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# Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Da - PowerPoint PPT Presentation

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.

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SOURCES

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

Jacob Biamonte and students

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.

• 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).

• 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

• 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??

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

Postulate 1:State Space

• 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

• 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)

• 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.

• 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…

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

• 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.

• 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.

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

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

• 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

• 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.

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

• 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

This is Heisenberg picture

This is Schroedinger picture

..in this class we are interested in Heisenberg’s view…..

• 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

• 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 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.

• 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

Postulate 3:Quantum Measurement

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

We recalculate to a new basis

1/2

The second with probability zero

Good base

Not a base

You cannot add more vectors that would be orthogonal together with blue or red vectors

Probability and Measurement measurements

• 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 states and measurements

• 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 states and measurements

• 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 states and measurements

• 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.

Towards QM Postulate 3 on measurement and general formulas states and measurements

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

An example how states and measurementsMeasurement Operators act on the state space of a quantum system

Measurement operators act on the state space of a quantum system

Initial state:

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 states and measurements: 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 states and measurementsprojective 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

Bloch Sphere states and measurements

Very General Formula states and measurements

• 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: are used in measurements 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 are used in measurements

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 measurements from there

This is calculate as in 2 slides earlier

Mixed States measurements from there

• 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 measurements from thereof 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 measurements from there

• 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

Trace of Matrix measurements from there

• 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:

The trace operation measurements from there

Partial Trace of a Matrix measurements from there

• 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 measurements from there

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 measurements from there

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 measurements from there

Probability of outcome k being in state j

Probability of being in state j

trace and density matrix

REMINDER: Ensemble point of view measurements from there

Probability of outcome k being in state j

Probability of being in state j

Postulate 3: measurements from thereQuantum Measurement

Now we can formulate precisely the Postulate 3

Now we use this notation for an Example of Qubit measurements from thereMeasurement

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

How the state vector changes in measurement? measurements from there

M0 |> =

= |0> (<0| 0 |0> + <0| 1 |1> = |0> 0

What happens to a system after a Measurement? measurements from there

• 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

Observable:

Can write:

scalar

Average value of a measurement:

scalar

Standard deviation of a measurement:

scalar

Phase Deviations

Global phase

Relative phase can be physically distinquished

Relative phase depends on the basis

Postulate 4: Deviations Composite Systems

Compound Systems Deviations

• 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=AB given by the tensor product of spaces A and B, with basis states labeled as |aibj.

Composition example Deviations

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 = 12|00 + 12|01 + 12|10 + 12|11

• entanglement

2 qubits areentangled if |  |1  |2 for any |1, |2

e.g. | = |00 + |11

Entanglement Deviations

• If the state of compound system C can be expressed as a tensor product of states of two independent subsystems A and B,c = ab,

• 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 Deviations

• 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.

Postulate 4: Deviations Composite Systems

Summary on Postulates Deviations

Hilbert Space

Evolution

Measurement

Tensor Product

Key Points to Remember: Deviations

• 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 Deviations

• 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 Deviations

• 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.

Distinguishability Deviations

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

Uncertainty DeviationsPrinciple