechanics. The use of long strings of "bits" in traditional computers encode either a zero or a one. In contrast with that a quantum computer uses quantum bits, or qubits. The difference can be explained as follows: a qubit is a quantum system that encodes the zero and the one into two distinguishable quantum states. However, because qubits behave quantumly, we can capitalize on the phenomena of "superposition" and "entanglement," which is not possible in the case of using "bits," as an encoding device.

### **5.3 Superposition and entanglement**

These two concepts might baffle us, since we do not come across the phenomena they describe in our everyday lives. Only in the event of our looking at the tiniest quantum particles, atoms, electrons, photons and the like, that we see these intriguing things, like **superposition** and **entanglement** [23].

#### *5.3.1 Superposition*

Superposition essentially subscribes to the principle that a quantum system can be in multiple states *at the same time*, that is, something can be "here" and "there," or "up" and "down" at the same time. Thus it is possible for Qubits to represent numerous possible combinations of *1* and *0* at the same time. To put qubits into superposition, researchers manipulate them using precision lasers or microwave beams. This possibility of simultaneously being in multiple states is the phenomenon of superposition. In its most basic form, this principle says that if a quantum system can be in one of two distinguishable states |xi and |yi, then according to this principle it can be in any state of the form α |xi + β |yi, where α and β are complex numbers with | α | <sup>2</sup> + | β | <sup>2</sup> = 1.

#### *5.3.1.1 Schrödinger's cat*

In 1935, Erwin Schrödinger conjured up a famous **thought experiment** of putting in place a cat in a superposition of both alive and dead states. He envisioned that a cat, a small radioactive source, a Geiger counter, a hammer and a small bottle of poison were sealed in a chamber. He also imagined that if one atom of the radioactive source decays, the counter will trigger a device to release the poison. This is where Schrödinger invoked the idea of entanglement so that the state of the cat will be entangled with the state of the radioactive source. He expected that sometime after the cat will be in superposition of both alive and dead states.

It is certainly counterintuitive to think of the possibility of an organism to be in such a superposition of both alive and dead states (**Figure 3**). It also dramatically reveals the baffling consequences of quantum mechanics.

#### *5.3.1.2 The double-slit experiment*

Another well-known example of quantum superposition is the double-slit experiment in which a beam of particles passes through a double slit and forms a wave-like interference pattern on a screen on the far side.

Based on this experiment quantum interference is explained by saying that the particle is in a superposition of the two experimental paths: one passage is through

**Figure 3.** *Schrödinger's cat.*

### *Some Foundational Issues in Quantum Information Science DOI: http://dx.doi.org/10.5772/intechopen.98769*

the upper slit and the second passage is through the lower slit. Correspondingly a quantum bit can be in a superposition of |0i and |1i. The implication seems to be that each particle passes simultaneously through both slits and interferes with itself. This combination of "both paths at once" is known as a superposition state [23].

It must be noted here that the particles. After going through the two slits, will turn into two sets of waves, **Figure 4**, in accordance with quantum mechanical principles. Moreover, at some points the two sets of waves will meet crest to crest, at other points the crest will meet the trough of the other wave. Accordingly two possibilities will arise: (i) in **Figure 5**, where crest meets crest, there will be **constructive interference** and the waves will make it to the viewing screen as a bright spot, and (ii) where crest meets trough, there will be **destructive interference** that cancel each other out and a black spot will appear on the screen. One should see below bright lines of light, where the waves from the two slits *constructively interfere*, alternating with dark lines where the wave *destructively interfere*, **Figure 6**.

**Figure 4.** *Double-slit experiment.*

**Figure 5.** *Interference pattern.*

**Figure 6.** *Constructive and destructive interference.*

A particle tends to appear more often at some places (regions of constructive interference) and do not appear very often at other places (regions of destructive interference). However, the likelihood of finding the particle at a particular point can be described only probabilistically.

#### *5.3.1.3 Superposition and quantum information science*

In the two experiments explained above we have seen one of the features of a quantum system (viz, Superposition) whereby several separate quantum states can exist at the same time by superposition.

The Quantum Information Science claims that each electron will exist spin up and spin down, until it is measured. Till measurement is done it will have no chance of being in either state. Only when measured, it is **observed** to be in a specific spin state. Common experience tells us that a coin facing up is in a specific state: it is a head or a tail. Irrespective of looking at the coin, one is sure while tossing the coin must be either facing head or otherwise tail.

In **quantum experience** the situation is not as simple and more unsettling: according to quantum mechanics, **material properties of things do not exist** until **they are measured**, i.e. until one **"looks"** (measure the particular property) at the coin, as if, it has no fixed face.

#### *5.3.2 The problem of measurement*

Delving into the issue of **quantum measurement** and taking the double-slit experiment as a case in point, the "wave" of a particle, e.g., an electron, should be interpreted as relating to the *probability* of finding the particle at a specific point on the screen. We cannot detect any wave properties of a single particle in isolation. When we repeat the experiment many times, we notice that the **wave function "collapses"** and the particle is detected as a point particle. Thus, in Quantum Information Science the problem of wave function collapse is the problem of measurement of finding the *probability*.

#### *5.3.3 Inherent uncertainty*

In 1927 Heisenberg shook the physics community with his **uncertainty principle**:

$$
\sigma\_{\mathbf{x}} \sigma\_{\mathbf{p}} \geq \frac{\hbar}{2} \tag{5}
$$

where *ħ* is the reduced Planck constant, *h*/2*π*.

The formula states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

The uncertainty principle, the wave-particle duality, the wave collapsing into a particle when we measure it together lead to the claim that the probability of the same particle being there in several places at the same time cannot be ruled out, i.e. 'smeared out' multiple positions at a time.

The smiley face shows, **Figure 7**, the location of the particle in one peak, but then there are many such places as the multiplicity of smiley faces show.

#### *5.3.4 Superposition and the power of a quantum computer*

We have already seen (in Section 5.2.1) that whereas classical computing uses "bits" for data processing, quantum computing uses qubits. We have also

*Some Foundational Issues in Quantum Information Science DOI: http://dx.doi.org/10.5772/intechopen.98769*

**Figure 7.** *A traveling wave.*

scrutinized that the practical difference between a bit and a qubit is: a bit can only exist in one of two states at a time, usually represented by a 1 and a 0, whereas a qubit can exist in both states at one time.

Moreover we have observed that the phenomenon of "superposition" allows the power of a quantum computer to grow exponentially with the addition of each bit. For example, two bits in a classical computer provides four possible combinations— 00, 01, 11, and 10, but only one combination at a time. Two bits in a quantum computer provides for the same four possibilities, but, because of superposition, the qubits can represent all four states at the same time, making the quantum computer four times as powerful as the classical computer. So, adding a bit to a classical computer increases its power linearly, but adding a qubit to a quantum computer increases its power exponentially: doubling power with the addition of each qubit.

## *5.3.5 Application of superposition in solving engineering problems*

The principle of superposition is useful for solving simple practical problems, but its main use is in the **theory of circuit analysis**.

For example, in quantum science, the **superposition theorem states** that the response **(voltage** or current) in any branch of a linear circuit which has more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are turned off (made zero). Alternatively, a circuit with multiple voltage and current sources is equal to the sum of simplified circuits using just one of the sources.

#### **5.4 Entanglement**

Entanglement in quantum mechanics is considered to be an extremely strong correlation and inextricable linkage that may found between different particles of the same kind and with the same physical property. It has been observed that such linkage and intrinsic connection, subsisting between Quantum particles, is so robust, that two or more quantum particles separated albeit by great distances, may be placed at opposite ends of the universe, can still interact with each other in perfect unison. This seemingly impossible connection led Einstein to describe entanglement as "spooky action at a distance."

This intriguing phenomenon demonstrates that it is possible for scientists and researchers to generate pairs of qubits that are "entangled,"which amounts also to saying that two members of a pair exist in a single quantum state. Thus, they claim that if we change the state of one of the qubits, it will bring about instantaneous change in the state of the other one in a predictable way, even if they are separated by very long distances.

The notion of entanglement was coined by Erwin Schrodinger in order to signify the peculiar properties of quantum correlations. In the *classical* world, "the whole is the sum of its parts", but the *quantum* world is very different. Schrödinger [24] says: *"the best possible knowledge of a* whole *does not necessarily include the best possible knowledge of all its* parts, *even though they may be entirely separate and therefore virtually capable of being 'best possibly known,' i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known, at least not in the way that it could possibly be known more completely, it is due to the interaction itself.*

*Attention has recently been called to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to* one *system, the representative obtained for the* other *system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are* entirely *arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it."*

For example, consider a pair of qubits. Suppose that each one is described by a state vector: the first one by ∣a⟩ and the second one by ∣b⟩. One might therefore think that the most general state of the two qubits should be represented by a pair of state vectors, ∣a⟩∣b⟩, with one for each qubit. Indeed, such a state is certainly possible, but there are other states that *cannot* be expressed in this form. The possible pair of states are also **separable** (often called **product states**). States that are not separable are said to be **entangled**. *Most* vectors are entangled and *cannot* be written as product states. This shows a peculiar feature of quantum states.

Example of entanglement when a measurement is made: a subatomic particle decays into an entangled pair of other particles. Essentially, quantum entanglement suggests that acting on a particle here, can instantly influence a particle far away. This is often described as **theoretical teleportation**. It has huge implications for quantum mechanics, quantum communication and quantum computing.

#### *5.4.1 Entanglement and quantum information science*

Quantum entangled states play a crucial role and have become the key ingredient in the field of Quantum Information Science.

It will be a fair question to ask as to why does the effect of **entanglement** matter? The answer to this is as follows: the behavior of the Quantum entangled states gives rise to seemingly paradoxical effects, viz. any measurement of a particle's properties results in an irreversible wave function collapse of that particle and changes the original **quantum** state. In the case of entangled particles such measurement will affect the **entangled system** as a whole.

Schrödinger, (unlike Einstein, the most skeptical about **entanglement** and considered it the fatal flaw in quantum theory, referring to it as "*spooky action-at-adistance*"), was much more prepared to accept quantum theory with the concept of entanglement and along with all its predictions, no matter how weird they might be. In his paper [24], which introduced quantum entanglement, Schrödinger wrote "I would not call it *one* but rather *the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".

#### *5.4.2 Application of entanglement*

The interpretation of quantum states, in particular the interpretation of so-called 'entangled states' exhibit peculiar **nonlocal** (explained below) statistical correlations for widely separated quantum systems. For example, the theory underlying the field of quantum information, dealing with "entanglement," has found

intriguing connections with different fields of physics, like condensed matter, quantum gravity, or string theory.

Quantum entanglement, more often is viewed as a physical resource, which enables us to communicate with perfect security, build very precise atomic clocks, and even teleport small quantum objects, dense coding and cryptography.

## *5.4.3 In what way entanglement enables us to communicate with perfect security*

Quantum entanglement offers a new modality for communications that is different from classical communications. It has been claimed that entanglement enhances security in secret sharing.

Quantum cryptography (it is a method of storing and transmitting data quantum mechanically in a particular form so that only those for whom it is intended can read and process it) to a great extent revolves around quantum computing. The entanglement concept is one tool used in quantum computing, e.g., in the use of transmitting data via entangled Qubit, which is a unit of quantum information that is stored in a quantum system. Quantum cryptography utilizes photons and depends on the laws of physics rather than very large numbers and the deciphering of cryptographic codes.

It appears that we are perched on the edge of a quantum communication revolution that will change transmission of information, information security and how we understand privacy.
