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MIT 6 453 - Lecture Notes

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MIT OpenCourseWarehttp://ocw.mit.edu 6.453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Lecture Number 2 Fall 2008 Jeffrey H. Shapiro c�2006, 2008 Date: Tuesday, September 9, 2008 Dirac-notation Quantum Mechanics. Introduction Last time you were introduced to—teased with, really—three examples of how quan-tum optical communication has distinctly non-classical features: quadrature noise squeezing, polarization entanglement, and teleportation. In this lecture, we begin laying the foundation f or understanding all three of these phenomena, and more. Our task is to present the essentials of Dirac- not ation quantum mechanics. No prior acquaintance with this material is assumed. There are three fundamental notio ns tha t we must establish: state, time evo lution of the state, and measurements. The first two will be completed in this lecture; the last will spill over into Lecture 3. Moreover, although these three concepts are easily stated, they will be accompanied by a variety of notational and mathematical details that will comprise most of today’s lecture. Quantum Systems and Quantum States Slide 3 defines a quantum system and the state of a quantum system. The first definition—that of a quantum system—requires no explanation. There are several points to be made, however, about the definition of the state of a quantum system. First, let us remember what it means to be t he state of a classical system. We’ll do so by means of two examples from classical physics, one from mechanics, and one from circuit theory. After that, we’ll review—and perhaps extend—what you know about vector spaces and linear operations on vectors. Here we will use the Dirac notation, but we also exhibit two special cases that will help illustrate the points being made. The St ate of a Point Mass The state, at time t0, of an m-kg point mass that is moving in three-dimensional space under the influence of an applied force is its position, ~r(t0), and its momentum, p~(t0). The state contains a ll information about the behavio r of the mass prior to time 1      t0 that is relevant to predicting its behavior for t > t0. In particular, if the applied force, F~(t), is known for t0 ≤ t ≤ t1, t hen ~x(t1) and p~(t1) can b e found by solving dp~(t) ~d~r(t) = F (t) and m = p~(t), for t0 ≤ t ≤ t1, (1) dt dt subject to the initial conditions t hat the position and momentum at time t0 be ~x(t0) and ~p(t0), respectively. The St ate of an RLC Circuit Consider the parallel RLC circuit shown in Fig. 1. The state of this circuit a t time t = t0 can be taken to be the charge on its capacitor, Q(t) = Cv(t), and the flux through its inductor, Φ(t) = LiL(t).1 + _ v(t)i(t) CLR iL(t) Figure 1: The state of this parallel RLC circuit can be taken to be the charge on its capacitor, Q(t) = Cv(t), and the flux through its inductor, Φ(t) = LiL(t). To find the state at some later time, we can use Kirchhoff’s current law and Kirchhoff’s voltage law—plus the v-i relations for the three circuit elements—to show that d2v(t)dv(t) di(t)RLC + L + Rv(t) = RL , for t0, (2) dt2 dt dtt ≥which can be solved, given i(t) for t0 ≤ t ≤ t1 and the initial conditions Q(t0)dv(t) i(t0) v(t0) Φ(t0) v(t0) =and = , (3) C dt C − RC − LC t=t0 to obtain v(t1) and dv(t)/dtt=t1. These, in turn, allow us to find |v(t1)dv(t) Q(t1) = Cv(t1) and Φ(t1) = LiL(t1) = Li(t1) − L − LC , (4) R dt t=t1 proving that knowledge of {Q(t0), Φ(t0)} and {i(t) : t0 ≤ t ≤ t1} is sufficient to determine {Q(t1), Φ(t1)}. 1Because C and L are known constants, it is eq uivalent to say that v(t0) and iL(t0) compr ise the state at time t0. Alternatively, we can take v(t0) and dv(t)/dtt=t0 to be the state. |2Vector Spaces A vector space is a set of elements (vectors), which we’ll denote {|·�}, and complex numbers (scalars) with vecto r addition and scalar multiplicatio n defined and obeying: • Vector addition is closed . If |x� and |y� are elements of a vector space, then so too is |x + y� ≡ |x�+ |y�. • Vector addition is commutative: |x� + |y� = |y�+ |x�. • Vector addition is associative: (|x� + |y�) + | z� = |x� + (|y� + |z�). • There exists an identity element, |0a�, such that |x� + |0a� = |x�. • There exists an additive inverse element, |–x�, such that |x� + |–x� = |0a�. Scalar multiplication is closed. If x� is a vect or and c is a scalar, then c• is also a vector. | |cx� ≡ | x� • Scalar multiplication is distributive: (c1+c2)|x� = c1|x�+c2|x�, and c(|x�+|y�) = c|x� + c|y�. • There is a n identity scalar, 1, such that 1|x� = |x�. • There is a zero scalar, 0, such that 0|x� = |0a�. As we progress through this lecture’s general mathematical development, we shall carry along the two running examples that we now introduce. Example 1: N-D Real Euclidean Space The elements of N-D real Euclidean space, RN, are conveniently represented as col-umn vectors,   x1 x2 |x� = x ≡ ...  , (5) xN where the {xn} and the scalars are real numbers. That the preceding vector space properties are satisfied by RN should be f amiliar to you fro m your linear algebra prerequisite for 6.453. Example 2: Complex-valued, Square-integrable Ti me Functions on [0, T ] The complex-valued, square-integrable time f unctions, |x� = {x(t) : 0 ≤ t ≤ T }, form a vector space L2[0, T ]. Here, by square-integrable, we mean that Z T 0 dt |x(t)| 2 < ∞. (6) You should verify that L2[0, T ] has the properties we have listed for a vector space. 3p p X Inner Product Spaces An inner product space


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