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

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Postulates of Quantum Mechanics

SOURCES

Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen

Jacob Biamonte and students

Linear Operators

Short review

- V,W: Vector spaces.
- A linear operatorA from V to W is a linear function A:VW. 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.

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).

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

Unitary Transformations

- A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U1 = 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??

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

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

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 AB) iff A is completely contained within B.
- Later, we may try to make these definitions more formal & precise.

B

A

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)

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.

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…

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

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.

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.

Distinguishability of States, more precisely

t

s

- Two state vectors s andt are (perfectly) distinguishable or orthogonal (write st) 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.

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.

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

Time Evolution

- Recall the Postulate: (Closed) systems evolve (change state) over time via unitary transformations.

t2 = Ut1t2 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

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 sS, 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.

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

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

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…..

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

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

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.

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

Computational Basis – a reminder

Observe that it is not required to be orthonormal, just linearly independent

We recalculate to a new basis

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

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

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

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”.

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

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.

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 elementsi,iare the probabilities of the basis states.
- The off-diagonal elements are “coherences”.
- The density matrix describes the state exactly.

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 |00| = |0||2 = ||2
- |1 with probability |11| = |1||2 = ||2

eigenvalue

m

Derived in next slide

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

Qubit example: calculate the density matrix

Conjugate and change kets to bras

Density matrix

Density matrix is a generalization of state

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

Very General Formula

- probability p(m) = |Pm|.

Any type of measurement operator

“insert Operator between Braket” rule

|Pm| = state (column vector)

|Pm| = number = probability

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

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.

General Measurement

Review:

To prove it it is sufficient to substitute the old base and calculate, as shown

Illustration of the formalisms used. You can calculate measurements from there

q

State Vector

Density State

Postulate 3, rough form

This is calculate as in 2 slides earlier

Mixed States

- Suppose one only knows of a system that it is in one of a statistical ensemble of state vectors vi (“pure” states), each with density matrix i and probability Pi.
- This is called a mixed state.
- This ensemble is completely described, for all physical purposes, by the expectation value (weighted average) of density matrices:
- Note: even if there were uncountably many state vectors vi, the state remains fully described by <n2 complex numbers, where n is the number of basis states!

Ensemble of quantum states

- Quantum states can be expressed as a density matrix:
- A system with n quantum states has n entries across the diagonal of the density matrix. The nth entry of the diagonal corresponds to the probability of the system being measured in the nth quantum state.
- The off diagonal correlations are zeroed out by decoherence.

Sum of these probabilities is one

For the ensemble

Unitary operations on a density matrix

- Unitary operations on a density matrix are expressed as:
- In other words the diagonal of density matrix is left as weights corresponding to the current states projection onto the computational basis after acted on by the unitary operator U
- Much like an inner product.

Old density matrix

New density matrix

adjoint

Trace of Matrix

- Trace of a matrix (sum of the diagonal elements):
- Unitary operators are trace preserving.
- The trace of a pure state is 1, all information about the system is known.
- Operators Commute under the action of the trace:

Partial Trace of a Matrix

- Partial Trace (defined by linearity)
- If you want to know about the nth state in a system, you can trace over the other states.

Partial trace

Density matrix

trace

Measurement of a density state for circuit with entanglement

H

Initial state:

Operate on the state space with an operator that preservers unitary evolution (H gate first bit):

Now act on system with CNOT gate:

We still define collections of measurement operators to act on the state space of our system:

Measurement of a density state

The probability that a result m occurs is given by the equation:

0000

0000

0000

1001

M3

½ tr =1/2

recall

For most of our purposes we can just use state vectors.

REMINDER: Ensemble point of view

Probability of outcome k being in state j

Probability of being in state j

Formula linking

trace and density matrix

REMINDER: Ensemble point of view

Probability of outcome k being in state j

Probability of being in state j

Postulate 3:Quantum Measurement

Now we can formulate precisely the Postulate 3

Now we use this notation for an Example of Qubit Measurement

0

1

0

1

We show here two methods to derive it. Check for homeworks and exam

1 0 0 0

1 0 0 0

[0*1*]

= [0*1*]

1 0 0 0

What happens to a system after a Measurement?

- After a system or subsystem is measured from outside, its state appears to collapseto exactly match the measured outcome
- the amplitudes of all states perfectly distinguishable from states consistent with that outcome drop to zero
- states consistent with measured outcome can be considered “renormalized” so their probabilities sum to 1

Only orthogonal state can be distinguished in a measurement

Projective Measurements: Average Values and Standard Deviations

Observable:

Can write:

scalar

Average value of a measurement:

scalar

Standard deviation of a measurement:

scalar

Compound Systems

- Let C=AB be a system composed of two separate subsystems A, B each with vector spaces A, B with bases |ai, |bj.
- The state space of C is a vector space C=AB given by the tensor product of spaces A and B, with basis states labeled as |aibj.

Composition example

The state space of a composite physical system is the tensor product of the state spaces of the components

- n qubits represented by a 2n-dimensional Hilbert space
- composite state is | = |1 |2 . . . |n
- e.g. 2 qubits:

|1 = 1|0 + 1|1|2 = 2|0 + 2|1| = |1 |2 = 12|00 + 12|01 + 12|10 + 12|11

- entanglement

2 qubits areentangled if | |1 |2 for any |1, |2

e.g. | = |00 + |11

Entanglement

- If the state of compound system C can be expressed as a tensor product of states of two independent subsystems A and B,c = ab,
- then, we say that A and B are not entangled, and they have individual states.
- E.g. |00+|01+|10+|11=(|0+|1)(|0+|1)
- Otherwise, A and B are entangled (basically correlated); their states are not independent.
- E.g. |00+|11

Entanglement results from axioms/postulates. It exists only for quantum states

Size of Compound State Spaces

- Note that a system composed of many separate subsystems has a very large state space.
- Say it is composed of N subsystems, each with k basis states:
- The compound system has kN basis states!
- There are states of the compound system having nonzero amplitude in all these kN basis states!
- In such states, all the distinguishable basis states are (simultaneously) possible outcomes (each with some corresponding probability)
- Illustrates the “many worlds” nature of quantum mechanics.

Key Points to Remember:

- An abstractly-specified system may have many possible states; only some are distinguishable.
- A quantum state/vector/wavefunction assigns a complex-valued amplitude (si) to each distinguishable state si(out of some basis set)
- The probability of state si is |(si)|2, the square of (si)’s length in the complex plane.
- States evolve over time via unitary (invertible, length-preserving) transformations.
- Statistical mixtures of states are represented by weighted sumsof density matrices =||.

Key points to remember

- The Schrödinger Equation
- The Hamiltonian
- The Evolution Matrix
- How complicated is a single Quantum Bit?
- Measurement
- Measurement operators
- Measurement of a state vector using projective measurement
- Density Matrix and the Trace
- Ensembles of quantum states, basic definitions and importance
- Measurement of a density state

Bibliography & acknowledgements

- Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2002
- V. Bulitko, On quantum Computing and AI, Notes for a graduate class, University of Alberta, 2002
- R. Mann,M.Mosca, Introduction to Quantum Computation, Lecture series, Univ. Waterloo, 2000 http://cacr.math.uwaterloo.ca/~mmosca/quantumcoursef00.htm D. Fotin, Introduction to “Quantum Computing Summer School”, University of Alberta, 2002.

Recall that M is measurement operator

We represent |2 in new base

Thus we have contradiction, states can be distinguished unless they are orthogonal

On the other hand

This lecture was taught in 2005, 2007

- There is more about trace and partial trace in future lectures and examples.

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