Basics of Quantum Theory

1
Basics of Quantum Theory
2
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 phys. systs.
• 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
3
Closed vs. Open Systems

• A subsystem is closed to the extent that no
  particles, information, energy, or entropy (terms
  to be defined) 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)

4
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 concrete description.
• That system (to the entity) is a concrete system.
• Other descriptions are abstract descriptions.
• 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.

5
System Descriptions

• Classical physics
• A system could be completely described by giving
  a single state S out of the set ? of all possible
  states.
• Statistical mechanics
• Instead, give a probability distribution function
  p??0,1 stating that the system is in state S
  with probability p(S).
• Quantum mechanics
• Give a complex-valued wavefunction ?? ? C,
  ?(S)?1, implying the system is instate S with
  probability ?(S)2.

6
States State Spaces

• A possible state S 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.
• So far, the concepts weve discussed can be
  applied to either classical or quantum physics
• Now, lets get to the uniquely quantum stuff

7
Distinguishability of States

• Classical quantum mechanics differ crucially
  regarding the distinguishability of states.
• In classical mechanics, there is no issue
• Any two states s,t are either the same (st), or
  different (s?t), and thats 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.

8
State Vectors Hilbert Space

• Let S be any maximal set of distinguishable
  possible states s, t, of an abstract system A.
• I.e., no possible state that is not in S is
  perfectly distinguishable from all members of S.
• Identify the elements of S with unit-length,
  mutually-orthogonal (basis) vectors in an
  abstract complex vector space H.
• The systems Hilbert space
• Postulate 1 Each possible state ? ofsystem A
  can be identified with a unit-length vector in
  the Hilbert space H.

t
s
?
9
(Abstract) Vector Spaces

• A concept from abstract linear algebra.
• A vector space, in the abstract, is any set of
  objects that can be combined like vectors, i.e.
• You can add them
• Addition is associative commutative
• Identity law holds for addition to zero vector 0
• You can multiply them by scalars (incl. ?1)
• Associative, commutative, and distributive laws
  hold
• Note There is no inherent basis (set of axes)
• The vectors themselves are the fundamental
  objects,rather than being just lists of
  coordinates

10
Hilbert spaces

• A Hilbert space H is a vector space in which the
  scalars are complex numbers, with an inner
  product (dot product) operation ? HH ? C
• See Hirvensalo p. 107 for defn. of inner product
• x?y (y?x) ( complex conjugate)
• x?x ? 0
• x?x 0 if and only if x 0
• x?y is linear, under scalar multiplication
  and vector addition within both x and y

Componentpicture
y
Another notation often used
x
x?y/x
bracket
11
Review The Complex Number System

• It is the extension of the real number system via
  closure under exponentiation.
• (Complex) conjugate
• c (a bi) ? (a ? bi)
• Magnitude or absolute value
• c2 cc a2b2

i
The imaginaryunit
c
b

?
a
Real axis
Imaginaryaxis
?i
12
Review Complex Exponentiation

• Powers of i are complex units
• Note
• e?i/2 i
• e?i ?1
• e3? i /2 ? i
• e2? i e0 1

e?i
i
?
?1
1
?i
13
Vector Representation of States

• Let Ss0, s1, be any maximal set of
  mutually distinguishable states, indexed by i.
• A basis vector vi identified with the ith such
  state can be represented as a list of numbers
• s0 s1 s2 si-1 si si1
• vi (0, 0, 0, , 0, 1, 0, )
• Arbitrary vectors v in the Hilbert space H can
  then be defined by linear combinations of the vi
• And the inner product is given by

14
Diracs Ket Notation

• Note The inner product definition is the
  same as the matrix product of x, as a
  conjugated row vector, times y, as a normal
  column vector.
• This leads to the definition, for state s, of
• The bra ?s means the row matrix c0 c1
• The ket s? means the column matrix ?
• The adjoint operator takes any matrix Mto its
  conjugate transpose M ? MT, so?s can be
  defined as s?, and x?y xy.

Bracket
15
Distinguishability of States, again

• State vectors s and t are (perfectly)
  distinguishable or orthogonal (write s?t)
  iff st 0. (Their inner product is zero.)
• State vectors s and t are perfectly
  indistinguishable or identical (write st)
  iff st 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 st. Note amplitudes are complex numbers.

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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.
• Postulate 2 For a system prepared in state t,
  any measurement that asks is it in state s?
  will say yes with probability P(st) st2
• After the measurement, the state is changed, in a
  way we will define later.

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A Simple Example

• Suppose abstract system S has a set of only 4
  distinguishable possible states, which well
  call s0, s1, s2, and s3, with corresponding ket
  vectors s0?, s1?, s2?, and s0?.
• Another possible state is then the unit 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.

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Linear Operators

• V,W Vector spaces.
• A linear operator A from V to W is a
  linear function AV?W. An operator on V is
  an operator from V to itself.
• Given bases for V and W, we can represent linear
  operators as matrices.
• An Hermitian operator H on V is a linear operator
  that is self-adjoint (HH).
• Its diagonal elements are real.

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Eigenvalues Eigenvectors

• v is called an eigenvector of linear operator A
  iff A just multiplies v by a scalar a, i.e. Avav
  
• eigen (German) means characteristic
• a, the eigenvalue corresponding to
  eigenvector v, is just the scalar that A
  multiplies v by
• a is degenerate if it is shared by 2
  eigenvectors that are not scalar multiples of
  each other
• Any Hermitian operator has all real-valued eigenv
  ectors, which form an orthogonal set

20
Observables

• A Hermitian operator H on the set 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.
• Postulate 3 Every measurable physical property
  of a system can be described by a corresponding
  observable H. Measurement outcomes correspond to
  eigenvalues of H.
• The measurement can also be thought of as a
  yes-no test that compares the state with each of
  the observables normalized eigenvectors.

21
Wavefunctions

• Given any set S?H of system states,
• Whether all mutually distinguishable, or not,
• a quantum state vector v can be translated to a
  wavefunction ?S?C, giving, for each state s?S,
  the amplitude ?(s) of that state.
• When s is some other state vector, and the
  actual state is v, then ?(s) is just sv.
• Whenever S includes a basis set, ? determines v.
• ? is called a wavefunction because its dynamics
  takes the form of a wave equation when S ranges
  over a space of positional states.

22
Time Evolution

• Postulate 4 (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 the amplitude of a particular state at
  t1 leads to a correspondingly small change in the
  amplitude of the corresponding state at t2!
• Chaotic 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 outcomes ? true chaotic/analog
  computing doesnt exist

23
Schrödinger's Wave Equation

• Start w. classical Hamiltonian energy
  equation H K P (K kinetic, P
  potential)
• Express K in terms of momentum K ½mv2 p2/2m
• Substitute H i??/t and p i??/x
• Apply to wavefunction ? over position states x

(Where ?/a ? ?/?a)
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Multidimensional Form

• For a system with states given by (x,t) where t
  is a global time coordinate, and x describes N/3
  particles (p0,,pN/3-1) with masses (m0,,mN/3-1)
  in a 3-D Euclidean space, where each pi is
  located at coordinates (x3i, x3i1, x3i2), and
  where particles interact with potential energy
  function P(x,t), the wavefunction ?(x,t) obeys
  the following (2nd-order, linear, partial)
  differential equation

25
Features of the wave equation

• Particles momentum state p is encoded by their
  wavelength ?, as per ph/?
• The energy of a state is given by the frequency
  f of rotation of the wavefunction in
  the complex plane Ehf.
• By simulating this simple equation, one can
  observe basic quantum phenomena, such as
• Interference fringes
• Tunneling of wave packets through potential
  energy barriers
• Demo of SCH simulator

26
Gaussian wave packet moving to the rightArray
of small sharp potential-energy barriers
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Initial reflection/refraction of wave packet
28
A little later
29
Aimed a little higher
30
A faster-moving particle
31
Compound Systems

• Let CAB 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 CA?B
  given by the tensor product of spaces A and B,
  with basis states labeled as aibj?.
• E.g., if A has state ?aca0a0 ? ca1
  a1?,while B has state ?bcb0b0 ? cb1 b1?,
  thenC has state ?c ?a??b ca0cb0a0b0?
  ca0cb1a0b1? ca1cb0a1b0? ca1cb1a1b1?

32
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 ?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?

33
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.

34
Unitary Transformations

• A matrix (or linear operator) U is unitary iff
  its inverse equals its adjoint U?1 U
• Some properties of unitary transformations
• Invertible, bijective, one-to-one.
• The set of row vectors is orthonormal.
• Ditto for the set of column vectors.
• Preserves vector length U? ?
• Therefore also preserves total probability over
  all states
• Corresponds to a change of basis, from one
  orthonormal basis to another.
• Or, a generalized rotation of? in Hilbert space

35
After a Measurement?

• After a system or subsystem is measured from
  outside, its state appears to collapse to exactly
  match the measured outcome
• the amplitudes of all states perfectly
  distinguishable from states consistent w. that
  outcome drop to zero
• states consistent with measured outcome can be
  considered renormalized so their probs. 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).

36
Pointer States

• For a given system interacting with a given
  environment,
• The system-environment interactions can be
  considered measurements of a certain observable
  of the system by the environment, and vice-versa.
• For each observable there are certain basis
  states that are characteristic of that
  observable.
• The eigenstates of the observable
• A pointer state of a system is an eigenstate of
  the system-environment interaction observable.
• The pointer states are the inherently stable
  states.

37
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.

38
Simulating the Schroedinger Wave Equation

• A Perfectly Reversible Discrete Numerical
  Simulation Technique

39
Simulating Wave Mechanics

• The basic problem situation
• Given
• A (possibly complex) initial wavefunction
  in an N-dimensional position basis,
  and
• a (possibly complex and time-varying) potential
  energy function ,
• a time t after (or before) t0,
• Compute
• Many practical physics applications...

40
The Problem with the Problem

• An efficient technique (when possible)
• Convert V to the corresponding Hamiltonian H.
• Find the energy eigenstates of H.
• Project ? onto eigenstate basis.
• Multiply each component by .
• Project back onto position basis.
• Problem
• It may be intractable to find the eigenstates!
• We resort to numerical methods...

41
History of Reversible Schrödinger Sim.
See http//www.cise.ufl.edu/mpf/sch

• Technique discovered by Ed Fredkin and student
  William Barton at MIT in 1975.
• Subsequently proved by Feynman to exactly
  conserve a certain probability measure
• Pt Rt2 It?1It1
• 1-D simulations in C/Xlib written by Frank at MIT
  in 1996. Good behavior observed.
• 1 2-D simulations in Java, and proof of
  stability by Motter at UF in 2000.
• User-friendly Java GUI by Holz at UF, 2002.

(Rreal, Iimag., ttime step index)
42
Difference Equations

• Consider any system with state x that evolves
  according to a diff. eq. that is 1st-order in
  time x f(x)
• Discretize time to finite scale ?t, and use a
  difference equation instead x(t ?t) x(t)
  ?t f(x(t))
• Problem Behavior not always numerically stable.
• Errors can accumulate and grow exponentially.

43
Centered Difference Equations

• Discretize derivatives in a symmetric fashion
• Leads to update rules like x(t ?t) x(t ?
  ?t) 2?t f(x(t))
• Problem States at odd- vs. even-numbered time
  steps not constrainedto stay close to each other!

2?tf
x1
g
x2

g
x3

g
x4

44
Centered Schrödinger Equation

• Schrödingers equation for 1 particle in 1-D
• Replace time ( also space) derivatives with
  centered differences.
• Centered difference equation has realpart at odd
  times that depends only onimaginary part at even
  times, vice-versa.
• Drift not an issue - real imaginaryparts
  represent different state components!

R1
g
?
I2
g
R3

g
I4
?
45
Proof of Stability

• Technique is proved perfectly numerically stable
  convergent assuming V is 0 and ?x2/?t gt ?/m
  (an angular velocity)
• Elements of proof
• Lax-Richmyer equivalence convergence?stability.
• Analyze amplitudes of Fourier-transformed basis
• Sufficient due to Parsevals relation
• Use theorem (cf. Strikwerda) equating stability
  to certain conditions on the roots of an
  amplification polynomial ?(g,?), which are
  satisfied by our rule.
• Empirically, technique looks perfectly stable
  even for more complex potential energy funcs.

46
Phenomena Observed in Model

• Perfect reversibility
• Wave packet momentum
• Conservation of probability mass
• Harmonic oscillator
• Tunneling/reflection at potential energy barriers
  
• Interference fringes
• Diffraction

47
Interesting Features of this Model

• Can be implemented perfectly reversibly, with
  zero asymptotic spacetime overhead
• Every last bit is accounted for!
• As a result, algorithm can run adiabatically,
  with power dissipation approaching zero
• Modulo leakage frictional losses
• Can map it to a unitary quantum algorithm
• Direct mapping
• Classical reversible ops only, no quantum speedup
• Indirect (implicit) mapping
• Simulate p particles on kd lattice sites using pd
  lg k qubits
• Time per update step is order pd lg k instead of
  kpd