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Postulates of Quantum Mechanics Lecture 3: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying quantum computing, it gives the basic laws according to which any quantum system (or a quantum computer) works. These postulates were agreed upon after a lot of trial and error. We won't be concerned about the physical motivation of these postulates. Most of the material for this lecture is taken from [1]. It is a very good reference for more details. 1 State of a system The first postulates specifies, what is meant mathematically by the state of a system. Postulate 1: A physically isolated system is associated with a Hilbert space, called the state space of the system. The system, at a particular time, is completely described by a unit vector in this Hilbert space, called the state of the system. Intuitively, Hilbert space is a vector space with enough structure so that we can apply the techniques of linear algebra and analysis on it. Exercise 1. Read more about Hilbert spaces. For this course, we will only be dealing with vector spaces over complex numbers with inner product defined over them. In particular, we will assume that our state space is Cn for some n. Here n is the dimension of this vector space. The simplest state space would be C2 (dimension being 2), the state space of a qubit. It will be spanned by two vectors, j0i and j1i. Exercise 2. Find another basis of C2. Any state in this system can be written as, j i = αj0i + βj1i; jαj2 + jβj2 = 1: The coefficients, α and β, are called the amplitude. Specifically, α (β) is the amplitude of the state j i for j0i (j1i) respectively. Note 1. Many people interpret this as, the state j i is in state j0i with probability jαj2 and in state j1i with probability jβj2. This is only a consequence of j i = αj0i + βj1i and not equivalent to it. Exercise 3. Why is it not equivalent? In general, if there are n different classical states, the quantum state would be a unit vector with or- thonormal basis fj0i; j1i; ··· ; jn − 1ig or fj1i; j2i; ··· ; jnig. 2 Evolution of a quantum system The next postulate specifies, how a closed quantum system evolves. You might already know this postulate in terms of the very famous Schr¨odinger'sequation. It is a partial differential equation which describes how a quantum state evolves with time. The evolution is described by a Hamiltonian H which depends on the system being observed. For us, it is a Hermitian matrix H. Given the Hamiltonian H, the equation dj i i = Hj i; dt describes how the quantum system will change its state with time. For readers who are already familiar with it, we have assumed that the Planck's constant can be absorbed in the Hamiltonian. This equation can be considered as the second postulate of quantum mechanics. But, we will modify it a little bit to get rid of partial differential equation and write it in terms of unitary operators. Exercise 4. Read about Schr¨odinger's equation. Suppose the quantum system is in state j (t1)i at time t1. Then using the Schr¨odinger'sequation, j (t2)i, the state at time t2, j (t2)i, is −iH(t2−t1) j (t2)i = e j (t1)i: Exercise 5. Show that the matrix e−iH(t2−t1) is unitary. Using the previous exercise, j (t2)i = U(t2; t1)j (t1)i: This gives us the \working" second postulate. Postulate 2: A closed quantum system evolves unitarily. The unitary matrix only depends on time t1 and t2. If the state at t1 is j (t1)i then the state at time t2 is, j (t2)i = U(t2; t1)j (t1)i: Note 2. Unitary operators preserve the norm. Do you remember any unitary operators considered in this course before? Exercise 6. Show that all the Pauli matrices and the Hadamard matrix H are unitary operators. Exercise 7. \Guess" the eigenvalues and eigenvectors of H. Check, if not, find the actual ones. 3 Measurement of the system We have talked about the state of the system and how it evolves. To be able to compute, we should be able to observe/measure the properties of this system too. It turns out that measurement is an integral part of quantum mechanics. Not only does it allow us to determine properties of the quantum system but it significantly alters the system too. Before we discuss the third postulate describing the measurements, try to recall if we have seen any measurement in this course before? Yes, we said that if the state is j i = αj0i + βj1i, then it will be in j0i with probability jαj2 and in j1i with probability jβj2. That meant, if we measure the state in the basis fj0i; j1ig, then the output will be 0 with probability jαj2 and similarly for 1. The final state will be j0i if the output is 0, and j1i if the output is 1. It is as though the state j i is projected onto the space spanned by j0i or j1i. This idea gives us the definition of projective measurements (a subclass of general measurements we will define later). Any partition of the vector space (where the state lives) is a possible measurement. Suppose P1;P2; ··· ;Pk are the projectors onto these spaces. A measurement on j i using these projection will give Pij i 2 state with probability kPij ik . We divide by kPij ik so that the resulting state is a unit vector. kPij ik Exercise 8. Check that this definition matches with one qubit projection in the standard basis defined above. 2 P More formally, a projective measurement is described by a Hermitian operator M = miPi. Here Pi's P i are projectors, s.t., i Pi = I and for all pairs fi; jg, PiPj = 0. In other words, Pi are orthogonal projectors which span the entire space. Exercise 9. Show that I Pi 0. Where A B means A − B is positive semidefinite matrix. 2 If we measure state j i with M, we get value mi with probability kPij ik = h jPij i and the resulting state is Pij i . kPij ik We will not answer why measurement happens this way. This and the subsequent definition of other kind of measurements is taken as a postulate. Though it agrees with the intuition we had about measurement (projecting into subspaces). When we say that the state is measured in the basis fv1; v2; ··· ; vng; it means the projections are, fP1 = jv1ihv1j;P2 = jv2ihv2j; ··· ;Pk = jvkihvkjg: In this case, it is easy to come up with the average value of the measurement. You will show in the assignment, the average value of measurement M on j i is h jMj i. As we hinted above, a more general class of measurements can be defined. This gives us our third postulate. Postulate 3: A state j i can be measured with measurement operators fM1;M2; ··· ;Mkg. The linear P ∗ ∗ operators Mi's should satisfy i Mi Mi = I. The probability of obtaining outcome i is p(i) = h jMi Mij i and the state after measurement is Mpij i . p(i) P ∗ Exercise 10. Prove that the condition i Mi Mi = I is equivalent to the fact that measurement probabilities sum up to 1. Exercise 11. Show that projective measurements are a special case of measurements defined in the postulate. Exercise 12. Find a measurement that is not projective. Notice that individual measurement operators are not unitary. We made the resulting vector a unit vector by dividing it with its norm. It turns out that given ancilla (additional quantum system) we can simulate any general measurement operator using unitary operators and projective measurements 4.1. 3.1 POVM For the complete specification of measurement postulate, we defined the probability of getting an outcome and the state of the system after the measurement. Sometimes, we are not interested in the state after the measurement, e.g., measurement is the last step in the algorithm. In that case there is an easier description of measurements. ∗ ∗ Notice that the probability only depends upon Mi Mi and not Mi. So we only need to specify Ei = Mi Mi. These Ei's are called the POVM elements. P Given fE1;E2; ··· ;Ekg, such that i Ei = I and 8 i : Ei 0. The POVM measurement on j i gives outcome i with probability h jEij i. Exercise 13. What are the POVM elements for the projective measurement. Exercise 14. Show that the state j i and the state eiθj i have the same measurement statistics for any measurement. 3 4 Composite Systems The final postulate deals with composite systems, systems with more than one part. In the last lecture, we motivated tensor product for the sake of describing multiple systems. So the use of tensor product in the final postulate does not come as a surprise. Postulate 4: Suppose the state space of Alice is HA and Bob is HB, then the state space of their combined system is HA ⊗ HB. If Alice prepares her system in state j i and Bob prepares it in jφi, then the combined state is j i ⊗ jφi, succinctly written as j ijφi. Similarly, if operator A is applied on Alice's system and operator B is applied on Bob's system, then operator A ⊗ B is applied to the combined system.
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