Chapter 1 Brief overview 1.1 Linear Alzebra 1. Tensor Product: A Tensor product is a method of multiplying two tensors (matrices), given by the general (pneumonic) form: (It is represented as ) ⊗ a11 a12 a13 a14 ... a11M a12M a13M a14M ... a a a a ... a M a M a M a M ... 21 22 23 24 21 22 23 24 a31 a32 a33 a34 ... a31M a32M a33M a34M ... a a a a ... M = a M a M a M a M ... 41 42 43 44 ⊗ 41 42 43 44 ....... ....... ....... ....... ....... ....... where, M is any matrix. If the matrices on the LHS have the dimensions (D D ) and (D D )respectively, 1r × 1c 2r × 2c then the result matrix, on the RHS will have the dimensions: [(D + D ) (D + D )] DRAFT COPY1r 2r × 1c 2c It is really important to note that the above is NOT a formal definition for a tensor product, but it is just a pneumonic form. The result in this case seems to have the same dimensions as the original matrix in the LHS (first matrix). Only when we expand the RHS by putting various values of M,we will get the matrix of the dimensions (as given above). 1.2 Basic Quantum mechanics We shall look at fundamentals of quantum mechanics in greater detail in the next chapter. Some important definitions are: 1. State of a system:1 The state of a quantum system is a vector in the infinite dimensional complex vector space known as the Hilbert Space. The state of a classical system is represented by its position and momentum in a phase space. But this is not possible in Quantum Mechanics because of the built in concept of the Heisenberg’s Uncertainty principle. So, if we close in into one definite value of position for a position, the possible values that its momentum can take is infinite. So, we can only close on a given area, and say , with some probability that the particle lies within that area. So, the (classical) state of the particle lies anywhere 1Adeeperpictureofthisisgiveninthequantummechanicssection.Fornow,thisdescriptionwoulddo. 9 1.3. QUBIT - BASIC UNIT OF QUANTUM INFORMATION CHAPTER 1. BRIEF OVERVIEW in that continuous area that we define. So, the state of the particle has infinite position and momentum coordinates. Hence, it is represented as a vector in an infinite dimensional complex vector space (Hilbert Space). The state of a N-independent particle system where the individual particle wave functions are ψ , ψ , ψ ... ψ , the combined state of the N particle system is given by: | 1 | 2 | 3 | N ψ = ψ ψ ψ ...... ψ | | 1⊗| 2⊗| 3 ⊗| N In general, if ψ1 , ψ2 , ψ3 , ..... are vectors in N1, N2, N2, ..... dimensional complex vector spaces respectively, then| ψ| is a| vector in a (N + N + N + .....) dimensional complex vector space. | 1 2 2 2. Dirac Bra-c-Ket Notation: The dirac ket notation is a well known and frequently used here. In this notation, every column vector −→ψ is represented as: ψ which is also called a Ket vector. Similarly, a row vector is represented as ψ which is called a Bra| vector. Hence the name: BracKet notation. | 3. Local and non Local processes: By “local ”process between two particles , it is meant that influ- ences between the particles must travel in such a way that they pass through space continuously; i.e. the simultaneous disappearance of some quantity in one place cannot be balanced by its appearance somewhere else if that quantity didn’t travel, in some sense, across the space in between. In particular, this influence cannot travel faster than light, in order to preserve relativity theory. 4. Canonical Commutation Relations: The some observables in Quantum mechanics do not commute. That is, they have a non zero commutator. The commutator for a pair of operators is defined as: [A, B]=AB BA ——— Commutator − The basic commutator relations in quantum mechanics are: [xi,pj]=i [xi,xj]=0 [pi,pj]=0 1.3 Qubit - basic unit of quantum information A ’bit’, a classical two state system, represents the smallest unit of information. A classical bit is represented by 1 or 0. It can be thought of as ’true’ and ’false’ or any two complementary quantities whose union is the universe, and intersection is the null set. There is a profound reason to why the smallest unit of information DRAFTis a 1 or 0, or ’true’ or ’false’. This is because any logical querry COPY can be split into a series of ’yes’ or ’no’ questions. That is, with a series of ’yes’ or ’no’ answers to questions, we can perform any logical query. This is why a ’bit’ (which can be thought of as a most general strcture of storing information) is a 1 or a 0. A quantum bit, is just an example for a two state quantum system. A qubit can can also be an electron (with spin up and down), an ammonia molecule, etc. In quantum mechanics, a two state quantum system does not mean that it has only two states. This is what distinguishes a qubit from a classical bit. The difference comes due to a very important quantum mechanical phenomena known as interferrence.Justlikehowa classical bit can take a value 0 or 1, a quantum bit can take values 0 or 1 , and any value produced by the interferrence between the states 0 and 1 (like α 0 + β 1 ). Since,| there| are infinite such superpositions (where each of the states have 0 |and 1 given| by some| probability| amplitude α and β respectively), a qubit can exist in infinite states. If each state can store a unit of information, then the qubit can hold infinite units of information. A Qubit, like any other two level quantum system, is (conventionally) represented by its state: ψ = α + β | |↑ |↓ ψ = α + + β | | |− ψ = α 0 + β 1 | | | where α 2 + β 2 =1. | | | | GO TO FIRST PAGE 10 ↑ CHAPTER 1. BRIEF OVERVIEW 1.4. MULTIPLE QUBITS This can be misleading, since it gives us the feeling that a classical bit can at most carry 2 units of information whereas a quantum bit can carry infinitely many units. But, if a measurement is done on the state of the Qubit, it collapses into one of the eigen states of the measurement. So, if measurement of ψ gives ’a’, then after this measurement, the state of the Qubit will remain a (the eigen state of the measurement| corresponding to the eigen value a). This new state a now, will got give| the same measurement results as a . In fact it will not respond to any other measurement.| Why this happens Postulates| of Quantum Mechanics. Hence, only a single→ unit of information can be retrived from a Qubit. 1.4 Multiple Qubits Any two state system (like the electron which has a spin) can be represented by a Qubit. But what about representing the state of two electrons (which are independent of each other), using a qubit? Such a repre- sentation, we saw in the beginning of the chapter, was possible. If the state of one electron is ψ and state | 1 of the other electron is ψ2 , then the system of two independent electrons can be collectively represented by the state ψ ,where: | | ψ = ψ ψ | | 1⊗| 2 Since, we take a direct product of the two states, we may represent the now state, which is the two qubit state, as: (using the convention: i j ij ) | ⊗| ≡| ψ = α 00 + α 01 + α 10 + α 11 | 00| 01| 10| 11| where : α 2 + α 2 + α 2 + α 2 =1 | 00| | 01| | 10| | 11| 2 and αij is the probability of the first qubit being in state i , and the second Qubit being in state j .If we want| | only one of them, we must sum over the other. If we| want the probability of system being in| state 2 i only, we must sum αij over all j’s. | | | 2 2 So, the probability of measuring the first qubit to be 0 is = α00 + α01 , and obviously, the measurement 2 | 2| | | α00 + α01 will collapse the state of the qubit into ψ = | | | | . | α 2 + α 2 | 00| | 01| Therefore, we can say that in ths 2 qubit system, we can retrieve 2 units of information. It certainly DRAFTcan deliver more information than a single qubit system, but COPY there are some difficulties too. In general, we need to carry the measurement process twice to determine the information stored in both the qubits. So, earlier we were carrying out the measurement only once, and now we need to do it twice. Can it be better? Can we get away in one measurement itself? In other words, can we store some amount of information about one qubit in another, such that we can guess both the qubits, by measuring only one of them? The answer is yes. We can do such a trick that can, with certainty, retrieve the information stored in one qubit, by measuring the other. Such a two qubit state is called a Bell State or the EPR pair. 00 + 11 The Bell State is given by : ψ = | | | √2 1 Here, the first Qubit being is measured to be 0 with the Probability (changing the state to : ψ = 00 ) 2 | | 1 and 1 with Probability (Changing the state to : ψ = 11 ) 2 | | Hence, P(measuring the first qubit to be 0) = P(measuring the second qubit to be 0) , also state after the measurement of the first qubit to be 0 = state after the measurement of the second qubit to be 0.