Introduction to Quantum Computing Petros Wallden Lecture 3: Basic Quantum Mechanics 26th September 2016 School of Informatics, University of Edinburgh
Resources 1. Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang 2. Lecture Notes available on http://qcintro.wordpress.com 1
Postulates of Quantum Mechanics - QM is mathematical framework for the development of physical theories - On its own QM doesnt tell you what laws a physical system must obey, but it does provide a mathematical and conceptual framework for the development of such laws. Postulates: 1. Predictions of results of measurements are probabilistic in nature. This information is encoded by a vector in a complex Hilbert space H which is the state space of QM. The state of a quantum system is, therefore, a vector in H. 2. Observables are represented by Hermitian (self-adjoint) operators that act on H 3. Given an observable A and a state ψ, the expected result of measuring the observable A (expected value) is A ψ = ψ A ψ 2
4. In the absence of any external influence, the state of the system changes smoothly in time according to the Scrödinger equation i d ψ t = H ψ dt t where H is the a Hermitian operator called Hamiltonian and it corresponds to the energy of the system. One can also prove that there exist unitary operators U(t) that evolve the state ψ(t) = U(t) ψ(0)
Measurements (technical) - Here we are interested on measurements that give m different outcomes, each of them with probability p(m) and the effect that this has on the state ψ of the system. For now, we are not concerned with the actual values of each of the different outcomes or with what is the physical significance/meaning of the measurement. - The measurement is represented with a collection {M 1,, M m } of operators that correspond to different outcomes, acting on the Hilbert space of the system measured. The index denotes the different outcomes that may occur. - The probability of each outcome depends on the premeasurement state ψ of the system. p(m) = ψ M mm m ψ - An important requirement that the operators M m must obey is the completeness m M mm m = 1 This property guarantees that the total probability that any of the possible outcomes occurs, is unity m p(m) = m ψ M mm m ψ = ( ) = ψ m M mm m ψ = 1 3
- The state of the system after the measurement occurs and the outcome m is obtained becomes ψ 1 M m ψ p(m) (Note that the new state has also unit norm) - The fact that the measurement changes the state is similar with what happens in a classical random event such as tossing a coin. The state of knowledge that we have, after tossing the coin but before looking the outcome is probability 1/2 heads and 1/2 tails. After looking the coin and we see that it is heads, our state of knowledge changes since we now know that the coin is heads. - However, there is an important difference. In the classical case, it is the state of our knowledge that gets updated, since the coin was heads before we looked. In the quantum case, it appears as if the actual state of the system (and not only our knowledge about it) that changes by the act of measurement. Example: Measurement of a Qubit Initial state ψ = a 0 + b 1 where a 2 + b 2 = 1 so that ψ is unit vector Measurement: M 0 = 0 0, M 1 = 1 1 Probabilities: p(0) = ψ M 0 M 0 ψ = ψ M 0 ψ = a 2
p(1) = ψ M 1 M 1 ψ = ψ M 1 ψ = b 2 State after measurement: (i) If we obtain outcome 0 1 M 0 ψ = a p(0) a 0 (ii) If we obtain outcome 1 1 M 1 ψ = b p(1) b 1 Projective Measurements An important special case of measurements (mostly we will encounter this type of measurements) - A measurement where all operators M m are projection operators, i.e. M mm m = M m = P m that are pairwise orthogonal P m P n = δ mn P m - The completeness then becomes m P m = 1 - Before the measurement the state is ψ - After the measurement the state depends on the outcome observed. Therefore after a measurement occurs, but before we see the outcome of the measurement, our knowledge about the state of the system is a mixture of different states, that each of them occurs with different probability
- In particular, it may be in m different states. With probability p(m) = ψ P m ψ the state is in the state 1/ p(m) (P m ψ ). We will give a formal description of this type states later - The change of the state from ψ to the one of the outcomes P m ψ is called collapse of the wavefunction. It is important to note that this change of the state of the system is different from the one given by the unitary evolution (Postulate 4). This is the source of the measurement problem (see next lecture) - We should now return to the question of what values do we obtain from a measurement. In Postulate 2 we established that observables are represented by Hermitian operators A. One can always rewrite a Hermitian operator A in terms of his eigenvalues a in the following way (spectral decomposition) A = a ap a where P a is a projection operator projecting at the subspace corresponding to the eigenvalue a - The measurement {P a } is a measurement of the observable A, where the possible values are the different eigenvalues a - The expected result of measuring this observable should be a p(a) a which is exactly a a ψ P a ψ = ψ a ap a ψ = ψ A ψ as given by Postulate 3
Distinguishing Pure Quantum States - Assume a fixed set of possible states { ψ 1,, ψ n } - Alice chooses one of these states ψ i, prepares it and sends it to Bob - Bob can make any measurement he wishes in order to identify the index i {1,, n} of the state sent - This type of setting is very common is quantum communication protocols. Moreover this type of problem highlights some of the most distinctive properties that quantum theory has Case I: States ψ i are orthogonal, i.e. ψ i ψ j = δ ij We perform a (projective) measurement that consist of the following operators P i = ψ i ψ i and P 0 = 1 i ψ i ψ i Exercise: Check that this measurement satisfies the completeness relation We can see easily that if the state ψ k is prepared, then p(i) = ψ k P i ψ k = δ ik and therefore Bob finds with probability one the correct index. Case II: Some of the states are not orthogonal 4
Theorem: Non-orthogonal pure states cannot be distinguished with certainty Proof by contradiction: - Consider two non-orthogonal states ψ 1 ψ 2 = 0 - Related to these are two measurement operators (not necessarily projective) E 1 = M 1 M 1 and E 2 = M 2 M 2 - If we can distinguish them perfectly it means that when Alice sends ψ 1 Bob has p(i = 1) = ψ 1 E 1 ψ 1 = 1 and when Alice sends ψ 2 Bob has p(i = 2) = ψ 2 E 2 ψ 2 = 1 - From i E i = 1 and ψ 1 E 1 ψ 1 = 1 we conclude that ψ 1 E 2 ψ 1 = 0 and thus E 2 ψ 1 = 0 - Since the two states are non-orthogonal we can write ψ 2 = α ψ 1 + β φ where ψ 1 φ = 0 is a unit vector - Then it follows ψ 2 E 2 ψ 2 = β 2 φ E 2 φ β 2 < 1 which contradicts our assumption and completes the proof
Density Matrices and Mixed States - So far we have represented quantum states as vectors ψ at a Hilbert space. We can also represent the states as operators, which we call density matrices ρ ψ = ψ ψ Examples: (1) 0 0 0 = [ ] 1 0 0 0 (2) + := 1/ 2 ( 0 + 1 ) + + = 1/2 [ ] 1 1 1 1 - Using this representation enable us to represent the state of quantum systems that are not completely known. Assume that the quantum state is one of a number of states ψ i, each of them with respective probabilities p i. We call {p i, ψ i } ensemble of states. The state of this system is described by the following density matrix ρ = i p i ψ i ψ i - When a density matrix cannot be expressed in terms of a single pure state, we say that it is a mixed state - The mixed states include two types of randomness: 5
(i) The classical randomness because we do not know which is the pure quantum state. This randomness is related with the lack of knowledge that we (the observers) have about the system and is fully analogous with the randomness of classical physics (epistemic). (ii) Fundamental quantum randomness. This is related with the fact that even if we know the exact pure quantum state (have maximum information about the system) the different outcomes occur probabilistically. - We can now describe better what happens during a (projective) measurement. Assume initial state ψ and measurement {P i } ρ initial = ψ ψ ρ final = i P i ψ ψ P i Note that p(i) = ψ P i ψ, and the state if result i is obtained is 1/ p(i)p i ψ. Writing the resulting state in density matrix form becomes 1/p(m)P i ψ ψ P i which leads to the above ρ final - The quantum state changes from the act of measurement even without knowing the outcome. It makes a pure state to a mixed state (i.e. from a completely known quantum state it results to an ensemble of quantum states) - There are certain properties that the allowed density matrices should obey. One can take these properties as a generalisation of what are the allowed quantum states.
A density matrix is a matrix (or operator) ρ that (1) is Hermitian ρ = ρ (2) positive semi-definite (i.e. has non-negative eigenvalues) (3) has unit trace Tr(ρ) = 1 Exercise: Check that these conditions are satisfied (i) for pure density matrices (ii) for density matrices of the form ρ = i p i ψ ψ Modifications needed for mixed states: - Unitary Evolution: ρ UρU - Expectation values: A ρ = Tr(ρA) - Measurement: p(m) = Tr(M mm m ρ) and the state becomes ρ initial ρ final = 1 p(m) M mρm m Different Ensembles can give rise to the same density matrix! [ ] 3/4 0 Example: ρ = 0 1/4 Ensemble 1: {p(0) = 3/4, 0, p(1) = 1/4, 1 } Ensemble 2: {p(a) = 1/2, a, p(b) = 1/2, b } where 3 a = 4 0 + 1 4 1 3 b = 4 0 1 4 1 Check that: ρ = 1 2 a a + 1 2 b b = 3 4 0 0 + 1 1 1 4