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, |0i and |1i.

Exercise 2. Find another basis of C2. Any state in this system can be written as,

|ψi = α|0i + β|1i, |α|2 + |β|2 = 1.

The coeﬃcients, α and β, are called the amplitude. Speciﬁcally, α (β) is the amplitude of the state |ψi for |0i (|1i) respectively.

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 {|0i, |1i, ··· , |n − 1i} or {|1i, |2i, ··· , |ni}.

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 Schrödinger’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 d|ψi i = H|ψi, 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.

Exercise 5. Show that the matrix e−iH(t2−t1) is unitary.

Using the previous exercise, |ψ(t2)i = U(t2, t1)|ψ(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 |ψ(t1)i then the state at time t2 is,

|ψ(t2)i = U(t2, t1)|ψ(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 |ψi = α|0i + β|1i, then it will be in |0i with probability |α|2 and in |1i with probability |β|2. That meant, if we measure the state in the basis {|0i, |1i}, then the output will be 0 with probability |α|2 and similarly for 1. The final state will be |0i if the output is 0, and |1i if the output is 1. It is as though the state |ψi is projected onto the space spanned by |0i or |1i. 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 |ψi using these projection will give Pi|ψi 2 state with probability kPi|ψik . We divide by kPi|ψik so that the resulting state is a unit vector. kPi|ψik 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 {i, j}, 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 |ψi with M, we get value mi with probability kPi|ψik = hψ|Pi|ψi and the resulting state is Pi|ψi . kPi|ψik 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 {v1, v2, ··· , vn}; it means the projections are,

{P1 = |v1ihv1|,P2 = |v2ihv2|, ··· ,Pk = |vkihvk|}.

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 |ψi is hψ|M|ψi. As we hinted above, a more general class of measurements can be deﬁned. This gives us our third postulate. Postulate 3: A state |ψi can be measured with measurement operators {M1,M2, ··· ,Mk}. The linear P ∗ ∗ operators Mi’s should satisfy i Mi Mi = I. The probability of obtaining outcome i is p(i) = hψ|Mi Mi|ψi and the state after measurement is M√i|ψi . 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 {E1,E2, ··· ,Ek}, such that i Ei = I and ∀ i : Ei 0. The POVM measurement on |ψi gives outcome i with probability hψ|Ei|ψi.

Exercise 13. What are the POVM elements for the projective measurement.

Exercise 14. Show that the state |ψi and the state eiθ|ψi 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 |ψi and Bob prepares it in |φi, then the combined state is |ψi ⊗ |φi, succinctly written as |ψi|φi. 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)(|ai ⊗ |bi) = A|ai ⊗ B|bi.

|00i+|11i Exercise 15. Let ψ = √ , calculate the value of hψ|X ⊗ Z |ψi. 2 1 2 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 an operator on ﬁrst system and B is an operator on second system, the combined operator is A1 ⊗ B2. The tensor product structure of the composite system gives rise to a 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 states. The most famous example of an entangled state is called the Bell state, 1 √ (|00i + |11i). 2 Exercise 16. Show that the Bell state can’t be written as |ψi ⊗ |φi.

Pl It is clear that every state in the composite system H1 ⊗ H2 can be written as i=1 |ψii ⊗ |φii (Why?).

Exercise 17. Prove a bound of dim(H1) × dim(H2) on l 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? These are very interesting questions and lot of research is currently being done to answer them.

4.1 General measurements using projective measurements The first application of fourth postulate will be to simulate generalized measurement using projective measurements and unitary operators. Suppose, we would like to perform measurements {Mi : 1 ≤ i ≤ k} on a Hilbert space H. Consider a state space M with basis {|1i, |2i, ··· , |ki}. Pick a fixed state |0i in the state space M and define a unitary U on the space H ⊗ M, X U|ψi|0i = Mi|ψi|ii. i Exercise 18. Show that U preserves the norm between states of the form |ψi|0i.

Exercise 19. Show that U can be extended to a unitary operator on the entire space.

Then the projective measurements are Pi = IH ⊗ |iihi|. Exercise 20. Show that the probability of obtaining i using the general measurement on |ψi is same as the probability of getting i when U|ψi|0i is measured with {Pi}.

4 Hence the probability p(i) of obtaining the outcome i matches with the generalized measurement. The combined state of the system using the measurement postulate is,

P U|ψi|0i M |ψi|ii i = i . pp(i) pp(i)

So, if outcome i is obtained, the state of system M is |ii and state of system H is M√i|ψi . Since the state and p(i) the probability of the outcome of the projective measurement matches with the generalized measurement, we are able to simulate general measurement using ancilla, unitary and projective measurements.

4.2 Quantum Teleportation

Let us look at another application of the fourth postulate, speciﬁcally entanglement, called quantum teleportation. 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 the help of only classical communication. This protocol is called the quantum teleportation protocol.

Exercise 21. 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 |ψi = α|0i + β|1i to Bob. Suppose the state Alice wants to transfer, |ψi, is the ﬁrst qubit and her part of Bell state is the second qubit. Alice applies CNOT gate to these two qubits. CNOT gate is a 2-qubit gate, which applies NOT gate to the second qubit if and only if the ﬁrst qubit is in state |1i.

Exercise 22. Write the matrix representation of CNOT. Show that CNOT is unitary.

Then she applies Hadamard gate to her ﬁrst qubit.

Exercise 23. What is the state of the three qubits now?

It can be shown that the resulting state is,

1 (|00i(α|0i + β|1i) + |01i(α|1i + β|0i) + |10i(α|0i − β|1i) + |11i(α|1i − β|0i)) . 2

Now Alice measures her two qubits and sends them to Bob.

Exercise 24. Convince yourself that Bob can recover |ψi using Pauli operators.

This completes the quantum teleportation. Alice is able to transfer one quantum bit using two classical bits of communication.

Exercise 25. 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.

5 5 Assignment

Exercise 26. Give suﬃcient condition for eA+B = eAeB.

Exercise 27. Show that for every unitary U, there exist Hermitian H, such that, U = eiH .

Exercise 28. Show that the average value of measurement M on |ψi is hψ|M|ψi.

Exercise 29. What are the projectors on the eigenspace of v1X + v2Y + v3Z where {v1, v2, v3} is a unit vector.

∗ Exercise 30. Remember that there exist POVM operators Ei = Mi Mi for measurement√ operators Mi. Given measurement operators Mi’s, show that there exists unitaries Ui, s.t., Mi = Ui Ei.

Exercise 31. Show that an operator which takes |ψi|0i to |ψi|ψi for all ψ is not a unitary operator. What does this show?

Exercise 32. Prove Schmidt decomposition using the singular value decomposition.

Exercise 33. Read about super-dense coding.

References

1. M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge, 2010. 2. S. Arora and B. Barak. Computational Complexity: A modern approach. Cambridge, 2009. 3. R. Lidl and H. Niederreiter. Finite Fields. Cambridge University Press, 1997. 4. B. Kleinberg. Course notes: Introduction to algorithms. http://www.cs.cornell.edu/courses/cs4820/2010sp/handouts/MillerRabin.pdf, 2010. 5. D. R. Simon. On the power of quantum computation. Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on: 116123, 1994. 6. A. Childs. Course notes: Quantum algorithms. https://cs.umd.edu/ amchilds/teaching/w13/qic823.html, 2013.