Quantum information theory by seny kamara


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Quantum Information Theory by Seny Kamara

Quantum information theory provides a foundation for such topics as quantum cryptography,
quantum error-correction and quantum teleportation. This paper seeks to provide an introduction
to quantum information theory for non-physicists at an undergraduate level. It covers basic
concepts in quantum mechanics as well as in information theory, and proceeds to explore some
results such as Von Neumann entropy, Schumacher coding and quantum error-correction.

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Quantum information theory by seny kamara

  1. 1. Quantum Information Theory ∗ Seny Kamara Dept. of Computer Science Purdue University December 10, 2001 Abstract Quantum information theory provides a foundation for such topics as quantum cryptography, quantum error-correction and quantum teleportation. This paper seeks to provide an introduc- tion to quantum information theory for non-physicists at an undergraduate level. It covers basic concepts in quantum mechanics as well as in information theory, and proceeds to explore some results such as Von Neumann entropy, Schumacher coding and quantum error-correction.1 IntroductionQuantum information theory is a new field that attempts to quantify and describe quantum me-chanical resources and the processes that act on them. In a short amount of time its study hasproduced many results that have had a strong impact on both quantum computing and the foun-dations of quantum mechanics. In particular, the discovery of quantum error-correcting codes hasmade the feasibility of an experimental quantum computer much more likely, and the study ofquantum communication channels and of quantum measurement has provided useful insight intothe laws of quantum mechanics. Section 2 offers an introduction to quantum mechanics, section 3 presents some basic conceptsin information theory and finally section 4 goes over some results from quantum information theory.2 Quantum MechanicsQuantum mechanics is the biggest hurdle for non-physicists interested in quantum computing. Thissection seeks to provide a brief introduction to some elements of quantum mechanics that are neededto understand quantum information theory. For a more thorough treatment see [1, 2]. Quantum mechanics is a theoretical framework with which physicists are able to describe theworld of sub-atomic particles. While the description that quantum mechanics offers is universallyaccepted as correct, the explanation it offers is the subject of many debates. The difficulty seems tolie in its understanding and interpretation rather than it use. The controversy largely stems from ∗ Current address: seny@cs.jhu.edu
  2. 2. the fact that to accept quantum theory as true, we need to accept properties about nature thatare very unintuitive. However we cannot dismiss it as it consistently produces the most accuratepredictions experimental predictions.2.1 Quantum StatesIn quantum mechanics, we study sub-atomic particles such as protons, neutrons, photons andelectrons. These quantum systems are differentiated by their degrees of freedom which are charac-teristics such as polarization, spin, and energy levels. We describe a particle by its state. The stateof a particle encompasses all of the information about a particle such as position, polarization, spinand momentum. In the mathematical abstraction of quantum mechanics this state is representedas a vector in a complex vector space. This vector is called a state vector, and this complex vec-tor space is a Hilbert space. Usually a particle Ψ’ s state vector is written using Dirac’s bra-ketnotation as |Ψ . Since a quantum system’ s state encompasses all its information, the first thing we have to doin order to gather information about it is to define what it is we want to know. Mathematically,this is done by defining a set of mutually orthogonal vectors in the system’ s Hilbert space that willform a basis. We then use this basis to describe, or to project, the system’s state vector onto itsbases. For example if we were interested in a system that can be in one of three position, we woulddefine a basis comprised of three vectors, each one describing one of the the system’ s possiblelocations. The state vector would then be described by a linear function - a vector - of these threebase vectors. So the dimension of a particle’ s state space is equal to its number of base states.2.2 SuperpositionSuperposition is is one of the most puzzling properties of quantum systems. It is the propertythat a quantum system can be in two or more distinct states. Not somewhere in between states,but possibly in all states. As previously mentioned, a quantum state can represent anything frompolarization to spin to position. So what this implies is that if we were interested in the position ofa particle, and this particle was in a superposition of different position states, we would find thatit was in multiple places at once. Using Dirac’ s bra-ket notation, a state |Ψ that is in a superposition of two states |φ and |ϕis expressed as |Ψ = α|φ + β|ϕ , α, β ∈ C (1) The terms α and β are called probability amplitudes and are further explained below.2.3 MeasurementsMeasurements play a very important role in quantum mechanics. When dealing with a quantumsystem, we do not know what state it is in until we make a measurement. For example, measuringa system |Ψ in a two dimensional state space spanned by two mutually orthogonal states |φ and|ϕ , as in state 1, we would find that |Ψ as either: |Ψ = |φ (2) 2
  3. 3. or: |Ψ = |ϕ (3) or in a superposition of the base states: |Ψ = α|φ + β|ϕ (4) When a system is in a base state as in state 2 or state 3, the act of measurement will always reveal|φ and |ϕ . However if the particle is in a superposition of states, as in state 4, a measurementwill result in either one of the base states φ or ϕ with probability α2 and β 2 respectively. Becauseit has to be either one, the probabilities α2 and β 2 have to sum to one.2.4 Probabilities and Probability AmplitudesQuantum mechanics is a statistical theory. This means that the information we have about aparticle is stochastic. For a state such as state 4, all we know before measurement is that |Ψ isin state |φ with probability α2 and in state |ϕ with probability β 2 . If α = 1 and β = 0 then weknow with certainty that |Ψ = |Φ . The terms α and β, which are the components of the state vector, are the amount of the basestates that are present in |Ψ . These terms are called probability amplitudes and are complexnumbers. Squaring them results in probabilities and their sum has to equal to unity.2.5 Pure and Mixed StatesWe say that a system is in a pure state if we know what its state is with certainty; and we say thatit is a mixed state if we do not. So you might ask what the difference is between a mixed state and a superposition of states. Asuperposition is a consequence of measurement while a mixed state is a consequence of our ignorance(much like a normal statistical scenario). This is a subtle difference that can sometimes lead toconfusion. For example, let’ s take a pure state such as |Ψ . We know |Ψ ’ s state with certaintysince the probability amplitude associated with it is equal to one. Now suppose we change our basis.You can think of this process in mathematical terms as a simple basis rotation or in more intuitiveterms as a change in what we want to know about our system (Gram-Schmidt decomposition). Inour new basis, |Ψ might now look more like state 4. But |Ψ ’ s state did not change, only the basischanged, so we still know |Ψ ’ s state, which makes it a pure state even though it is a superposition.The fact that we are now expressing |Ψ as a superposition only tells us that if we measure it inthis basis, we will get different results with certain probabilities, however the state of |Ψ itself isstill well defined. A mixed state however means that we do not know exactly what the state of asystem is, regardless of the basis we choose to measure it in. So say we know that |Ψ ’ s state isequal to |φ with probability Pφ and |ϕ with probability Pϕ , we are simply ignorant of which it is.|φ and |ϕ do not even have to be basis states and could be superpositions themselves.2.6 QubitsIn quantum information and computing we use two-state quantum systems. We label one state |0and the other |1 and call them qubits. For example, if we use a photon as a qubit, we label a 3
  4. 4. vertical polarization as |1 and horizontal as |0 . These qubits are analogous to classical bits exceptthat they have the property of superposition and entanglement.2.7 Bra-Ket (Dirac) NotationDirac notation is the standard notation used in quantum mechanics and quantum computing, so itis important to understand and recognize its meanings. While confusing at first, it is quite usefuland intuitive once understood. The first element of Dirac notation is the ket: |∗ . This is used to represent vectors in Hilbertspace so we use it to represent our quantum states. Since it describes a vector, it is equivalent to .all the other representations of vectors, namely the column vector or matrix representation: . .. To every ket is associated a bra: ∗|. This is a vector’ s row column equivalent after complexconjugation: [· · · ]∗ . The purpose of all this is to be able to define an inner product, which in turn enables us todefine a vector space. Using kets and bras, the inner product can now be defined as a bra-ket: ∗|∗ To see why, suppose we have two systems: |Ψ = α|0 + β|1 (5) |Ω = γ|0 + δ|1 (6) they can both be represented as column vectors: α |Ψ = (7) β γ |Ω = (8) δ Now to compute their inner product we invert |Ω to its bra and multiply them: α Ω|Ψ = [γ ∗ δ ∗ ] (9) β Ω|Ψ = γ × α + δ∗ × β ∗ (10) As usual, if two vectors are orthogonal, their inner product will vanish and if they are equal itwill be equal to 1. In addition note that bras and kets do not commute so |Ψ Φ| is different than Ψ|Φ .2.8 Tensor Products and Qubit RegistersUntil now we have dealt with single qubits. But suppose we have an ensemble of qubits - a register- and we wish to find the state of the entire register. To do this we use the tensor product (⊗),also referred to as Kronecker product. What this enables us to do is build a new state out of manysmaller states. When building a register of n bits. Each bit is a two dimensional space and theregister is a 2n dimensional space. So given two two-dimensional states |Ψ and |Φ defined as: 4
  5. 5. |Ψ = α|0 + β|1 (11) |Φ = γ|0 + δ|1 (12) their tensor product is: |Ψ ⊗ |Φ = αγ|00 + αδ|01 + βγ|10 + βδ|11 (13) which is a 22 dimensional space.2.9 Linear OperatorsThe way we describe the evolution of a quantum system is with linear operators. These operatorstake a quantum system from one state to another. We express them as matrices - Hermitianmatrices to be precise. So for example, applying an operator P to a system |Ψ we get: P |Ψ1 = |Ψ2 (14) More explicitly defining |Ψ and P as follows: |Ψ = α|0 + β|1 (15) 0 1 P = (16) 1 0 we get: 0 1 α P |Ψ = (17) 1 0 β β P |Ψ = (18) α The operator P, which inverses a vector, is commonly referred to as the Pauli-X operator. In quantum computing we concern ourselves with linear operators. This means that they satisfythe following properties for both kets and bras: P a|Ψ = aP |Ψ (19) P (α|Ψ + β|Φ ) = αP |Ψ + βP |Φ (20)2.9.1 Pauli OperatorsThe Pauli operators are a set of four operators that are very common in quantum mechanics andquantum computing. They are defined as follows: 5
  6. 6. 0 1 σx = (21) 1 0 0 −i σy = (22) i 0 1 0 σz = (23) 0 −1 1 0 I = (24) 0 12.9.2 Hadamard GateThe Hadamard gate is usually used to change a system to a superposition of states. It is definedas follows: 1 1 1 H=√ (25) 2 1 −1 1 1 and it brings |0 to √ 2 (|0 + |1 ) and |1 to √ 2 (|0 − |1 ).2.9.3 Projection OperatorsAnother useful type of operator are the projection operators. A projection operator simply projectsa vector onto its base states. Say we have a state |Ψ that we want to project onto a basis definedby vectors |0 and |1 we define the projection operator as: P = |0 0| + |1 1| (26) if we apply it to Ψ we get: P |Ψ = 0|Ψ |0 + 1|Ψ |1 (27) Now assuming that |Ψ looks like state 5 in the basis defined by |0 and |1 , the inner productof |Ψ and base state |0 will yield: 0|Ψ = α 0|0 + β 0|1 (28) 0|Ψ = α (29) Since |0 and |1 are base states, they are mutually orthogonal so their inner product vanishes. So the inner product of |Ψ and |0 returns the probability amplitude that |Ψ associates with|0 . Obviously the same is true for the inner product of |Ψ and |1 . So equation 27 gives: P |Ψ = α|0 + β|1 (30) So we have a taken a state |Ψ and projected it on a basis defined by |0 and |1 . 6
  7. 7. 2.9.4 Density OperatorsDensity operators, also referred to as density matrices, are a way of representing ensembles ofquantum states. By ensemble we mean a system comprised of other quantum systems. Given nqubits each in a state |Ψi , i ∈ {1 . . . n} and each with probability pi of being selected form theensemble, we define the density matrix as: ρ= pi |Ψi Ψi | (31) i So if we have two systems |Ψ and |Φ defined as: |Ψ = α|0 + β|1 (32) |Φ = γ|0 + δ|1 (33) and with a probability function P = { 1 , 1 }, their density matrix is: 2 2 1 ∗ ∗ α 1 γ ρ = [α β ] + [γ ∗ δ ∗ ] (34) 2 β 2 δ 1 α∗ α α∗ β γ∗γ γ∗δ ρ = ∗α β∗β + (35) 2 β δ∗γ δ∗δ α2 +γ 2 α∗ β+γ ∗ δ ρ = 2 2 (36) β ∗ α+δ ∗ γ β 2 +δ 2 2 2 So what this matrix is giving us is the probability of measuring a certain state if we were toblindly pick a qubit from the ensemble and measure it.2.10 EntanglementEntanglement is a correlation that can exist between two quantum systems. This correlation hasbeen at the center of one of the most important debates in modern physics. In 1935 Einstein,Podolsky and Rosen published a paper describing what they felt was a paradox in the theoryof quantum mechanics [3]. This paradox, known as the EPR paradox, used entanglement to setquantum mechanics against special relativity. The outcome of this debate (which some still contendis not over) was the acceptance of the fact that our reality is non-local. This conclusion hasramifications not only in physics but in philosophy as well. More can be found in [4, 5]. Entanglement is one of, if not the most, important resource that quantum mechanics has tooffer us in terms of computing. Even more so than superposition. Besides being a major componentof quantum algorithms such as Shor’s factoring algorithm [6] and Grover’s search algorithm [7], itis the main reason we are able to conduct quantum teleportation [8], quantum key distribution[9, 10, 11] and quantum error-correction [12]. We say that two states are entangled when the outcome of a measurement on one, is dependenton the outcome of a measurement on the other. Given a two qubit state |Ψ defined as: |Ψ = α|01 + β|10 (37) 7
  8. 8. we can see that if we conduct a measurement on the first qubit and find that it is in state |0 ,then the second qubit can only be in state |1 . Similarly, if we obtain |1 on the first measurement,then the second qubit can only return |0 when measured. This is called an anti-correlated entangledstate since the state that we will obtain upon measuring the second qubit will always be the oppositeof what we obtained when measuring the first one. There are other kinds of entanglement such as: |Ψ = α|00 + β|11 (38) which are not anti-correlated, but correlated nonetheless. In addition, we can entangle morethan two qubits together [13]. We can also quantify entanglement in order to find out how entangledtwo systems are. The way we describe an entangled state mathematically is that it cannot be constructed from atensor product of the qubits we wish to entangle. For example, say we wish to entangle two qubits|Ψ and |Φ defined as: |Ψ = α|0 + β|1 (39) |Φ = γ|0 + δ|1 (40) Their tensor product yields: |Ψ ⊗ |Φ = αγ|00 + αδ|01 + βγ|10 + βδ|11 (41) The difference between entangling two qubits and simply associating them is the sum of twoterms (the middle terms in a correlated pair, and the first and fourth terms in an anti-correlatedpair). Intuitively this implies that entanglement is more (or in this case less) than just the sum ofits parts.3 Information TheoryInformation theory was developed in the 50’ s by Claude Shannon [14]. He found a mathematicalmodel in which to quantify the notion of information. In addition to finding a way to measureinformation, he developed a framework with which we could use it. Today, information theory isused in many scientific and engineering fields including computer science, electrical engineering andmathematics. It is especially important in in cryptography, data compression and communications.The following sections give a brief introduction to some basic topics in information theory. For amore advanced treatment see [15].3.1 Shannon EntropyThe first concept in information theory is entropy. This is how we measure information. Ininformation theory every event is represented as a random variable whose outcomes are associatedwith a certain probability. So we say that we want to find how much information is containedin a random variable X with a probability mass function P . The information in this variableis proportional to its uncertainty. A more intuitive understanding of the relationship betweenuncertainty and information can be acquired by the following scenario: Assume you were told thattomorrow it would not rain in the desert. This statement - not raining in the desert - contains very 8
  9. 9. little information since you would expect as much. It is a very likely scenario so it has very littleuncertainty. However, if you were told that it would rain tomorrow in the desert, the statementwould contain a very large amount of information as that is a very rare and unlikely event. So given a random variable X and a probability function p(x) = P r(X = x) we define theentropy H(X) of X as: H(X) = − p(i) log(p(x)) (42) i Here we assume log is log2 . This is a measure of the uncertainty of an event as a function of the probability of its outcomes. It turns out that the entropy of a variable X is also the number of bits it takes to encode it.3.2 Mutual InformationThe mutual information is a measure of how much information one random variable possesses aboutanother. Given two random variables X and Y , with individual probability distributions p(x) andp(y) and a joint probability distribution p(x, y), we define the mutual information as: p(x, y) I(X; Y ) = p(x, y) log (43) x y p(x)p(y) This tells us the amount of information remaining in X once we know Y .3.3 Source CodingOne of the first uses of information theory was in devising ways to encode data that was to betransmitted through a communication channel. In this scenario, there are two problems we areinterested in: using the least amount of resources, and making sure that the data is not corruptedat the other end. This implies that we need to find ways to compress and correct our message.Essentially source coding is about compressing a message into the minimum number of bits. As-suming a resource is a bit, we want to find the smallest amount of bits it will take to encode anarbitrary message. Say we have a source that generates a message M comprised of n bits. In addition, the sourcehas no memory of what it has previously output, so the value of each new bit has to be independentof the ones that preceded it. Now say that each bit has a probability p of being a 0 and 1 − p ofbeing a 1 and we model the source as a random variable X. Given that we have n bits, we can assume that the number of 1’ s and 0’ s in M will be: N0 = np (44) N1 = n(1 − p) (45) We call the messages that actually contain N0 0’s and N1 1’s typical sequences and group themin a set T . They are typical because they are the most likely sequences that S could emit. As ngrows large, the probability that the source emits a typical sequence (a sequence that belongs toT ) grows closer to unity. 9
  10. 10. Since all bits are independent of each other (the source is memoryless) the probability of amessage M is: Pr [ M ] = Πn Pr [ Mi ] i=1 (46) Since the probability of a bit being a 0 is p and a 1 is 1 − p we have: Pr [ M ] = pnp × (1 − p)n(1−p) (47) Taking the log and negating both sides we get: − log Pr [ M ] = −np log p − n(1 − p) log(1 − p) (48) − log Pr [ M ] = nH(X) (49) so we know that the probability of a typical message M is 2−nH(X) . We also know that thesum of the probabilities of all the typical messages cannot exceed 1. Say nT is the total number oftypical sequences, we have: nT × Pr [ M ] ≤ 1 (50) nH(X) nT ≤ 2 (51) So we have at most 2nH(X) typical messages. Since we can assume that the source will emit a message that belongs to T , we index thesequences in T , and send the index instead of the message M . And since T has at most 2nH(X)sequences, we can encode the index in nH(X) bits. So using this method, we have encoded amessage M that was n bits long into a new message that is only nH(X) bits long.4 Quantum Information Theory4.1 Von Neumann EntropyWhile Shannon entropy (H) is used in classical information theory, quantum information theoryuses Von Neumann entropy S, defined as: S(ρ) = −trρ log ρ (52)4.2 Schumacher CodingSchumacher encoding is the quantum equivalent of noiseless source coding. The overall idea issimilar to its classical counterpart, however instead of using typical sequences to encode messages,quantum source coding uses typical subspaces. Given a quantum source emitting n quantum systems each in state |Ψi with probability pi ,we wish to compress the sequence. The emitted systems are letters and the entire sequence is amessage. In order to perform Schumacher encoding, however, the states emitted by the quantumsource (i.e. the letters) must be pure and mutually orthogonal. For a sequence of n qubits each with a probability pi to be in a pure state |Ψi , we define amessage M as: 10
  11. 11. M = |Ψ1 ⊗ |Ψ2 ⊗ |Ψ3 ⊗ · · · ⊗ |Ψn (53) This message is in a space H ⊗ that is derived from applying the tensor product to the individualspaces of the qubits that compose it: H ⊗ = H1 ⊗ H2 ⊗ H3 ⊗ · · · ⊗ Hn (54) In addition M has density matrix: ρ= ρi (55) i where ρi is the density matrix of |Ψi . We define a likely message as one that, for a total of n qubits, has mi qubits in state |Ψi , with: mi = pi n (56) We further define the set of likely messages as λ: the set of all messages that have mi qubits instate |Ψi and the set of unlikely messages γ = ¬λ. Following a similar reasoning as in the classicalsource coding, we know that there are δ likely messages with: δ = 2nS(ρ) (57) Since each letter can be described by at least S(ρ) qubits, and the message is n letters long, weneed at least nS(ρ) qubits to describe a message. So we have a set λ that includes the δ messages that are most likely to appear from the quantumsource and a set γ that includes all the other messages. We index each message in λ from 1 to δ.Now since the quantum source can only emit pure orthogonal states, all the messages are mutuallyorthogonal. This is because the message space H ⊗ was built from smaller qubit spaces that wereall mutually orthogonal. So now we can use the messages to define two new bases Λ and Γ whichare subspaces of the message space H ⊗ . In addition: dim(Λ) = δ (58) ⊗ dim(Γ) = dim(H ) − dim(Λ) (59) So in order to compress the message emitted by our quantum source, we take the entire messageas an ensemble and measure it in the message space H ⊗ . Since the subspace Λ is comprised ofthe most likely messages, a measurement on an unknown message will probably result in one ofour likely messages. Since we have indexed this entire set, we simply send the index instead of themessage, and since there are δ sequences in the likely set, we can encode any index in log2 δ qubits.So we have effectively compressed our message of n states to log2 δ qubits. And we have seen thatthe maximum compression one can achieve on an ensemble of distinguishable states is simply theVon Neumann entropy of that sequence. So Schumacher encoding gives an information theoreticmeaning to the Von Neumann entropy. It is the information contained in a sequence. 11
  12. 12. 4.3 Holevo Information (χ)The amount of information in a sequence of distinguishable (i.e. orthogonal) states is equal to theVon Neumann entropy of that sequence. However what is the information content of a sequence ofnon-distinguishable states? This is given by the Holevo information χ: χ( ) = S(ρ) − pi S(ρi ) (60) i There are two properties about χ that should be emphasized: 1) because distinguishability (orthogonality) is a special case of non-distinguishability (non-orthogonality), the Holevo information reduces to the Von Neumann entropy for distinguishablestates. 2) the Holevo information of non-distinguishable states is less than the Holevo information ofdistinguishable states. Besides giving us a more general measure of entropy, the Holevo information also provides anupper bound on the mutual information that is available about quantum systems. For a moredetailed explanation see [16, 17].4.4 Quantum Error CorrectionUp until now everything we have covered was idealized and devoid of any practical limitations andimperfections. Unfortunately over time quantum systems have a tendency to interact with theirenvironment. This interaction leads to errors. So in order to perform non-trivial calculations weneed to isolate the components of our quantum computer from its environment. Of course this isimpossible but there is a lot of research being done to find ways to circumvent this interaction.One of them is quantum error-correcting codes.4.4.1 Error ModelsThe first step in building good error-correcting codes is to understand what kinds of errors we arefaced with. In quantum computation we divide the possible errors in two groups: dissipation anddecoherence. The first is due to a quantum system spontaneously emitting energy. If using anatom as a qubit, with the excited state as |1 and the non-excited state as |0 , dissipation couldlead the atom to spontaneously emit a quanta of energy and drop a level. This is equivalent to ourqubit spontaneously changing from |1 to |0 , which is a bit flip. Decoherence on the other handcauses a quantum system to become entangled with its environment and can lead to changes in thequbit’ s phase. So there are three possible kinds of errors: phase shifts, bit flips and a combinationof both. Any possible error can be decomposed into a combination of these basic error types. Wemodel these basic error types mathematically using the Pauli matrices from section 2.9.1 For example given a state |Ψ : |Ψ = α|0 + β|1 (61) applying the Pauli-X operator to |Ψ we get a bit flip: |Ψx = α|1 + β|0 (62) 12
  13. 13. and if we apply the Pauli-Z operator we get a phase shift: |Ψz = α|0 − β|1 (63) Similarly we could apply the Pauli-Y operator to obtain a bit flip and a phase shift or theIdentity operator to get no error. So when we need to reproduce the effect of an error on a single qubit, we simply apply a linearcombination of the error operators on that qubit. And in order to apply errors to a register of qubits we just calculate the tensor product of theerror operator with the Identity matrices. For example, say we want to apply a bit flip to the thirdqubit of a three qubit register. We would first compute the correct bit flip operator: Ef lip = I ⊗ I ⊗ σx (64) and then apply it to the register.4.4.2 Three Qubit Bit Flip Error-Correcting CodeThe three qubit code is the simplest quantum error-correcting code. Many more sophisticatedones have been designed, but this one illustrates the underlying concepts well. This scheme usesredundancy to protect one qubit against bit flip errors by encoding each bit as three entangledphysical qubits: |0 L = |000 P (65) |1 L = |111 P (66) α|0 L + β|1 L = α|000 P + β|111 P (67) Now whether we encoded our original qubit as state 65, 66 or 67, there are only four scenariospossible: either none, bit one, bit two or bit three gets flipped. Assuming our original bit was asuperposition, as state 67, after a bit flip, we would be left with one of the following states: |ΨA = α|000 P + β|111 P (68) |ΨB = α|100 P + β|011 P (69) |ΨC = α|010 P + β|101 P (70) |ΨD = α|001 P + β|110 P (71) Now we define four projection operators as in section 2.9.3: P0 = |000 000| + |111 111| (72) P1 = |100 100| + |011 011| (73) P2 = |010 010| + |101 101| (74) P3 = |001 001| + |110 110| (75) 13
  14. 14. These projections are non-destructive and will not cause a superposition to collapse, so we canapply each one sequentially on the logical qubit. Assuming that the qubit was in state 68, meaningthat there were no errors, applying P0 results in: Ψ|P0 |Ψ = (α 000| + β 111|) · (|000 000| + |111 111|) · (α|000 + β|111 ) (76) Ψ|P0 |Ψ = 1 (77) The set of three qubit systems {|000 , |001 , . . . , |111 } are all mutually orthogonal, so theirinner product vanishes. This means that we will obtain 1 when we apply P0 to a qubit that has noerror, and 0 when we apply it to any other qubit; similarly we will obtain 1 when we apply P1 toa qubit whose first bit is flipped, and 0 when we apply it to another and so on and so forth. Thisresult from the projector is called the error syndrome and it tells us where the error occurred. Allwe have to do now is flip the appropriate bit. Note that using this scheme, we only know wherethe error occurs, but we cannot determine what the original value was.4.4.3 Three Qubit Phase Flip Error-Correcting CodeIn this section we describe a way to correct a phase flip. A phase flip, represented by the Pauli-Yoperator flips the relative phase between the base states of a system. A phase flip is a purelyquantum mechanical effect and has no classical equivalent. This means that it affects only qubitsthat are in a superposition. It acts as follows: |0 + |1 → |0 − |1 (78) To correct this we encode each logical qubit as a superposition of physical states using aHadamard gate. This gives: 1 |0 L = √ (|0 + |1 ) (79) 2 1 |1 L = √ (|0 − |1 ) (80) 2 1 1 1 (|0 + |1 )L = √ √ (|0 + |1 ) + √ (|0 − |1 ) (81) 2 2 2 The effect of a phase flip (applying a Pauli-Y operator) on these new qubits is: 1 1 √ (|0 + |1 ) → √ (|0 − |1 ) (82) 2 2 1 1 √ (|0 − |1 ) → √ (|0 + |1 ) (83) 2 2 1 1 1 1 1 1 √ √ (|0 + |1 ) + √ (|0 − |1 ) →√ √ (|0 + |1 ) − √ (|0 − |1 ) (84) 2 2 2 2 2 2 14
  15. 15. 1 Now if we label state 79 as | + ++ , state 80 as | − −− and state 81 as √2 (| + ++ + | − −− ),we find that the phase flip simply flipped | + ++ to | − −− and vice versa. This means that nowwe can apply the three qubit bit flip code on | + ++ and | − −− .References [1] R. Shankar. Principles of Quantum Mechanics. Plenum Press, New York, 1994. [2] A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1995. [3] A. Einstein, B. Podolsky, and N. Rosen. Can quantum mechanical description of reality be considered complete? Physical Review, 47:777, 1935. [4] J. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1(3):195–200, 1964. [5] J. Bell. Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, 1993. [6] P. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997. [7] L. Grover. Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters, 79(2):325–328, July 1997. [8] C. Bennet, G. Brassard, C. Crepeau, R. Josza, A. Peres, and W. Wooters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70:1895–1899, 1993. [9] C. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers Systems and Signal Processing, pages 175–179, 1984.[10] C. Bennet. Quantum cryptography using any two nonorthogonal states. Physical Review Letters, 68(21):3121–3124, May 1992.[11] A. Ekert. Quantum cryptography based on bell’s theorem. Physical Review Letters, 67(6):661– 663, August 1991.[12] P. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52:2493–2496, 1992.[13] D. Greenberger, M. Horne, and A. Shimony. Bell’ s theorem without inequalities. American Journal of Physics, 58:1131, 1990.[14] C. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379– 423, 1949.[15] T. Cover and J. Thomas. Elements of Information Theory. Wiley & Sons, 1991.[16] J. Preskill. Caltech Physics 229 Lecture Notes. http://www.theory.caltech.edu/people/preskill/ph229/.[17] M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. 15