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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. Gates on Multi-Qubit State, a reminder.

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Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

Postulates of Quantum Mechanics

SOURCES

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

Jacob Biamonte and students



Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

Example of Complex quantum system of 3 qubits – other realization of Toffoli, composed of 2-qubit gates

  • All gates are at most 2-qubit

  • Only CNOT as 2-qubit gates

  • It has 6 not 5 interaction gates


Linear operators
Linear Operators realization of Toffoli, composed of 2-qubit gates

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.


Eigenvalues eigenvectors
Eigenvalues & Eigenvectors realization of Toffoli, composed of 2-qubit gates

  • 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.

  • x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other.

  • Any Hermitian operator has all real-valued eigenvectors, which are orthogonal (for distinct eigenvalues).


Exam problems
Exam Problems realization of Toffoli, composed of 2-qubit gates

  • 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.


Unitary transformations
Unitary Transformations realization of Toffoli, composed of 2-qubit gates

  • 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 mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

A great breakthrough realization of Toffoli, composed of 2-qubit gates


Postulates of quantum mechanics lecture objectives
Postulates of Quantum Mechanics realization of Toffoli, composed of 2-qubit gatesLecture 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 mechanics connections
Physical Systems realization of Toffoli, composed of 2-qubit gates-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
Postulate 1: realization of Toffoli, composed of 2-qubit gatesState Space


Systems and subsystems
Systems and Subsystems realization of Toffoli, composed of 2-qubit gates

  • 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


Closed vs open systems
Closed vs. Open Systems realization of Toffoli, composed of 2-qubit gates

  • 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
Concrete vs. Abstract Systems realization of Toffoli, composed of 2-qubit gates

  • 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
States & State Spaces realization of Toffoli, composed of 2-qubit gates

  • 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…


Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

An example of a state space realization of Toffoli, composed of 2-qubit gates


Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

Schroedinger’s Cat and Explanation of Qubits realization of Toffoli, composed of 2-qubit gates

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
Distinguishability of States realization of Toffoli, composed of 2-qubit gates

  • 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 space1
Postulate 1: realization of Toffoli, composed of 2-qubit gates 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.


  • Revised postulate 1
    Revised Postulate 1 realization of Toffoli, composed of 2-qubit gates


    Distinguishability of states more precisely
    Distinguishability of States, more precisely realization of Toffoli, composed of 2-qubit gates

    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.


    State vectors hilbert space
    State Vectors & Hilbert Space realization of Toffoli, composed of 2-qubit gates

    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: realization of Toffoli, composed of 2-qubit gatesEvolution


    Postulate 2 evolution1
    Postulate 2: realization of Toffoli, composed of 2-qubit gates 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


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Example of evolution realization of Toffoli, composed of 2-qubit gates


    Time evolution
    Time Evolution realization of Toffoli, composed of 2-qubit gates

    • 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


    Wavefunctions
    Wavefunctions realization of Toffoli, composed of 2-qubit gates

    • 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.

      • When sis another state vector, and the real state is t, then (s) is just s†t.

      •  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.


    Schr dinger s wave equation
    Schrödinger’s Wave Equation realization of Toffoli, composed of 2-qubit gates

    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
    Features of the wave equation realization of Toffoli, composed of 2-qubit gates

    • 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
    Heisenberg and Schroedinger views of Postulate 2 realization of Toffoli, composed of 2-qubit gates

    This is Heisenberg picture

    This is Schroedinger picture

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


    The schr dinger equation
    The Schrödinger Equation realization of Toffoli, composed of 2-qubit gates

    • 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 schr dinger equation1
    The Schrödinger Equation realization of Toffoli, composed of 2-qubit gates

    • 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.


    The hamiltonian matrix in schroedinger equation
    The Hamiltonian Matrix in Schroedinger Equation realization of Toffoli, composed of 2-qubit gates

    • 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 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
    The Evolution Matrix in the Schrodinger Equation realization of Toffoli, composed of 2-qubit gates

    • 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.


    Postulate 3 quantum measurement
    Postulate 3: realization of Toffoli, composed of 2-qubit gatesQuantum Measurement


    Computational basis a reminder
    Computational Basis – a reminder realization of Toffoli, composed of 2-qubit gates

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

    We recalculate to a new basis


    Example of measurement in different bases
    Example of measurement in different bases realization of Toffoli, composed of 2-qubit gates

    1/2

    The second with probability zero


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Good base

    Not a base


    A simplified bloch sphere to illustrate the bases and measurements
    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
    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.


    A simple example of distinguishable non distinguishable states and measurements
    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
    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


    Observables1
    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
    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
    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


    Duals and inner products are used in measurements
    Duals and Inner Products are used in measurements states and measurements

    <|

    This is inner product not tensor product!

    (

    )

    Remember this is a number

    We prove from general properties of operators


    Duals as row vectors
    Duals as Row Vectors states and measurements

    To do bra from ket you need transpose and conjugate to make a row vector of conjugates.


    General measurement
    General Measurement states and measurements

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


    Illustration of some formalisms used you can calculate measurements from there
    Illustration of some formalisms used, you can calculate measurements from there

    q

    State Vector

    Density State


    Postulate 3 rough form
    Postulate 3, rough form measurements from there

    This is calculate as in previous slide


    The measurement problem
    The Measurement Problem measurements from there

    Can we deduce postulate 3 from 1 and 2?

    Joke. Do not try it. Slides are from MIT.


    More examples how measurement operators act on the state space of a quantum system
    More examples 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:

    Operate on the state space with an operator that preservers unitary evolution:

    Define a collection of measurement operators for our state space:

    Act on the state space of our system with measurement operators:


    Mixed states
    Mixed States space of a quantum system

    • 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 expectationvalue (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!


    Measurement of a state vector using projective measurement
    Measurement of a state vector using projective measurement space of a quantum system

    Operate on the state space with an operator that preservers unitary evolution:

    Define observables:

    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


    The density matrix and the trace ensembles of quantum states basic definitions and importance 1
    The Density Matrix and the Trace space of a quantum systemEnsembles of quantum states, basic definitions and importance(1)

    • 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.


    The density matrix and the trace ensembles of quantum states basic definitions and importance 2
    The Density Matrix and the Trace space of a quantum systemEnsembles of quantum states, basic definitions and importance (2)

    • Unitary operations on a density matrix are expressed as:

    • In other words the diagonal 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


    The density matrix and the trace ensembles of quantum states basic definitions and importance
    The Density Matrix and the Trace space of a quantum systemEnsembles of quantum states, basic definitions and importance

    • 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 (defined by linearity)

    • If you want to know about the nth state in a system, you can trace over the other states.


    Measurement of a density state
    Measurement of a density state space of a quantum system

    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:


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    REMINDER: Ensemble point of view space of a quantum system

    Probability of outcome k being in state j

    Probability of being in state j


    Measurement of a density state1
    Measurement of a density state space of a quantum system

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

    p(m)=

    M3

    recall

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


    Postulate 3 quantum measurement1
    Postulate 3: space of a quantum systemQuantum Measurement

    Now we can formulate precisely the Postulate 3


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Now we use this notation for an Example of Qubit space of a quantum systemMeasurement


    What happens to a system after a measurement
    What happens to a system after a Measurement? space of a quantum system

    • 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

    • This “collapse” seems nonunitary (& nonlocal)

      • However, this behavior is now explicable as the expected consensus phenomenon that would be experienced even by entities within a closed, perfectly unitarily-evolving world (Everett, Zurek).


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Distinguishability space of a quantum system

    Recall that M is measurement operator

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

    On the other hand


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Projective Measurements: Average Values and Standard Deviations

    Observable:

    Can write:

    Average value of a measurement:

    Standard deviation of a measurement:



    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Phase Deviations


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Postulate 4: Deviations Composite Systems


    Compound 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
    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
    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


    Entanglement1
    Entanglement Deviations


    Entanglement2
    Entanglement Deviations



    Quantum entanglement
    Quantum Entanglement Deviations

    We assume that we can factorize as tensor product of |a> and |b>

    Leads to contradiction


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Superdense Coding Deviations



    Postulate 4
    Postulate 4 Deviations


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da
    Example to calculate state of a composite system from previous state of it (problem possible for final exam)


    Size of compound state spaces
    Size of Compound State Spaces previous state of it (problem possible for final exam)

    • 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.


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Postulate 4: previous state of it (problem possible for final exam) Composite Systems


    Summary on postulates
    Summary on Postulates previous state of it (problem possible for final exam)

    Hilbert Space

    Evolution

    Measurement

    Tensor Product


    Key points to remember
    Key Points to Remember: previous state of it (problem possible for final exam)

    • 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 remember1
    Key points to remember previous state of it (problem possible for final exam)

    • 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
    Bibliography & acknowledgements previous state of it (problem possible for final exam)

    • 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.


    Additional slides
    Additional Slides previous state of it (problem possible for final exam)


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    General Measurements in compound spaces previous state of it (problem possible for final exam)


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Uncertainty previous state of it (problem possible for final exam)Principle


    Postulates of quantum mechanics sources angela antoniu david fortin artur ekert michael frank kevin irwig anuj da

    Positive Operator-Valued Measurements (POVM) previous state of it (problem possible for final exam)