Lecture 4: 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 on 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 too. 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 is completely described by a unit vector in this Hilbert space. Intuitively, Hilbert space is the vector space with enough structure so that we can apply algebra and analysis on it. For this course, we will only be dealing with vector spaces over complex numbers with inner product defined over them. Exercise 1. Read more about Hilbert spaces. It is not clear which Hilbert space should we be taking. For us, the simplest state space would be C, the state space of a qubit. It will be spanned by two vectors, 0 and 1. Exercise. Find another basis of C. Any state in this system is, ψ = α 0 + β 1, α + β = 1. The coefficients α, β are called the amplitude. Specifically, α (β) is the amplitude of the state ψ for 0 ( 1 ) respectively. Note 1. Many people interpret this as, the state ψ is in state 0 with probability α and in state 1 with probability β. This is only a consequence of ψ = α 0 + β 1 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 orthonormal basis { 0, 1,, n 1 } or { 1,,, n }. Evolution of the system The next postulate specifies how a closed quantum system evolves. This is the very famous Schrödinger s equation. It is a partial differential equation which describes the how a quantum state evolves with time. The evolution is described by a Hamiltonian, which is a Hermitian matrix for us. Given the Hamiltonian H, i d ψ = H ψ, dt describes how the quantum system will change its state with time. For readers who are already familiar with it, we have assumed that Planck s constant can be absorbed in the Hamiltonian. This can be thought of 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 Schrödinger s equation. Suppose the quantum system is in state ψ(t 1 ) at time t 1. Then using the previous equation, ψ(t ), the state at time t is, ψ(t ) = e ih(t t1) ψ(t 1 ). Exercise 5. Show that the matrix e ih(t t1) is unitary. Using the previous exercise, ψ(t ) = U(t, t 1 ) ψ(t 1 ). This gives us the working second postulate. Postulate : A closed quantum system evolves unitarily. The unitary matrix only depends on time t 1 and t. If the state at t 1 is ψ(t 1 ) then the state at time t is, ψ(t ) = U(t, t 1 ) ψ(t 1 ). Note. Unitary operators preserve the norm. What operators do you know which are unitary? Exercise 6. Show that all the Pauli matrices and H is unitary. 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, have we seen any measurement in this course before? Yes, we said that if the state is ψ = α 0 + β 1 then it will be in 0 with probability α and in 1 with probability β. That meant, if we measure the state in the basis { 0, 1 }, then the output will be 0 with probability α and similarly for 1. The final state will be 0 if the output is 0, and 1 if the output is 1. It is as though the state ψ is projected by 0 0 or 1 1. 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 P 1, P,, P k are the projectors onto these spaces. A measurement on kψ using these projection will give state Pi ψ P with probability P i ψ i ψ. We divide by P i ψ so that the resulting state is a unit vector. Exercise 8. Check that this definition matches with one qubit projection in the standard basis defined above. More formally, a projective measurement is described by a Hermitian operator M = i m ip i. Here P i s are projectors, s.t., i P i = I and for all pairs P i P j = 0. In other words, P i are orthogonal projectors which span the entire space. Exercise 9. Show that 0 P i I. Where A B means A B is positive semidefinite matrix.
If we measure state ψ with M. We get value m i with probability P i ψ = ψ P i ψ and the resulting P state is i ψ P. i ψ We will not answer why this happens. This and the subsequent definition of projective 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 v 1, v,, v n ; it means the projections are v 1 v 1, v v,, v k v k 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 ψ is ψ M ψ. As we hinted above, a more general class of measurements can be defined. This gives us our third postulate. Postulate 3: A state ψ can be measured with measurement operators {M 1, M,, M k }. The linear operators M i s should satisfy i M i M i = I. Exercise 10. Prove that the condition i M i M i = I is equivalent to the fact that measurement probabilities sum up to 1. If probability of obtaining outcome i is = ψ M i M i ψ and then the state after measurement is M i ψ. Exercise 11. Show that projective measurements are a special case of measurements defined in the postulate. Exercise 1. 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 (another 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 with what probability we get the outcome and what is the state of the system after that. Sometimes, we are not interested in the state after the measurement (say 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 M i M i and not M i. So we only need to specify E i = M i M i. These E i s are called the POVM elements. Given {E 1, E,, E k }, such that i E i = I and i : E i 0. The POVM measurement on ψ gives outcome i with probability ψ E i ψ. Exercise 13. What are the POVM elements for the projective measurement. Exercise 14. Show that ψ and e iθ ψ have the same measurement statistics. 4 Composite Systems The final postulate deals with composite systems. In the last lecture, we motivated tensor products 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 H A and Bob is H B then the state space of their combined system is H A H B. If Alice prepares her system in state ψ and Bob prepares it in φ then the combined state is ψ φ succinctly written as ψ φ. 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. This follows from the property, (A B)( a b ) = A a B b. 3
Exercise 15. Calculate the quadratic form of the Bell state, 00 + 11, on the operator X 1 Z. Generally, it is quite clear which part of the system belongs to which party. In case of confusion, we will use subscripts to resolve it. So if A is on first system and B is applied on second system, the combined operator is A 1 B. The tensor product structure of the composite system gives rise to very interesting property called entanglement. As explained before, there are states in the composite system which cannot be decomposed into the states of their constituent systems. Such states are called entangled. The most famous example of an entangled state is called the Bell state, 1 ( 00 + 11 ). Exercise 16. Show that the Bell state can t be written as ψ φ. It is clear that every state in the composite system H 1 H can be written as n i=1 ψ i φ i (Why?). Exercise 17. Prove a bound of dim(h 1 ) dim(h ) on n for any state in the composite system. Can you give a better bound? Read about Schmidt decomposition for a better bound. We have defined when a state is entangled and when is it not. But how can we quantify entanglement? In other words, how entangled is an state? This is a very interesting question. 4.1 General measurements using projective measurements We would like to perform measurements {M i : 1 i k} on a system H. Consider a state space M with basis { 1,,, k }. Pick a fixed state 0 in the state space M and define a unitary U, U ψ 0 = i M i ψ i. Exercise 18. Show that U preserves the norm between states of the form ψ 0. Exercise 19. Show that U can be extended to a unitary operator on the entire space. Then the projective measurements are P i = I H i i. Exercise 0. Show that the probability of obtaining i using the general measurement on ψ is same as the probability of getting i when U ψ 0 is measured with {P i }. Hence the probability of obtaining the outcome i matches. The combined state of the system using the postulate is, P i U ψ 0 = M i ψ i. So, if outcome i is obtained, the state of system M is i and state of system H is Mi ψ. Since the state and the probability both match, we are able to simulate general measurement using ancilla, unitary and projective measurements. 4
5 Quantum Teleportation To motivate ourselves, let us look at one of the applications of quantum computing. Quantum teleportation is a technique to transfer quantum bits without using quantum communication. In other words, suppose Alice and Bob have quantum computers but don t have a channel which can transfer quantum bits. Using entanglement, we can transfer quantum bits from one party to another with just classical communication. This is called quantum teleportation. Exercise 1. Why could this be useful? The protocol requires the use of entanglement. Alice and Bob can meet before and keep one part (qubit) of the Bell state with each of them. Suppose Alice wants to transfer state ψ to Bob. Suppose the state Alice wants to transfer (state ψ ) is the first qubit and the part of Bell state is the second qubit. Alice applies CNOT gate to these two qubits. CNOT gate is a -qubit gate, which applies NOT gate to the second qubit if and only if the first qubit is in state 1. Exercise. Write the matrix representation of CNOT. Show that CNOT is unitary. Then she applies Hadamard gate to the second qubit. Exercise 3. Suppose ψ = α 0 + β 1, what is the state of the three qubits now? It can be shown that the resulting state is, 1 ( 00 (α 0 + β 1 ) + 01 (α 1 + β 0 ) + 10 (α 0 β 1 ) + 11 (α 1 β 0 )). Now Alice measures her two qubits and sends them to Bob. Exercise 4. Convince yourself that Bob can recover ψ using Pauli operators. This completes the quantum teleportation. Alice is able to transfer one quantum bit using two classical bits of communication. Exercise 5. We said that we can transfer and not copy the quantum bit. Why? (look at question 31 of assignment) In quantum computing we can t copy qubits, this is known as no-cloning theorem. 6 Assignment Exercise 6. Give sufficient condition for e A+B = e A e B. Exercise 7. Show that for every unitary U, there exist Hermitian H, such that, U = e ih. Exercise 8. Show that the average value of measurement M on ψ is ψ M ψ. Exercise 9. What are the projectors on the eigenspace of v 1 X + v Y + v 3 Z where {v 1, v, v 3 } is a unit vector. Exercise 30. Remember that there exist POVM operators E i = M i M i for measurement operators M i. Given measurement operators M i s, show that there exists unitaries U i, s.t., M i = U i Ei. Exercise 31. Show that an operator which takes ψ 0 to ψ ψ for all ψ is not a unitary operator. What does this show? Exercise 3. Prove Schmidt decomposition using the singular value decomposition. Exercise 33. Read about superdense coding. References 1. M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge, 010.. S. Arora and B. Barak. Computational Complexity: A modern approach. Cambridge, 009. 5