Lecture 2: Basics of Quantum Mechanics Scribed by: Vitaly Feldman Department of Mathematics, MIT September 9, 2003 In this lecture we will cover the basics of Quantum Mechanics which are required to under-stand the process of quantum computation. To simplify the discussion we assume that all the Hilbert spaces mentioned below are finite-dimensional. The process of quantum computation can be abstracted via the following four postulates. Postulate 1. Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. The state of the system is completely described by a unit vector in this space. Qubits were the example of such system that we saw in the previous lecture. In its physical realization as a polarization of a photon we have two basis vectors: |�� and |↔� representing the vertical and the horizontal polarizations respectively. In this basis vector p olarize d at angle θ can be expressed as cos θ |↔� − sin θ |��. An important property of a quantum system is that multiplying a quantum state by a unit iθcomplex factor (eiθ ) yields the sam e complex state. Therefore e |�� and |�� represent essentially the same state. Notation 1. State χ is denoted by χ� (often called a ket) is a column vector, e.g., |⎛ ⎞1/2 ⎝ i√3/2 ⎠ 0 χ�† = (often called a bra) denotes a conjugate transpose of χ�. In the previous example we | �χ|would get (1/2, −i√3/2, 0). It is easy to verify that �χ χ� = 1 and|�x| |y� ≤ 1. Postulate 2. Evolution of a closed quantum system is described by a unitary transformation. If ψ� is the state at time t, and ψ�� is the state at time t�, then ψ�� = U ψ� for some unitary operator | | | |U which depends only on t and t�. Definition 1. A unitary operator is a linear operator that takes unit vectors to unit vectors. For every ψ, �ψ U †U ψ� = 1 and therefore U †U = I. Here by A† we denote the adjoint operator | |of A, that is, the operator that satisfies (�x A†)† = A x� for every x.| |Definition 2. A Hermitian operator is an operator that satisfies A† = A. 1� � � � � � � � � � 2 P. Shor – 18.435/2.111 Quantum Computation – Lecture 2 Commonly used operators on qubits are Pauli matrices I, σx, σy , σz and Hadamard transform H described as follows. 1 0�+σx = 0 Maps: |0� → |1�; 0�; |√2|1� �1 0 |1� → |σy =0 −i Maps: |0� → i |1�;i 0 |1� → −i |0� 1 0 σz =0� −1 Maps: |0� → |0�; |1� → −|1� 1 1 Maps: |0� → |0�+√2|1� ; |1� → |0�−|1�√2 H = √12 1 −1 Postulate 2 stems from Sr¨odinger equation for physical systems, namely di� |ψ� = H|ψ�dt where H is a fixed Hermitian operator known as the Hamiltonian of a closed system. Postulate 3. Quantum measurements are described by a collection {Mm} of measurement opera-tors. These are operators acting on a state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is ψ� immediately before the measurement then the probability that result m occurs is given by |p(m) = �ψ Mm†Mm ψ� ,| |and the state of the system after the measurement is Mm� |ψ� . �ψ|Mm†Mm |ψ� The measurement operators satisfy the completeness equation, Mm†Mm = I . m The completeness equation expresses the fact that probabilities sum to one: 1 = p(m) = �ψ|Mm†Mm ψ� .|m m We will mostly se e the following types of measurements. Suppose v1�, v2�, . . . , vd� form an | |orthonormal basis. Then {Mi = |vi��vi|} is a quantum measurement. From state |ψ� in this |measurement we will obtain vi|vi�� |ψ� with probability |�vi. vi|� |ψ�| |ψ�| 2 Definition 3. A projector is a Hermitian matrix with eigenvalues 0 and 1. The subspace with eigenvalue 1 is the subspace associated with this operator.� � � � � � � � � � � � 3 P. Shor – 18.435/2.111 Quantum Computation – Lecture 2 Suppose S1, S2, . . . , Sk are orthogonal subspaces that span the state space. Then {Pi} is a quantum measurement where Pi is the projector onto Si. We can write = α1 ψ1� + α2 + αk ψk � ,|ψ� | |ψ2� + ··· |where Then this measurement takes ψ� to ψi� with probability .|ψi� ∈ Si. | | |αi| 2 Postulate 4. The state space of a composite quantum system is the tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered 1 through n, and system number i is prepared in the state ψi�, then the joint state of the total system is ψn�.| |ψ1�⊗|ψ2�⊗···⊗|Definition 4. Let S1 and S2 be Hilbert spaces with bases e1�, . . . , ek � and f1�, . . . , fl� respectively. | | | |Then a tensor product of S1 and S2 denoted S1 ⊗S2 is a kl-dimensional space consisting of all the linear combinations of all the possible pairs of original bases elements, that is, of {|ei�⊗ fj �}i≤k,j≤l|w� is often contracted to w� or vw�).(|v� ⊗ | |v�| |In a more concrete matrix repres entation the tensor product of two vectors is the Kronecker product of vectors. For example, ⎛ ⎞ 3 5√2 ⎜⎜⎜⎝ ⎟⎟⎟⎠ 14 5√2 −3 5√2 3√2 = 1 4√2 ⊗ 5 − 5 −4 5√2 The tensor product satisfies the property that the product of two unit vectors is a unit vector. This is to verify as follows. Let v1� = ai ei� and v2� = bj fj � be two unit vectors. Then | | | |v2� = ai bj fj � = aibj ei� fj � .|v1� ⊗ | |ei� ⊗ | | |Therefore, 2 2 22 = |aibj | 2 = |bj | 2 =||v1� ⊗ |v2�| |ai| ||v1�| ||v2�| Another important property of the tensor product space is that it contains vectors which are not tensor product themselves. For example, it can be easily verified that the vector 1 ( e1� e2� f1�)√2 | |f2� − | |is not a tensor product itself. Such vectors are called