Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj D

1. Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students

2. Linear Operators Short review • V,W: Vector spaces. • A linear operatorA from V to W is a linear function A:VW. An operator onV is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An operator A on V is Hermitian iff it is self-adjoint (A=A†). • Its diagonal elements are real.

3. Eigenvalues & Eigenvectors • v is called an eigenvector of linear operator Aiff A just multiplies v by a scalar x, i.e.Av=xv • “eigen” (German) = “characteristic”. • x, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by. • the eigenvalue x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other. • (Two different eigenvectors have the same eigenvalue) • Any Hermitian operator has all real-valued eigenvectors, which are orthogonal (for distinct eigenvalues).

4. Exam Problems • Find eigenvalues and eigenvectors of operators. • Calculate solutions for quantum arrays. • Prove that rows and columns are orthonormal. • Prove probability preservation • Prove unitarity of matrices. • Postulates of Quantum Mechanics. Examples and interpretations. • Properties of unitary operators

5. Unitary Transformations • A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U1 = U† • Some properties of unitary transformations (UT): • Invertible, bijective, one-to-one. • The set of row vectors is orthonormal. • The set of column vectors is orthonormal. • Unitary transformation preserves vector length: |U| = | | • Therefore also preserves total probability over all states: • UT corresponds to a change of basis, from one orthonormal basis to another. • Or, a generalized rotation of in Hilbert space Who an when invented all this stuff??

6. A great breakthrough

7. Postulates of Quantum MechanicsLecture objectives • Why are postulates important? • … they provide the connections between the physical, real, world and the quantum mechanics mathematics used to model these systems • Lecture Objectives • Description of connections • Introduce the postulates • Learn how to use them • …and when to use them

8. Physical Systems-Quantum MechanicsConnections Postulate 1 Isolated physical system   Hilbert Space Postulate 2 Evolution of a physical system   Unitary transformation Postulate 3 Measurements of a physical system   Measurement operators Postulate 4 Composite physical system   Tensor product of components

9. Postulate 1:State Space

10. Systems and Subsystems • Intuitively speaking, a physical system consists of a region of spacetime& all the entities (e.g. particles & fields) contained within it. • The universe (over all time) is a physical system • Transistors, computers, people: also physical systems. • One physical system A is a subsystem of another system B (write AB) iff A is completely contained within B. • Later, we may try to make these definitions more formal & precise. B A

11. Closed vs. Open Systems • A subsystem isclosed to the extent that no particles, information, energy, or entropy enter or leave the system. • The universe is (presumably) a closed system. • Subsystems of the universe may be almost closed • Often in physics we consider statements about closed systems. • These statements may often be perfectly true only in a perfectly closed system. • However, they will often also be approximately true in any nearly closed system (in a well-defined way)

12. Concrete vs. Abstract Systems • Usually, when reasoning about or interacting with a system, an entity (e.g. a physicist) has in mind a description of the system. • A description that contains every property of the system is an exact or concretedescription. • That system (to the entity) is a concrete system. • Other descriptions are abstractdescriptions. • The system (as considered by that entity) is an abstract system, to some degree. • We nearly always deal with abstract systems! • Based on the descriptions that are available to us.

13. States & State Spaces • A possible stateS of an abstract system A (described by a description D) is any concrete system C that is consistent with D. • I.e., it is possible that the system in question could be completely described by the description of C. • The state space of A is the set of all possible states of A. • Most of the class, the concepts we’ve discussed can be applied to eitherclassical or quantum physics • Now, let’s get to the uniquely quantum stuff…

14. An example of a state space

15. Schroedinger’s Cat and Explanation of Qubits Postulate 1 in a simple way: An isolated physical system is described by a unit vector (state vector) in a Hilbert space (state space) Cat is isolated in the box

16. Distinguishability of States • Classicalandquantum mechanicsdiffer regarding the distinguishability of states. • In classical mechanics, there is no issue: • Any two states s, t are either the same (s = t), or different (s  t), and that’s all there is to it. • In quantum mechanics (i.e.in reality): • There are pairs of states s  t that are mathematically distinct, but not 100% physically distinguishable. • Such states cannot be reliably distinguished by any number of measurements, no matter how precise. • But you can know the real state (with high probability), if you prepared the system to be in a certain state.

17. Postulate 1: State Space • Postulate 1 defines “the setting” in which Quantum Mechanics takes place. • This setting is the Hilbert space. • The Hilbert Space is an inner product space which satisfies the condition of completeness (recall math lecture few weeks ago). • Postulate1:Any isolatedphysical space is associated with a complex vector spacewith inner product called the State Space of the system. • The system is completely described by a state vector, a unit vector, pertaining to the state space. • The state space describes all possible states the system can be in. • Postulate 1 does NOT tell us either what the state space is or what the state vector is.

18. Distinguishability of States, more precisely t s  • Two state vectors s andt are (perfectly) distinguishable or orthogonal (write st) iffs†t = 0. (Their inner product is zero.) • State vectors s and t are perfectly indistinguishable or identical (write s=t) iff s†t = 1. (Their inner product is one.) • Otherwise, s and t are both non-orthogonal, and non-identical. Not perfectly distinguishable. • We say, “the amplitude of state s, given state t, is s†t”. • Note: amplitudes are complex numbers.

19. State Vectors & Hilbert Space t s  • Let S be any maximal setof distinguishable possible states s, t, …of an abstract system A. • Identify the elements of S with unit-length, mutually-orthogonal (basis) vectors in an abstract complex vector space H. • The “Hilbert space” • Postulate 1: The possible states of Acan be identified with the unitvectors of H.

20. Postulate 2:Evolution

21. Postulate 2: Evolution • Evolution of an isolated system can be expressed as: where t1, t2 are moments in time and U(t1, t2) is a unitary operator. • U may vary with time. Hence, the corresponding segment of time is explicitly specified: U(t1, t2) • the process is in a sense Markovian (history doesn’t matter) and reversible, since Unitary operations preserve inner product

22. Example of evolution

23. Time Evolution • Recall the Postulate: (Closed) systems evolve (change state) over time via unitary transformations. t2 = Ut1t2 t1 • Note that since U is linear, a small-factor change in amplitude of a particular state at t1 leads to a correspondingly small change in the amplitude of the corresponding state at t2. • Chaos (sensitivity to initial conditions) requires an ensemble of initial states that are different enough to be distinguishable (in the sense we defined) • Indistinguishable initial states never beget distinguishable outcome

24. Wavefunctions • Given any set Sof system states (mutually distinguishable, or not), • A quantum state vector can also be translated to a wavefunction : S  C, giving, for each state sS, the amplitude (s) of that state. •  is called a wavefunction because its time evolution obeys an equation (Schrödinger’s equation) which has the form of a wave equation when S ranges over a space of positional states.

25. Schrödinger’s Wave Equation for particles We have a system with states given by(x,t) where: • t is a global time coordinate, and • xdescribes N/3 particles (p1,…,pN/3) with masses (m1,…,mN/3) in a 3-D Euclidean space, • where each piis located at coordinates (x3i, x3i+1, x3i+2), and • where particles interact with potential energy function V(x,t), • the wavefunction (x,t) obeys the following (2nd-order, linear, partial) differential equation: Planck Constant

26. Features of the wave equation • Particles’ momentum state p is encoded implicitly by the particle’s wavelength:p=h/ • The energy of any state is given by the frequency  of rotation of the wavefunction in the complex plane: E=h. • By simulating this simple equation, one can observe basic quantum phenomena such as: • Interference fringes • Tunneling of wave packets through potential barriers

27. Heisenberg and Schroedinger views of Postulate 2 This is Heisenberg picture This is Schroedinger picture ..in this class we are interested in Heisenberg’s view…..

28. The Solution to the Schrödinger Equation • The Schrödinger Equation governs the transformation of an initial input state to a final output state . • It is a prescription for what we want to do to the computer. • is a time-dependent Hermitian matrix of size 2n called the Hamiltonian • is a matrix of size 2n called the evolution matrix, • Vectors of complex numbers of length 2n • Tτis the time-ordering operator

29. The Schrödinger Equation • n is the number of quantum bits (qubits) in the quantum computer • The function exp is the traditional exponential function, but some care must be taken here because the argument is a matrix. • The evolution matrix is the program for the quantum computer. Applying this program to the input state produces the output state ,which gives us a solution to the problem. We discussed this power series already Formula from last slide

30. The Hamiltonian Matrix in Schroedinger Equation • The Hamiltonian is a matrix that tells us how the quantum computer reacts to the application of signals. • In other words, it describes how the qubits behave under the influence of a machine language consisting of varying some controllable parameters (like electric or magnetic fields). • Usually, the form of the matrix H needs to be either derived by a physicist or obtained via direct measurement of the properties of the computer.

31. The Evolution Matrix in the Schrodinger Equation • While the Hamiltonian describes how the quantum computer responds to the machine language, the evolution matrix describes the effect that this has on the state of the quantum computer. • While knowing the Hamiltonian allows us to calculate the evolution matrix in a pretty straightforward way, the reverse is not true. • If we know the program, by which is meant the evolution matrix, it is not an easy problem to determine the machine language sequence that produces that program. • This is the quantum computer science version of the compiler problem. Or “logic synthesis” problem. Evolution matrix Hamiltonian matrix

32. Postulate 3:Quantum Measurement

33. Computational Basis – a reminder Observe that it is not required to be orthonormal, just linearly independent We recalculate to a new basis

34. Example of measurement in different bases 1/2 The second with probability zero

35. You can check from definition that inner product of |0> and |1> is zero. • Similarly the inner product of vectors from the second basis is zero. • But we can take vectors like |0> and 1/2(|0>-|1>) as a basis also, although measurement will perhaps suffer. Good base Not a base

36. A simplified Bloch Sphere to illustrate the bases and measurements You cannot add more vectors that would be orthogonal together with blue or red vectors

37. Probability and Measurement • A yes/no measurement is an interaction designed to determine whether a given system is in a certain state s. • The amplitude of state s, given the actual state t of the system determines the probability of getting a “yes” from the measurement. • Important: For a system prepared in state t, any measurement that asks “is it in state s?” will return “yes” with probability Pr[s|t] = |s†t|2 • After the measurement, the state is changed, in a way we will define later. Inner product between states

38. A Simple Example of distinguishable, non-distinguishable states and measurements • Suppose abstract system S has a set of only 4 distinguishable possible states, which we’ll call s0, s1, s2, and s3, with corresponding ket vectors |s0, |s1, |s2, and |s3. • Another possible state is then the vector • Which is equal to the column matrix: • If measured to see if it is in state s0,we have a 50% chance of getting a “yes”.

39. Observables • Hermitian operator A on V is called an observable if there is an orthonormal (all unit-length, and mutually orthogonal) subset of its eigenvectors that forms a basis of V. There can be measurements that are not observables Observe that the eigenvectors must be orthonormal

40. Observables • Postulate 3: • Every measurable physical property of a system is described by a corresponding operator A. • Measurement outcomes correspond to eigenvalues. • Postulate 3a: • The probability of an outcome is given by the squared absolute amplitude of the corresponding eigenvector(s), given the state.

41. Density Operators • For a given state |, the probabilities of all the basis states si are determined by an Hermitian operator or matrix  (the density matrix): • The diagonal elementsi,iare the probabilities of the basis states. • The off-diagonal elements are “coherences”. • The density matrix describes the state exactly.

42. Towards QM Postulate 3 on measurement and general formulas Pm| p(m) A measurement is described by an Hermitian operator (observable) M = m Pm • Pm is the projector onto the eigenspace of M with eigenvalue m • After the measurement the state will be with probability p(m) = |Pm|. • e.g. measurement of a qubit in the computational basis • measuring | = |0 + |1 gives: • |0 with probability |00| = |0||2 = ||2 • |1 with probability |11| = |1||2 = ||2 eigenvalue  m Derived in next slide

43. An example how Measurement Operators act on the state space of a quantum system Measurement operators act on the state space of a quantum system Initial state: Hadamard Operate on the state space with an operator that preservers unitary evolution: = |0+|1 Define a collection of measurement operators for our state space: Half probability of measuring 0 Act on the state space of our system with measurement operators: = p0 Half probability of measuring 1 But now it is a part of formal calculus

44. Qubit example: calculate the density matrix Conjugate and change kets to bras Density matrix Density matrix is a generalization of state

45. Measurement of a state vector using projective measurement Operate on the state space with an operator that preservers unitary evolution: Define observables: Vector is on axis X Act on the state space of our system with observables (The average value of measurement outcome after lots of measurements): This type of measurement represents the limit as the number of measurements goes to infinity Here 3 may be enough, in general you need four

46. Bloch Sphere

47. Very General Formula • probability p(m) = |Pm|. Any type of measurement operator “insert Operator between Braket” rule |Pm| = state (column vector) |Pm| = number = probability

48. More examples and generalizations: Duals and Inner Products are used in measurements <| This is inner product not tensor product! ( ) Remember this is a number We prove from general properties of operators

49. Review: Duals as Row Vectors To do bra from ket you need transpose and conjugate to make a row vector of conjugates. Do this always if you are in trouble to understand some formula involving operators, density matrices, bras and kets.

50. General Measurement Review: To prove it it is sufficient to substitute the old base and calculate, as shown