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Quantum Cryptography and Quantum Teleportation

  • Rajendra K. BeraEmail author
Chapter
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Part of the Undergraduate Lecture Notes in Physics book series (ULNP)

Abstract

This chapter is meant to be an appetizer and lightly relies on the reader’s intuition to understand the mathematical steps involved. The chapter directly introduces two quantum algorithms: (1) How to encrypt messages (cryptography), which if snooped upon during transmission to a recipient, will be detected; and (2) how to teleport the state of a quantum object. Along the way just enough intuitively understandable but weird and exclusive aspects of quantum mechanics as compared to classical mechanics are introduced.

1.1 Introduction

To understand quantum mechanics and quantum computing, you will need to have some working knowledge of complex numbers, linear algebra, familiarity with complex matrices, and matrix operations. It would be a good idea for you to refresh your ability to calculate eigenvalues and eigenvectors of a matrix after you have read this chapter and are still interested in reading the remaining chapters of this book.

In this chapter, we will directly introduce you to two quantum algorithms: (1) How to encrypt messages (cryptography), which if snooped upon during transmission to a recipient, will be detected (in no case will the snooper be able to read the message correctly); and (2) how to teleport the state of a quantum object. Along the way, you will be introduced to some intuitively understandable aspects of quantum mechanics, just enough for you to understand the two algorithms. Interestingly, classical physics has no solution for either problem. So, quantum mechanics does allow you to live in a magical world. To practice this magic, you will need to get used to some weird ways in which Nature works exclusively at the quantum level, a level most often seen at the atomic and sub-atomic levels. Before proceeding further, let me introduce you to some words of wisdom from the great physicist Richard Feynman. Prefacing a lecture on quantum mechanics, he said,

Do not take the lecture too seriously… just relax and enjoy it. 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… into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.1

His words put me at ease and encouraged me to learn quantum mechanics and quantum computing. The fact is, human intuition, not intellect, is ill-equipped to deal with the mysteries of the quantum world; it is best understood in the abstract language of mathematics. Why? Feynman answers:

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.2

In quantum mechanics, if your mathematics is right, you can ignore common sense and intuition. If in doubt, check your mathematics. This checking is not an intelligent activity, because, in principle, it can be mechanized. This we know from Alan Turing and his amazing paper3 in 1936 in which he showed that a machine, the Universal Turing Machine (UTM), can mimic an unintelligent person, who tirelessly and with absolute concentration performs calculations as instructed, in a step-by-step manner, i.e., according to a given algorithm. This Turing Person has at its disposal unlimited time, paper, pencil, and energy. Note that the UTM does only those tasks that a human might do in executing an algorithm without the help of insight. There is a strong belief, so far unrefuted, among computer scientists that what is human-computable is machine-computable. This is the famous Church–Turing Thesis: “The class of functions computable by a Turing machine corresponds exactly to the class of functions which we would naturally regard as being computable by an algorithm.” It is also a statement about the limitations of the human mind.

1.2 Hello to Some Weirdness in Quantum Mechanics

We now come to a few weird things in quantum mechanics. Quantum physicists use the terms superposition and measurement in very specific ways. You are perhaps already aware that in physics there are a set of key words, e.g., force, work, temperature, entropy, enthalpy, etc., which all qualified physicists understand in the same way, and these words and their various relationships are expressible by precise mathematical formulas. Hence, if in doubt, check the math before jumping to the conclusion that you have made a Nobel Prize-winning discovery.

In quantum mechanics, the term superposition is used to state that matter or energy at the quantum level can be in two different states at the same time, e.g., an electron can be in spin-up and spin-down states at the same time or a photon can be vertically polarized and horizontally polarized at the same time. When you measure the state of a quantum entity (say, the electron or the photon) in a superposed state, you will see only one or the other of the two states it is concurrently in. For example, you will see the electron in the spin-up state or the spin-down state, but you will not be able to predict beforehand what the measurement will be. You will never see some combination of spin-up and spin-down states. Likewise, for the photon. You will never be able to measure a photon in some combined state of vertical and horizontal polarization and you will not be able to predict the measurement outcome beforehand with certainty. That is because, and as far as we can tell, Nature decides the measurement outcome at the last moment by tossing a biased coin (i.e., probabilistically). The biasness of the coin has a precise mathematical relationship with how the quantum states are superposed in the quantum entity being measured. All the probabilities we talk about in quantum mechanics has to do with measurement and not with the way an undisturbed quantum system evolves. It is possible to take a quantum entity, e.g., an electron or a photon and put it into a state of superposition of our choice. But if we are given a quantum entity of unknown state, we can never determine its state by any means or make an exact replica of it.4 All this is very strange, but is it?

See Fig. 1.1. On the left is shown an electron in its two possible spin-up and spin-down states and in its state of superposition. It is difficult to get a feel for quantum superposition and measurement because of its emotionless abstraction. The picture on the right is very different. If you look at it several times, you (the measurement apparatus) will randomly see (measure) either a pretty girl or an old wrinkled woman (the collapsed state of the picture). It is the same picture! Your brain will ensure that at no time will you see both the pretty girl and the old woman or some weird combination of the two. This picture, not in detail, provides a glimpse of quantum superposition and measurement through the interplay of eye and mind.
Fig. 1.1

(Left) Lieven Vandersypen, Dot-to-Dot Design. IEEE Spectrum, September 2007. pp. 42–47. https://sites.cs.ucsb.edu/~cappello/IEEE/Spec_20070901_Sep_2007.pdf; (Right) German postcard from 1888. Wikimedia Commons. https://commons.wikimedia.org/wiki/File:German_postcard_from_1888.png. Popularly titled as My wife and my mother-in-law. (In public domain). If you look at the eyes in the figure, you will see the mother-in-law; if you look at the bonnet you will see the wife.

There is a third term in quantum mechanics that refers to an enigmatic quantum phenomenon called entanglement. This is an intriguing state of being in which two quantum entities are so deeply correlated that they behave as one composite entity, no matter how far apart they are in space. Indeed, distance has no meaning for entangled entities. If the state of one is changed, the state of the other is instantly adjusted to be consistent with quantum mechanical laws. If a measurement is made on one, the other automatically and instantaneously collapses to a predefined state. Einstein derisively called such action at a distance “spooky.” He wrote to Max Born in March 1947, “I cannot seriously believe in [quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.”5 After Einstein’s death, it was proven that he was wrong.6 Entanglement is real, and it is a joint characteristic of two or more quantum entities when entangled. Just as a quantum entity can be put in a desired state of superposition, so can two quantum entities be put in a state of entanglement, which means they acquire a group property. In quantum computing, entanglement plays an unusually big role.

1.3 Time for Some Mathematics

To proceed further, we need some mathematics. The important thing about mathematics is that it is a language that allows you to communicate with precision and compactness in a systematic manner. The most important characteristic of modern mathematics is that it is presented as an axiomatic study, whereas the sciences are axiomatic only to the extent they utilize mathematics. The key to axiomatic reasoning is the idea that the truth of a statement must be shown to follow logically from the truth of other statements, which have already been shown to be true by this method. Deductive reasoning dominates mathematics. The biggest dilemma in creating an axiomatic system is how do we get started? We crank up the system by accepting one or more statements to be true without demanding a proof of their truthfulness, i.e., we rely upon our intuition (the Achilles heel of mathematics). These statements are called postulates or axioms. We try to keep such statements as few as possible and do our best to see that the statements are extremely unlikely to lead to contradictions. Briefly, by a deductive system we mean:
  • We have selected one or more concepts that we feel are very primitive and that we agree to accept them without definition. These are the undefined concepts of the system (e.g., point and straight line in Euclidean geometry).

  • We have selected some statements concerning the undefined concepts that we feel express very primitive truths about the undefined concepts and that we are going to accept without proof. These are the axioms of the system.

  • Using undefined concepts and axioms we can begin the process of defining new concepts in terms of the undefined concepts. We call these defined concepts.

  • And establish the truth of new statements about these concepts based on the axioms. We call the new true statements theorems.

By an axiomatic (also called formal) system, we mean a system comprising a set of symbols; a grammar for combining the symbols into statements; a set of axioms, or statements that are accepted without proof; and rules of inference for deriving new statements (theorems). A proof is a listing of the sequence of inferences that derive a theorem. It is vital that a proof be formally (i.e., mechanically) verifiable. Thus, there is a correspondence between an axiomatic system and a computational system whereby a proof is essentially a string (usually a binary string) processing computation. This is shown in Table 1.1.
Table 1.1

Correspondence between axiomatic and computational systems

Axiomatic system

Computational system

Axioms

Program input or initial state

Rules of inference

Program interpreter

Theorem(s)

Program output

Derivation

Computation

Source Lewis [19]

A proof in axiomatic mathematics is an impeccable argument that uses only the methods of pure logical reasoning. The reasoning is such that it enables one to infer the validity of a given mathematical assertion from the pre-established validity of other mathematical assertions or the axioms. Once a mathematical assertion has been established by this procedure, it is called a theorem. Axiomatic mathematics is about axioms, theorems, and proofs.

To implement an axiomatic system, we set up a typographical system so that statements appear as mere strings of symbols according to some typographical rules. Through an appropriately chosen translation rules, symbol strings can be rewritten as binary strings and manipulated by digital computers (essentially, Universal Turing Machines).

1.3.1 Quantum Operators that Act on a Qubit

In the world of quantum mechanics, physical systems are described by an abstract mathematical object called the state vector (or the wave function) \( \left| \psi \right\rangle \). (Note the unusual notation \( \left| \psi \right\rangle \) first introduced by Paul Dirac. The symbols: \( \left\langle \cdot \right| \) is called bra and represents a column vector; \( \left| \cdot \right\rangle \) is called ket and represents a row vector; the dot inside the symbols is a placeholder for labels. The nomenclature is more fully described in Chap.  3, Tables.) People are still trying to understand the exact status of \( \left| \psi \right\rangle \) in quantum theory. However, such ignorance has not prevented physicists from moving forward. Further, for the purposes of this chapter, it is enough to know two (of four7) postulates or axioms of quantum mechanics. First is the natural evolution of a quantum system, i.e., of the state function \( \left| \psi \right\rangle \). It evolves in a deterministic manner according to the linear Schrödinger equation. Second is the measurement of the system by a process called wave packet reduction (or more dramatically as wave function collapse). The problem with measurement is that nobody yet knows how to precisely define the act of measurement. Whenever it happens, the quantum system collapses according to a probabilistic postulate. This makes measurements of quantum systems a source of major conceptual difficulties, but not enough to impede quantum mechanics from making breathtaking advances.

The unusual difference between quantum mechanics and classical mechanics is that in the former we have two different postulates for the evolution of the same mathematical object, and in the latter we have only one. So, in quantum mechanics we sometimes have difficulty in knowing which postulate to apply. This leads us to the problem of decoherence—the problem of the instability of coherence—since we are not always quite sure what constitutes a measurement, it is possible that during its evolution (say while doing computations) a quantum system may get decohered and create errors in the computations. The reason why quantum computers still have a long way to go in terms of robustness is that superposition and entanglement are extremely fragile states. Any interaction with the environment (i.e., anything external to the quantum system being studied) and the quantum system may decohere. Preventing decoherence from taking hold before a calculation is completed remains the biggest challenge.

The quantum mechanical counterpart of the classical binary bit, which at any time is in state 0 or 1, is the qubit (short for quantum bit, so named by Schumacher in [26]).8 A qubit can be in a superposition of 0 and 1, i.e., it can concurrently be 0 and 1 (like “my wife and my mother-in-law in Fig. 1.1). The state 0 of a qubit is represented by \( \left| 0\right\rangle \) and state 1 by \( \left| 1\right\rangle \), called its eigenstates, and the general superposed state of a qubit is represented by the unit vector \( \left| \psi \right\rangle \)  = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \), where a and b are complex numbers constrained by the relation |a|2 + |b|2 = 1. If such a superposition is measured with respect to the basis {\( \left| 0\right\rangle \), \( \left| 1\right\rangle \)}, the probability that \( \left| \psi \right\rangle \) will collapse to \( \left| 0\right\rangle \) is |a|2 and the probability that it will collapse to \( \left| 1\right\rangle \) is |b|2.

The state of a qubit (the simplest quantum entity we can think of) given by \( \left| \psi \right\rangle \) = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \) can be altered using one or more unitary operators. These operators have some simple and easy-to-remember properties. First, by definition, a unitary operator U has an inverse, which means that you can undo an operation, and that inverse is the same as its conjugate transpose U, hence UU = I = UU, where I is the identity operator, i.e., it leaves things it operates on unchanged, much like multiplying a number by 1. You really need to know only 4 unitary operators. These are:

1-qubit unitary operators

Operating on \( \left|\psi \right\rangle \) = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \)

Identity

I:

\( \left| 0\right\rangle \)\( \left| 0\right\rangle \)

\( \left| 1\right\rangle \)\( \left| 1\right\rangle \)

I\( \left| \psi\right\rangle \) = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \)

Negation

X:

\( \left| 0\right\rangle \)\( \left| 1\right\rangle \)

\( \left| 1\right\rangle \)\( \left| 0\right\rangle \)

X\( \left| \psi\right\rangle \) = a\( \left| 1\right\rangle \) + b\( \left| 0\right\rangle \)

ZX

Y:

\( \left| 0\right\rangle \) → −\( \left| 1\right\rangle \)

\( \left| 1\right\rangle \)\( \left| 0\right\rangle \)

Y\( \left| \psi \right\rangle \) = −a\( \left| 1\right\rangle \) + b\( \left| 1\right\rangle \)

Phase shift

Z:

\( \left| 0\right\rangle \)\( \left| 0\right\rangle \)

\( \left| 1\right\rangle \) → −\( \left| 1\right\rangle \)

Z\( \left| \psi \right\rangle \) = a\( \left| 0\right\rangle \) − b\( \left| 1\right\rangle \)

The rightmost column shows how each operator affects \( \left| \psi \right\rangle \). Any other unitary operator M required to operate on a qubit can be created by a linear combination of these four operators as
$$ M = \alpha I + \beta X + \gamma Y + \delta Z, $$
where α, β, γ, and δ are complex constants of one’s choice. Second, U is diagonalizable, and its eigenvectors are orthogonal. U only rotates the vector \( \left| \psi \right\rangle \) it operates on; it does not change the length of the vector, which remains 1, i.e., in \( \left| \psi \right\rangle \) = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \), a and b may change but only by maintaining |a|2 + |b|2 = 1.
There is one frequently used gate, called the Hadamard gate, H = (X + Z)/√2 that operates on a qubit to accomplish the following:
$$ \begin{array}{*{20}l} {{\text{Hadamard }}H :} \hfill & {\left| 0 \right\rangle \to \left( {1 /\sqrt 2 } \right)\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)} \hfill & {\text{or}} \hfill & {H\left| \psi \right\rangle = \left( {(a + b)\left| 0 \right\rangle + (a{-}b)\left| 1 \right\rangle } \right) /\sqrt 2 } \hfill \\ {} \hfill & {\left| 1 \right\rangle \to \left( {1 /\sqrt 2 } \right)\left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)} \hfill & {} \hfill & {} \hfill \\ \end{array} $$

Note that the length of H\( \left| \psi \right\rangle \) remains unity since |a + b|2/2 + |a − b|2/2 = 1.

1.3.2 A Quantum Operator that Acts on a Qubit Pair

The general state of a 2-qubit system is given by the linear combination
$$ \left| \psi \right\rangle = a\left| {00} \right\rangle + b\left| {01} \right\rangle + c\left| {10} \right\rangle + d\left| {11} \right\rangle , $$
Here, the notation \( \left| xy \right\rangle \) depicts the state where the first qubit’s state is \( \left| x \right\rangle \) and of the second is \( \left| y \right\rangle \). The 2-qubit system has only four eigenstates: \( \left| 00 \right\rangle \), \( \left| 01 \right\rangle \), \( \left| 10 \right\rangle \), and \( \left| 11 \right\rangle \). Further, |a|2 + |b|2 + |c|2 + |d|2 = 1 to ensure that \( \left| \psi \right\rangle \) remains a unit vector. There is a 2-qubit unitary operator, Cnot, called the controlled-not, that acts on \( \left| \psi \right\rangle \) such that
$$ \begin{array}{*{20}c} {{\text{Controlled} {-} {\text{not}}}\;C_{\textit{not}} :} & {\left| {00} \right\rangle \to \left| {00} \right\rangle } & {} & {} \\ {} & {\left| {01} \right\rangle \to \left| {01} \right\rangle } & {} & {} \\ {} & {\left| {10} \right\rangle \to \left| {11} \right\rangle } & {} & {} \\ & {\left| {11} \right\rangle \to \left| {10} \right\rangle } & {\text{or}} & {C_{\textit{not}} \left| \psi \right\rangle = a\left| {00} \right\rangle + b\left| {01} \right\rangle + c\left| {11} \right\rangle + d\left| {10} \right\rangle } \\ {} & {} & {} \\ \end{array} $$

The Cnot operator flips the second (target) qubit if the first (control) qubit is \( \left| 1 \right\rangle \) and does nothing if the control qubit is \( \left| 0 \right\rangle \) This operation also entangles the two qubits (more on this in Chap.  7). Note that the length of Cnot\( \left| \psi \right\rangle \) remains unity, i.e., |a|2 + |b|2 + |c|2 + |d|2 = 1.

Important remark: It can be shown that by stringing together 1-qubit operations and the 2-qubit controlled-not operation, it is possible to build a quantum computer capable of doing anything a classical computer can do.9 We also note that there are only two ways to manipulate a quantum system: (1) make a measurement which would irreversibly and probabilistically collapse the system into one of its eigenstate or (2) use unitary operators to deterministically evolve the system.

1.4 Encryption and Key Distribution

For our limited purpose of cryptography, we need the 1-qubit unitary Hadamard operator, H, that operates on one qubit at a time and to remember that measurement is a non-unitary operation that collapses the quantum system being measured in a probabilistic and irreversible way. Further, the inability to copy an unknown quantum state is a key difference between ordinary and quantum information. This fact has made quantum information theory very attractive to cryptographers.

The exchange of secret messages in a completely secure manner using non-quantum mechanical means requires a perfect cypher. Such a cypher known as the Vernam cypher10 or one-time pad was invented in 1917 by Gilbert S. Vernam. Unfortunately, it requires a key equal in size to the plaintext message. Shannon’s information theory11 shows that we cannot do better. For keys to be shorter, the cyphertext must compromise and contain some information about the plaintext message. Thus, for perfect security we have the problem of distributing the key itself, which must be done over a secure channel such as by a trusted courier. In many situations, such as banking transactions where the volume of information is very large, this is impractical. Therefore, alternative, but less secure methods such as the RSA public key cryptosystem12 are often used. Exchanging keys securely is therefore a truly crucial step in cryptography. It is the secure exchange of keys that we discuss here.

In 1984, Charles H. Bennett and Gilles Brassard described the first completely secure quantum key distribution algorithm, now known as the BB84 protocol,13 in which quantum states are used to establish a random secret key for cryptography. Thus, they were able to circumvent the restriction of Shannon’s theory and permit keys shorter than the message. The BB84 protocol exploits two unique quantum mechanical aspects—the ability to generate perfectly random numbers (using the Hadamard operator) and the fact that, in general, any observation (measurement) disturbs (collapses) the quantum system being observed. Thus, if there is an eavesdropper attempting to intercept a message being transmitted, his or her presence will be felt as a disturbance in the communication channel. BB84 does not use quantum entanglement. Since the protocol makes it possible to generate keys which are perfectly random, it is impossible for anyone who does not know it to decode any message, even if it is sent publicly.

  • BB84 protocol

Suppose the proverbial quantum denizens Alice and Bob want to communicate privately, and Eve is interested in eavesdropping. The available means of communication are an ordinary bidirectional open channel (e.g., a telephone) and a unidirectional quantum channel. Both channels can be eavesdropped by Eve. The quantum channel allows Alice to send individual particles (say, photons) to Bob who can measure their quantum state. Eve can attempt to measure the state of these photons and can resend them to Bob. To establish the key, Alice begins by sending Bob a sequence of encoded photons. To encode the photons, she randomly uses one of the following two bases:
$$ 0 \to \left| \uparrow \right\rangle ,\;1 \to \left| \to \right\rangle \quad {\text{or}}\quad 0 \to \left| \nwarrow \right\rangle ,\;1 \to \left| \nearrow \right\rangle $$
where the arrows indicate the polarized state of the photon used to encode the binary digits 0 (up arrow or left inclined arrow) and 1 (horizontal arrow or right inclined arrow). (Physicists do have a sense of humor in picking symbols for quantum states!) Bob measures the state of the photons he gets by randomly picking either basis. After the photons have been transmitted and measured, Alice and Bob communicate over the open channel, the basis they used for coding and decoding of each photon. (This amounts to sending a string of symbols, which are meaningless without the results themselves.) On average, 50% of the time their bases will match. Alice and Bob use those photons as the key for which their bases agree and discard the other photons. So far there is no quantum advantage.

Can Eve steal the key? Suppose Eve measures the state of the photons sent by Alice and resends new photons with the measured state to Bob. However, Eve will get her measurement basis wrong, on average, 50% of the time, since Eve does not know the basis sequence used by Alice. Thus, when Bob measures a resent photon with the correct basis (Alice’s basis), there will be a 25% probability that he will measure the wrong value. This is because Eve, by measuring the photons en route, would have collapsed them to her measured value. Thus, Eve is bound to introduce a high rate of error that Alice and Bob can detect by communicating a sufficient number of parity bits of their keys over the open channel. So, not only will Eve’s version of the key will be, on average, 25% incorrect, but that someone is eavesdropping will be apparent to Alice and Bob. If eavesdropping is detected, Alice and Bob simply discard the key and send a new one. Only when both are certain that their key was not compromised do they use it for encryption. Their encrypted messages can now be sent over bidirectional open channels. Quantum cryptography’s great advantage is that it solves the key distribution problem by taking advantage of the fact that measurement of a quantum system, no matter how delicately made, causes a collapse of the system’s wave function in an unpredictable manner. There was hardly any serious mathematics involved here! Indeed, one small step in mathematics was one giant leap in cryptography!

On April 24, 2014, Nature reported, “This week, China will start installing the world’s longest quantum-communications network, which includes a 2,000-km link between Beijing and Shanghai. And a study jointly announced this week by the companies Toshiba, BT and ADVA, with the UK National Physical Laboratory in Teddington, reports ‘encouraging’ results from a network field trial, suggesting that quantum communications could be feasible on existing fibre-optic infrastructure.”14 On September 29, 2018, the Chinese satellite Micius successfully beamed down a small data packet of encryption keys encoded in photons to a ground station in Xinglong, a couple of hours’ drive to the northeast of Beijing. Within an hour Micius, as it passed over Austria, delivered another such data packet to a station near the city of Graz. “The video encryption was conventional, not quantum, but because the quantum keys were required to decrypt it, its security was guaranteed. This made it the world’s very first quantum-encrypted intercontinental video link.”15 China’s ambition is to become a global leader in secure quantum communication by 2030.

1.5 Teleportation

Teleportation is the ability to transmit the quantum state of an entity, say, a particle, using classical bits and to re-construct the exact quantum state at the receiver. In 1993, Charles H. Bennett led a group which showed how a particle of unknown quantum state can be teleported.16

For our limited purpose of teleportation, we need the Hadamard operator that operates on one qubit at a time, and the Cnot operator that acts on two qubits at a time and in the process entangles them. Note also that it is impossible to make a duplicate of a quantum entity without knowing its complete state, but we can prepare a quantum entity in as many copies as we like in a state of our choice. Teleportation allows the transfer of an unknown state of a first quantum entity to a second quantum entity but only by changing the state of the first entity. Instinctively, one perhaps realizes that teleportation may be realized by manipulating a pair of entangled particles; if we could impose a specific quantum state on one member of an entangled pair of particles, then we would be instantly imposing a predetermined quantum state on the other member of the entangled pair.

Briefly, this is how it works. Initially, two entangled photons (p2 and p3) propagate toward two remote regions of space. Photon p2 reaches Alice, while photon p3 reaches Bob. A third photon p1 in state \( \left| \phi \right\rangle \) is then provided to Alice. Alice now possesses p1 and p2. The goal is to put Bob’s photon p3 into state \( \left| \phi \right\rangle \) without transporting any photon between Bob and Alice. (Recall p2 and p3 are entangled; hence, any change in the state of one will bring about an instantaneous change in the other.)

Obviously, Alice cannot perform any measurement on photon p1 currently in state \( \left| \phi \right\rangle \) because it would destroy the state of the photon. So, she entangles the two photons p1 and p2 in her possession (this, of course, entangles all three photons) and then performs a “combined measurement” on them. (Rest assured this can be done.) She, then, communicates the result of her measurement to Bob (using classical means—telephone, email, etc.).

Bob now applies to his photon a unitary operation depending on the classical information he has received from Alice. This operation puts his photon in exactly the state \( \left| \phi \right\rangle \), the initial state of p1 and thus realizes teleportation. Since all the photons are entangled, the two photons in Alice’s possession are no longer in their original states, and hence there is no duplicate of \( \left| \phi \right\rangle \) in existence! Note that the whole operation is mixed because it involves a combination of transmission of quantum information (through the entangled state) and classical information (phone call from Alice to Bob), which cannot travel faster than the speed of light. So, teleportation cannot be done faster than the speed of light.

Let us now see the mathematics behind it. Alice has qubit p1 of unknown state \( \left| \phi \right\rangle \) = a\( \left| 0\right\rangle \) + b\( \left| 1\right\rangle \). She wishes to send the state of this qubit to Bob through classical channels. Alice and Bob, respectively, possess qubit p2 and p3 which are entangled in the state \( \left| \psi_{0} \right\rangle \) = (1/√2) (\( \left| 00 \right\rangle \)+ \( \left| 11 \right\rangle \)) (it has the special name Bell state after John Bell; see Chap. 4, Sect.  4.5). The first qubit here is p2 and the second p3. At any time, the state of the 3-qubit system is a linear combination of some 3-qubit eigenstates, each of form \( \left| xyz \right\rangle \) where x, y, z, respectively, belong to qubit p1, p2, and p3. The initial state, χ0 of our 3-qubit system is
$$ \begin{aligned} \left| {\chi_{0} } \right\rangle = \left| {\phi \,\psi_{0} } \right\rangle & = \left( {a\left| 0 \right\rangle + b\left| 1 \right\rangle } \right)\left( {1 /\sqrt 2 } \right)\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right) \\ & = \left( {1 /\sqrt 2 } \right)\left( {a\left| 0 \right\rangle \left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right) + b\left| 1 \right\rangle \left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right)} \right) \\ & = \left( {1 /\sqrt 2 } \right)\left( {a\left| {000} \right\rangle + a\left| {011} \right\rangle + b\left| {100} \right\rangle + b\left| {111} \right\rangle } \right) \\ \end{aligned} $$
of which Alice controls the first two qubits and Bob controls the third qubit. Alice now applies Cnot to the first two qubits (p1 and p2) in her possession using p1 as the control. This puts the 3-qubit system in the state (note the changes in the second qubit):
$$ \left| {\chi_{1} } \right\rangle = \left( {1/\sqrt 2 } \right)\left( {a\left| {000} \right\rangle + a\left| {011} \right\rangle + b\left| {110} \right\rangle + b\left| {101} \right\rangle } \right) $$
She now applies the Hadamard operator on the first qubit. This puts the 3-qubit system in the state
$$ \begin{aligned} \left| {\chi_{2} } \right\rangle & = \left( {1/ 2 } \right)\left( {a\left( {\left| {000} \right\rangle + \left| {011} \right\rangle + \left| {100} \right\rangle + \left| {111} \right\rangle } \right)} \right. \\ & \quad \left. { + b\left( {\left| {010} \right\rangle + \left| {001} \right\rangle - \left| {110} \right\rangle - \left| {101} \right\rangle } \right)} \right) \\ & = \left( {1/2 } \right)\left( {\left| {00} \right\rangle + \left( {a\left| 0 \right\rangle + b\left| 1 \right\rangle } \right) + \left| {01} \right\rangle \left( {a\left| 1 \right\rangle + b\left| 0 \right\rangle } \right)} \right. \\ & \left. {\quad + \left| {10} \right\rangle \left( {a\left| 0 \right\rangle - b\left| 1 \right\rangle } \right) + \left| {11} \right\rangle \left( {a\left| 1 \right\rangle - b\left| 0 \right\rangle } \right)} \right). \\ \end{aligned} $$
Alice then makes a combined measurement of the first two qubits to get one of the 2-qubit eigenstates \( \left| 00 \right\rangle \), \( \left| 01 \right\rangle \), \( \left| 10 \right\rangle \), or \( \left| 11 \right\rangle \) with equal probability. Depending on the result of the measurement, the quantum state of Bob’s entangled qubit is projected to
$$ a\left( {\left| 0 \right\rangle + b\left| 1 \right\rangle ,} \right.\,a\left( {\left| 1 \right\rangle + b\left| 0 \right\rangle ,} \right.\,a\left( {\left| 0 \right\rangle - b\left| 1 \right\rangle ,} \right.\;{\text{or}}\;a\left( {\left| 1 \right\rangle - b\left| 0 \right\rangle ,} \right.\,{\text{respectively}}. $$

Bob must now wait for Alice to send her measurement, which she does by encoding it in two classical bits using a classical communication system which cannot communicate faster than light.

Note that Alice’s measurement has irretrievably altered the state of p1 from its original state \( \left| \phi \right\rangle \), which she is trying to send to Bob. When Bob receives Alice’s 2 bits, he knows how the state of his half of the entangled pair compares to the original state of Alice’s qubit (see Table below).

Result sent

Bob’s qubit

Decoder

Output

00

a\( \left| 0 \right\rangle \) + b\( \left| 1 \right\rangle \)

I

a\( \left| 0 \right\rangle \) + b\( \left| 1 \right\rangle \)

01

a\( \left| 1 \right\rangle \) + b\( \left| 0 \right\rangle \)

X

a\( \left| 0 \right\rangle \) + b\( \left| 1 \right\rangle \)

10

a\( \left| 0 \right\rangle \)b\( \left| 1 \right\rangle \)

Z

a\( \left| 0 \right\rangle \) + b\( \left| 1 \right\rangle \)

11

a\( \left| 1 \right\rangle \)b\( \left| 0 \right\rangle \)

Y

a\( \left| 0 \right\rangle \) + b\( \left| 1 \right\rangle \)

Operator

Operation

Identity

I

\( \left| 0 \right\rangle \)\( \left| 0 \right\rangle \)

\( \left| 1 \right\rangle \)\( \left| 1 \right\rangle \)

Negation

X

\( \left| 0 \right\rangle \)\( \left| 1 \right\rangle \)

\( \left| 1 \right\rangle \)\( \left| 0 \right\rangle \)

ZX

Y

\( \left| 0 \right\rangle \) → −\( \left| 1 \right\rangle \)

\( \left| 1 \right\rangle \)\( \left| 0 \right\rangle \)

Phase shift

Z

\( \left| 0 \right\rangle \)\( \left| 0 \right\rangle \)

\( \left| 1 \right\rangle \) → −\( \left| 1 \right\rangle \)

Bob can now reconstruct the original state \( \left| \phi \right\rangle \) of the unknown qubit p1 by applying the appropriate decoder to p3 in his possession as shown in the table’s decoder column.

The amazing thing about quantum teleportation is that it permits the transfer of quantum information into inaccessible space and into a quantum memory without revealing or destroying the stored quantum information.

The following sample of technological advances related to teleportation and cryptography indicate the tremendous role they will play in the future of secure communications. Teleportation of a laser beam consisting of millions of photons was achieved in 1998. In June 2002, an Australian team reported a more robust method of teleporting a laser beam. Teleportation of trapped ions (calcium and beryllium) was achieved in June 2004 by two groups and reported in Nature.17 Teleportation of single molecules may take some time. In 2006, a team at the Niels Bohr Institute, Copenhagen, teleported information stored in a laser beam into a cloud of atoms. Thus, for the first time teleportation between light and matter was achieved. One is the carrier of information and the other is the storage medium.18 In 2014, physicists demonstrated a device that can teleport quantum information to a solid-state quantum memory over telecom fiber, a crucial capability required of any future quantum Internet.19 On June 28, 2019, researchers from Yokohama National University in Japan reported teleporting quantum information securely inside a diamond.20 Using quantum teleportation, they were able to transfer the state of a photon polarization into a carbon spin in diamond. The breakthrough could help us better share and store sensitive information.

China launched the world’s first quantum satellite Micius (mentioned in Sect. 1.4, is named after the ancient Chinese scientist and philosopher Micius) on August 15, 2016.21 At the heart of the satellite is a crystal that produces pairs of entangled photons, whose properties remain entwined; however far apart they are separated. The satellite can fire the partners in these pairs to ground stations in Beijing and Vienna, and use them to generate a secret key. In June 2017, Chinese researchers announced that they had beamed photons between the satellite Micius and the two distant ground stations, and were successful in maintaining the entangled quantum state at a record-breaking distance of more than 1,200 km.22 Quantum theory predicts that entanglement can persist at any distance.

1.6 Concluding Remarks

This chapter was meant to be an appetizer and lightly relied on your intuition to understand the mathematical steps involved. If it has made you curious, do move on to the next chapter where the real stuff begins. But before doing so, do have a quick look at a text book that describes linear algebra, complex numbers, complex matrices and their operations, and the significance of eigenvalues and eigenvectors23 of a matrix. Mathematics and computation are about abstract symbol manipulation that unintelligent computers can be programmed to do. So, turn yourself into a robot, learn these manipulations, then switch on your intelligent self, and go to Chap.  2. Imagine if you can teleport with so little mathematics, what more you might achieve if you really knew mathematics.

Footnotes

  1. 1.

    Feynman [15], Chap. 6.

  2. 2.

    Feynman [15].

  3. 3.

    Turing [31].

  4. 4.

    Wootters and Zurek [33].

  5. 5.

    As quoted in: Mermin [20]. “Einstein maintained that quantum metaphysics entails spooky actions at a distance; experiments have now shown that what bothered Einstein is not a debatable point but the observed behaviour of the real world.”

  6. 6.

    See the following three papers to develop a perspective on how physicists came to understand entanglement: Einstein et al. [14], Bell [6], and Aspect et al. [3].

  7. 7.

    All four postulates are formally described in Chap. 2, Sect.  2.7.

  8. 8.

    Schumacher [26].

  9. 9.

    See Barenco et al. [4]. See also Chap.  7 of this book.

  10. 10.

    Developed by Gilbert S. Vernam of AT&T. This is the only known totally secure cypher. Vernam was granted a patent protecting the cypher: Secret Signaling System. US Patent No. 1,310, 719, patented July 22, 1919.

  11. 11.

    Shannon [27]. See also: Goldreich [18]; Black et al. [9], pp. 189–244, Sect. 5.

  12. 12.

    Rivest et al. [25]. See also: Allenby and Redfern [2]. Rivest, Shamir, and Adleman received the Turing award for 2002 for their contributions to public key cryptography. http://www.acm.org/announcements/turing_2002.html.

  13. 13.

    Bennett and Brassard [7]. The first quantum cryptography ideas were proposed by Stephen Wiesner in the late 1960s, but unfortunately were not accepted for publication at the time! It was eventually published in 1983, Wiesner [32]. Bennett and Brassard built upon Wiesner’s work. A simple proof of the security of the BB84 protocol was provided by Shor and Preskill [28]. See also: Brassard and Crépeau [11] and Brassard [10].

  14. 14.

    Qiu [23]. See also: Muralidharan [21].

  15. 15.

    Giles [17].

  16. 16.

    Bennett et al. [8].

  17. 17.

    Riebe et al. [24] and Barrett et al. [5].

  18. 18.

    Polzik et al. [22].

  19. 19.

    Bussieres et al. [12].

  20. 20.

    Tsurumoto et al. [30].

  21. 21.

    See, e.g., Alba [1] and Gibney [16].

  22. 22.

    Castelvecchi [13].

  23. 23.

    For an easy to understand reference for eigenvalues and eigenvectors, see Chap.  6, http://math.mit.edu/~gs/linearalgebra/ila0601.pdf in Strang [29].

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Acadinnet Education Services IndiaBangaloreIndia

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