EEL 4930

1
EEL 4930 6 / 5930 5, Spring 06Physical Limits
of Computing
http//www.eng.fsu.edu/mpf

• Slides for a course taught byMichael P. Frankin
  the Department of Electrical Computer
  Engineering

2
Physical Limits of ComputingCourse Outline
Currently I am working on writing up a set of
course notes based on this outline,intended to
someday evolve into a textbook

• Course Introduction
• Moores Law vs. Modern Physics
• Foundations
• Required Background Material in Computing
  Physics
• Fundamentals
• The Deep Relationships between Physics and
  Computation

• IV. Core Principles
• The two Revolutionary Paradigms of Physical
  Computation
• V. Technologies
• Present and Future Physical Mechanisms for the
  Practical Realization of Information Processing
• VI. Conclusion

3
Part II. Foundations

• This first part of the course quickly reviews
  some key background knowledge that you will need
  to be familiar with in order to follow the later
  material.
• You may have seen some of this material before.
• Part II is divided into two chapters
• Chapter II.A. The Theory of Information and
  Computation
• Chapter II.B. Required Physics Background

4
Chapter II.B. Required Physics Background

• This chapter covers All the Physics You Need to
  Know, for purposes of this course
• II.B.1. Physical Quantities, Units, and
  Constants
• II.B.2. Modern Formulations of Mechanics
• II.B.3. Basics of Relativity Theory
• II.B.4. Basics of Quantum Mechanics
• II.B.5. Thermodynamics Statistical Mechanics
• II.B.6. Solid-State Physics

5
Section II.B.4Basics of Quantum Mechanics

• Systems, Hilbert Spaces, Measurement,
  Observables, Time Evolution, Entanglement,
  Quantum Information

6
Section II.B.4 Basics of Quantum Mechanics

• We break this down into subsections as follows
• (a) Systems, subsystems, states, descriptions
• (b) State vectors and Hilbert spaces
• (c) Measurement and Observables
• (d) Unitary time evolution and wave equation
• (e) Compound systems and Entanglement
• (f) The nature of Quantum Information

7
Subsection II.B.4.aSystems, Subsystems, States,
Descriptions

• Merge this section into the discussion of forms
  in the Information module?

8
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.
  systems
• One physical system A is a subsystem of another
  system B (write A?B) iff As region is
  completely contained within Bs.
• Later, we will make these definitions a bit more
  general, formal precise.

B
A
9
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)

10
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.

11
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., the system in question could be fully
  fleshed out by the more detailed description of
  C.
• The state space of the abstract system A is the
  set of all possible states of A.
• So far, all the concepts weve discussed can be
  applied to either classical or quantum physics
• Now, lets get to the uniquely quantum stuff

12
System Descriptions in Classical and Quantum
Physics

• Classical physics
• A concrete instance of a system could be
  completely described by giving a single state S
  out of the set ? of all possible states.
• Statistical mechanics
• A description can be to give a probability
  distribution function p??0,1 stating only
  that the system is in state S with probability
  p(S).
• Note this is more abstract than giving the exact
  state.
• Constraint on p The probabilities sum to 1.
• Quantum mechanics
• A description can be a complex-valued
  wavefunction ?? ? C, where ?(S)?1, implying
  that the system is in state S with probability
  ?(S)2.
• This is more concrete than a probability
  distribution,but less concrete than an exact
  classical state.
• Constraint Squared magnitudes sum to 1.

S
S
S
p
1
0
S
?
i
-1
1
0
-i
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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.
• They are slightly different but still
  overlapping
• Such states cannot be reliably distinguished by
  any kind of measurement, 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 to begin with.
• We know that these slightly different,
  overlapping states really exist because we can
  see their different statistical properties given
  many identically-prepared systems (instances of
  the form).

14
Subsection II.B.4.bState Vectors and Hilbert
Spaces

• Vector Spaces, Complex Numbers, Hilbert Spaces,
  Ket Notation, Distinguishability

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State Vectors Hilbert Space

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

t
s
?
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(Abstract) Vector Spaces

• A concept from abstract linear algebra.
• Other concepts in abstract algebra Groups,
  rings, fields, algebras
• 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 a unique null
  vector 0
• You can multiply them by scalars (including ?1)
• Associative, commutative, and distributive laws
  hold
• Note A vector space has no inherent basis (set
  of axes)!
• The vectors themselves are considered to be
  fundamental, geometric objects, rather than being
  just lists of coordinates
• E.g., in the below example, vector v in a
  2-dimensional real vector space can be expressed
  as a linear combination of any given pair of
  basis vectors i, j.

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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 either x or y

implies
Componentpicture
y
Another notation often used
x
bracket
x?y/x
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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

The imaginaryunit
i
c
b
?1
1
a
Real axis
Imaginaryaxis
?i
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Review Complex Exponentiation

• Raising any real number xgt0 to the ith power
  yields a unit-magnitude complex number
• Note
• e?i/2 i
• e?i ?1
• e3? i /2 ? i
• e2? i e0 1

Eulers relation
(Where ? ln x.)
i
e?i
?
1
?1
?i
If we want, we can say that angles are
logarithmicquantities, identify the angle 1 rad
with the logarithmic unit Loge, and write xi
ExpiL CosL i SinL,where LLogx, and
the Cos and Sin functions now take
dimensioned arguments (indefinite log quantities).
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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

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Diracs Ket Notation

• Notice that in the expression for inner product
• Its 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 vector c0 c1
• The ket s? means the column vector
• The adjoint operator takes any matrix M to its
  conjugate transpose M ? MT, so ?s can be
  defined as s?, and we have x?y xy.

Bracket
Innerproduct
Bra ?x
Ket y?
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Distinguishability of States, More Formally

• State vectors s and t are (perfectly) mutually
  distinguishable or orthogonal (write s?t) iff st
  0.
• Their inner product has zero magnitude.
• State vectors s and t are perfectly
  indistinguish-able or equivalent (write st) iff
  st 1.
• Their inner product has unit magnitude.
• Otherwise, s and t are non-orthogonal to each
  other, and also inequivalent.
• We say they are not perfectly distinguishable.
• In such a case we say, the amplitude of state s,
  given state t, is st.
• Note Amplitudes are complex numbers!

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Subsection II.B.4.cMeasurement and Observables

• Measurement Postulate, Schrödingers Cat,
  Operators, Eigenvalues, Eigenvectors, Observables

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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 discuss later.

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A Simple Example

• Suppose abstract system S has a set of only 4
  distinguishable possible states,
• well call them s0, s1, s2, and s3, with
  corresponding ket vectors s0?, s1?, s2?, and
  s3?.
• 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!

26
Schrödingers Cat

• A thought experiment that illustrates the
  unintuitive nature of quantum states.
• An apparatus is set up tokill a cat if an atom
  decays in a certain time (50 prob.).
• The system enters the quantum superposition
  state live cat? dead cat?.
• We cant say that the cat is really either
  alive or dead until we open the box and observe
  it.
• Even then, the true state can validly be
  considered to be we see live cat? we see
  dead cat?.
• Outwardly-spreading entanglement ? Many-worlds
  picture

27
Linear Operators

• Given V,W Vector spaces,
• Definition 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,
  AV?V.
• 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 matrix elements are all real.

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Eigenvalues Eigenvectors

• Vector 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
• The eigenvalue a corresponding to eigenvector v,
  is just the scalar that A multiplies v by
• The eigenvalue a is called degenerate if it is
  shared by at least two independent eigenvectors
  (ones that arent just scalar multiples of each
  other).
• The multiplicity of a is the number of
  independent eigenvectors that share it.
• The eigenvectors of any Hermitian operator H are
  all real-valued and mutually orthogonal.

29
Observables

• A Hermitian operator H on the vector space V is
  called an observable if there is an orthonormal
  (all unit-length, and mutually orthogonal) subset
  of its eigenvectors that forms a basis for V.
• Postulate 3 Every measurable physical property
  of a system can be described by a corresponding
  observable H. The different possible outcomes of
  the measurement correspond to different
  eigenvalues of H.
• The measurement can also be thought of as a set
  of yes-no tests that compares the state with each
  of the observables normalized eigenvectors.

30
Subsection II.B.4.dTime Evolution and the
Schrödinger Wave Equation

• Wavefunctions, Unitary Transformations, Time
  Evolution Operator, Schrödinger Equation

31
Wavefunctions

• Given any set S?H of system states,
• whether all mutually distinguishable, or not,
• Any quantum state vector v in the systems
  Hilbert space can be translated to a
  corresponding wave-function ?S?C,
• This gives, for each state s?S, the amplitude
  ?(s) of that state, given that the actual
  system state is v.
• If s corresponds to state vector s, then?(s)
  sv.
• If S includes a basis set, ? also uniquely
  determines v.
• The function ? is called a wavefunction
  because
• As well see, its dynamics takes the form of a
  wave equation when S ranges over a space of
  positional states.

32
Quantum Dynamics

• A dynamics is a law that determines how states
  change over time.
• E.g., the Euler-Lagrange equations or Hamiltons
  equations, given a suitable Lagrangian or
  Hamiltonian.
• We have seen that quantum states are unit vectors
  in a Hilbert space.
• How then should such states evolve?
• Let us begin by supposing that the dynamical law
  for quantum time-evolution should have the
  following properties
• Linear - It should involve a linear
  transformation of the vector space.
• This is the simplest type of dynamics, and it
  appears sufficient
• Norm-conserving The sum of squared component
  magnitudes should be constant
• Since squared magnitude probability, and total
  probability must always be 1
• Invertible The dynamics should be one-to-one
• Reflects the apparent reversibility of physics
• As evidenced by the success of Hamiltonian
  dynamics and the 2nd law of thermodynamics.
• Continuous The state should only change
  infinitesimally in infinitesimal times.
• This is another apparent property of the world
• Time-independent The law should not change over
  time
• Apparently true, also, its kind of what we mean
  by a law to begin with

33
Unitary Transformations

• A matrix (or linear operator) U is called unitary
  iff its inverse equals its adjoint, that is, U?1
  U
• Some nice properties of unitary transformations
• They are invertible and bijective (one-to-one and
  onto).
• The set of row vectors comprises an orthonormal
  basis.
• Ditto for the set of column vectors.
• Preserves vector length U? ?
• Therefore, also preserves total probability over
  all states
• Implements a change of basis,
• from one orthonormal basis to another.
• Can be thought of as a kind of generalized
  rotation of? in Hilbert space.
• Mathematical fact Any norm-preserving,
  invertible linear transformation of a complex
  vector space is unitary!
• Thus our dynamics taking us from t1 to t2 must be
  a unitary operator

34
The Time Evolution Operator

• Since the dynamics is supposed to be
  time-independent, U(t1?t2) cant depend on the
  absolute value of t1 or t2, but can only depend
  on ?t t2 - t1.
• Thus we can write U(t1?t2) U(?t).
• Also, note that U(2?t)U(?t)U(?t) U(?t)2,
• And more generally U(n?t) U(?t)n.
• Thus, all ?t values can likewise be viewed as
  multiples of some infinitesimal time increment
  dt.
• Therefore, U(?t) Ud?t for some infinitesimal
  (near-identity) base unitary Ud, and with ?t
  expressed in dt units.
• Well see its convenient to express the base
  unitary Ud itself as the exponential of a matrix,
  Ud eM
• This will let us write U(?t) eM?t.
• Note that ?t can even remain dimensioned here, so
  long as we arrange for M to have dimensions of
  inverse time.

35
The Exponential of a Matrix

• Recall the Taylor series for theexponential
  function, ex, for x?R
• We can use this equation to also define the
  exponential of a matrix M in terms of powers of
  M
• A useful theorem Any eigenvector v of M having
  eigenvalue a is also an eigenvector of eM, with
  the eigenvalue ea

36
Time-Evolution Unitaries and Hermitian
Hamiltonian Operators

• Let v be an eigenvector of the matrix M we just
  discussed, where Ud eM.
• Thus, v is also an eigenvector of Ud.
• We can ask, what is vs eigenvalue u under Ud?
• Since Ud is length-preserving, u must be some
  complex unit, u ei?, so that we will have u
  1.
• Otherwise, we wouldnt have Udv uvv v.
• Thus, vs eigenvalue under M must be i? for some
  real number ?.
• By the theorem on the previous slide, this is the
  only way that its eigenvalue under Ud can have
  the form ei?!
• In other words, all of Ms eigenvalues are
  imaginary.
• Thus, we can write M iH, where H is a matrix
  with all real eigenvalues
• i.e., H is an Hermitian matrix.
• So, we can write U(?t) eiH?t.
• Note that H is an observable whose value is
  time-independent, since its eigenvectors remain
  unchanged under U(?t).
• We thus identify Hs eigenvalue (after converting
  its frequency units to energy units by
  multiplying by ?) as measuring the systems total
  energy.
• Energy is the conserved quantity associated with
  time-independence
• Thus, H is the operator equivalent of the
  Hamiltonian energy function!

37
Time Evolution

• Postulate 4 (Closed) systems evolve (change
  state) over time ?t via the unitary
  transformation U(?t) given by ExpiH?t, where H
  is the Hamiltonian energy observable.
• Note that since U is linear, a small-factor
  change in the amplitude of any particular state
  at t1 necessarily leads to only a correspondingly
  small change in the amplitude of the any 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 is physically
  impossible!

(U-1 U)
38
Deriving the General Form of the Schrödinger
Equation

• Let ?t t, and lets differentiate U(t) eiHt
  with respect to time
• Now, apply the first and last formulas in the
  above equation to an initial state ?(0)
• But now, U(t)?(0) ?(t), so we have
• This key differential equation (theSchrödinger
  equation) directly tells us how any given
  instantaneous wavefunction ?(t) evolves over time
  t.
• We can also write it more concisely in operator
  forms

or
(sign is arbitrary)
39
Schrödinger's Wave Equation

• How do we turn this generic operator equation
  into something that tells us about particles
• Start w. classical Hamiltonian energy
  equation H EK EP (K kinetic, P
  potential)
• Express (nonrelativistic) EK in terms of momentum
  p EK ½mv2 p2/2m
• Substitute H i??t and p -i??x
• Apply to wavefunction ? over position states x

(Where ?a ? ?/?a)
40
Consistency with Hamiltons Equations

• Recall the generic Hamiltons equations
• Are our quantum definitions of the Hamiltonian
  energy and momentum operators consistent with
  them?
• Almost, it seems But the belowderivation is
  most likely illegal anyway

Oops, sign iswrong!
41
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 EP(x,t), the wavefunction ?(x,t) obeys
  the following (2nd-order, linear, partial)
  differential equation

42
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

43
Gaussian wave packet moving to the rightArray
of small sharp potential-energy barriers
44
Initial reflection/refraction of wave packet
45
A little later
46
Aimed a little higher
47
A faster-moving particle
48
Relativistic Wave Equations

• Unfortunately, despite its many practical
  successes, the literal Schrödingers equation is
  not relativistically invariant.
• That is, it does not retain the same form in a
  boosted frame.
• However, solutions to the free Schrödingers
  equation (where V0) can be given a
  self-consistent relativistic interpretation.
• Let p -i??x be relativistic momentum,
• Let m i??t be rest mass in the particles frame
  of ref.
• Taking the derivative along an isospatial, i.e.,
  the proper time t axis
• Let E i??t be relativistic energy of the
  particle
• Then, E2 p2 m2 is easily shown to be true for
  plane-wave solutions
• Lines of constant phase angle are the isochrones
  of the moving particle.
• And everything transforms properly to a new
  reference frame.
• In fact, the solutions to the free Schrödingers
  equation closely correspond to solutions to the
  relativistic Klein-Gordon equation ?µ?µ
  m2f(xµ) 0.
• This describes a free, massive scalar particle.

49
Relativistic Spacetime Wavefunction Examples

• Here is an electron wavefunction (m0 9.1?10-28
  g) over spacetime for various velocity
  eigenstates, approaching the speed of light
• Scale of these images
• The width of the physical space is 16 pm (lt 1/5 H
  atom)
• The vertical time interval shown is 54 zs (very
  short!)
• The arrow shows the world-line of a point moving
  at the given velocity
• The lines of constant color (phase) are the
  isochrones of the electrons rest frame
• Notice that the angled arrows cross fewer
  isochrones than the straight one!
• This is time dilation, seen directly in the
  electron wavefunction over spacetime!

ß 0 ? 1
ß 0.50? 0.87
ß 0.98? 0.20
ß 0.90? 0.44
t
x
50
Normal Frame Viewvs. Mixed Frame View
Rightwardmovingelectronwave-functionas
seenin fixedstandardframe(x, t)
t', x'0
t', x'0
t', x'0
t', x'0
t
x
ß 0 ? 1ß/? 1
ß 0.60? 0.80 ß/? 0.75
ß 0.98? 0.20ß/? 4.92
ß 0.90? 0.44 ß/? 2.06
Fixedelectronwave-functionfrom
aleftwardmovingmixedframe(x, t')
Something is still not quite right in the below
t, x'0
t, x'0
t, x'0
t, x'0
t'
x
Electron moving throughits internal time (t')
Electron moving mostlythru our space (x)
51
Subsection II.B.4.eCompound Quantum Systemsand
Entanglement

• A Few Basic Concepts

52
Compound Quantum Systems

• Let CAB be a system composed of two separate
  subsystems A,B 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? ai?bj?.
• Well formally define tensor products later on
• 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?

(Use distributive law)
53
Entanglement

• If the state of a 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 definite individual states.
• E.g. 00?01?10?11?(0?1?)?(0?1?)
• Otherwise, A and B are entangled (quantumly
  correlated) their states are not independent.
• E.g. 00?11?

(State has entropy 0 but mutual information 2
bits!)
54
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!
• Many possible states of the compound system will
  have 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
• This illustrates the many worlds nature of
  quantum mechanics.
• And the enormous number of possible worlds
  involved.

55
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 to be renormalized so that their
  probs. sum to 1
• This collapse appears 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).

56
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.
• Any each observable, there are certain basis
  states that are characteristic of that
  observable.
• These are just 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.

57
Key Points to RememberAbout Quantum Mechanics

• An abstractly-specified system may have many
  possible states not all pairs are
  distinguishable.
• A quantum state/vector/wavefunction ? assigns a
  complex-valued amplitude ?(s) to each state s.
• The probability of state s is ?(s)2, the square
  of ?(s)s length in the complex plane.
• Quantum states evolve over time via unitary
  (invertible, length-preserving) transformations.

58
Subsection II.B.4.fThe Nature of Quantum
Information

• Generalizing classical information theory
  concepts to fit quantum reality

59
Density Operators

• For any given state ??, the probabilities of all
  the basis states si are determined by an
  Hermitian operator or matrix ? (called the
  density matrix)
• Note that the diagonal elements ?i,i are just the
  probabilities of the basis states i.
• The off-diagonal elements are called
  coherences.
• They describe the quantum entanglements that
  exist between basis states.
• The density matrix describes the state ??
  exactly!
• It (redundantly) expresses all of the quantum
  info. in ??.

60
Mixed States

• Suppose the only thing one knows about the true
  state of a system that it is chosen from a
  statistical ensemble or mixture of state vectors
  vi (called pure states), each with a derived
  density matrix ?i, and a probability Pi.
• In such a situation, in which ones knowledge
  about the true state is expressed as probability
  distribution over pure states, we say the system
  is in a mixed state.
• Such a situation turns out to be completely
  described, for all physical purposes, by simply
  the expectationvalue (weighted average) of the
  vis density matrices
• Note Even if there were uncountably many vi
  going into the calculation, the situation remains
  fully described by O(n2) complex numbers, where n
  is the number of basis states!

61
Von Neumann Entropy

• Suppose our probability distribution over states
  comes from the diagonal elements of some density
  matrix ?.
• But, we will generally also have additional
  information about the state hidden in the
  coherences.
• The off-diagonal elements of the density matrix.
• The Shannon entropy of the distribution along the
  diagonal will generally depend on the basis used
  to index the matrix.
• However, any density matrix can be (unitarily)
  rotated into another basis in which it is
  perfectly diagonal!
• This means, all its off-diagonal elements are
  zero.
• The Shannon entropy of the diagonal probability
  distribution is always minimized in the diagonal
  basis, and so this minimum is selected as being
  the true (basis-independent) entropy of the mixed
  quantum state ?.
• It is called the von Neumann entropy.

62
V.N. entropy, more formally

• The trace Tr M just means the sum of Ms diagonal
  elements.
• The ln of a matrix M just denotes the inverse
  function to eM. See the logm function in
  Matlab
• The exponential eM of a matrix M is defined via
  the Taylor-series expansion ?i0 Mi/i!

(Shannon S)
(Boltzmann S)
63
Quantum Information Subsystems

• A density matrix for a particular subsystem may
  be obtained by tracing out the other
  subsystems.
• Means, summing over state indices for all systems
  not selected.
• This process discards information about any
  quantum correlations that may be present between
  the subsystems!
• Entropies of the density matrices so obtained
  will generally sum to gt that of the original
  system. (Even if the original state was pure!)
• Keeping this in mind, we may make these
  definitions
• The unconditioned, reduced or marginal quantum
  entropy S(A) of subsystem A is the entropy of
  the reduced density matrix ?A.
• The conditioned quantum entropy S(AB)
  S(AB)-S(B).
• Note this may be negative! (In contrast to the
  classical case.)
• The quantum mutual information I(AB)
  S(A)S(B)-S(AB).
• As in the classical case, this measures the
  amount of quantum information that is shared
  between the subsystems
• Each subsystem knows this much information
  about the other.

64
Tensors and Index Notation

• For our purposes, a tensor is just a generalized
  matrix that may have more than one row and/or
  column index.
• We can also define a tensor recursively as a
  number or a matrix of tensors.
• Tensor signature An (r,c) tensor has r row
  indices and c column indices.
• Convention Row indices are shown as subscripts,
  and column indices as superscripts.
• Tensor product An (l,k) tensor T times an (n,m)
  tensor U is a (ln,km) tensor V formed from all
  products of an element of T times an element of
  U
• Tensor trace The trace of an (r,c) tensor T with
  respect to index k (where 1 k r,c) is given
  by contracting (summing over) the kth row index
  together with the kth column index

Example a (2,2)tensor T in which all 4indices
take on values from the set 0,1
(I is the set of legal values of indices rk and
ck) ?
65
Quantum Information Example
AB AB

• Consider the state vAB 00?11? of compound
  system AB.
• Let ?AB vv.
• Note that the reduced density matrices ?A ?B are
  fully classical
• Lets look at the quantum entropies
• The joint entropy S(AB) S(?AB) 0 bits.
• Because vAB is a pure state.
• The unconditioned entropy of subsystem A is S(A)
  S(?A) 1 bit.
• The entropy of A conditioned on B is S(AB)
  S(AB) - S(A) -1 bit!
• The mutual information I(AB) S(A) S(B) -
  S(AB) 2 bits!

00? 01? 10? 11?
66
Quantum vs. Classical Mutual Info.

• 2 classical bit-systems have a mutual information
  of at most one bit,
• Occurs if they are perfectly correlated, e.g.,
  00, 11
• Each bit considered by itself appears to have 1
  bit of entropy.
• But taken together, there is really only 1 bit
  of entropy shared between them
• A measurement of either extracts that one bit of
  entropy,
• Leaves it in the form of 1 bit of incompressible
  information (to the measurer).
• The real joint entropy is 1 bit less than the
  apparent total entropy.
• Thus, the mutual information is 1 bit.
• But, 2 quantum bit-systems (qubits) can have a
  mutual info. of two bits!
• Occurs in maximally entangled states, such as
  00?11?.
• Again, each qubit considered by itself appears to
  have 1 bit of entropy.
• But taken together, there is no entropy in this
  pure state.
• A measurement of either qubit leaves us with no
  entropy, rather than 1 bit!
• If done carefully see next slide.
• The real joint entropy is thus 2 bits less than
  the apparent total entropy.
• Thus the mutual information is (by definition) 2
  bits.
• Both of the apparent bits of entropy vanish if
  either qubit is measured.
• Used in a communication tech. called quantum
  superdense coding.
• 1 qubits worth of prior entanglement between two
  parties can be used to pass 2 bits of classical
  information between them using only 1 qubit!

67
Why the Difference?

• Scenario Entity A hasnt yet measured B and C,
  which (A knows) are initially correlated with
  each other, quantumly or classically
• A has measured B and is now correlated with both
  B and C
• A can use his new knowledge to uncompute
  (compress away) the bits from both B and C,
  restoring them to a standard state

OrderABC
Classical
Quantum
Knowing he is in state 0?1?, A can unitarily
rotate himself back to state 0?. Look ma, no
entropy!
A, being in a mixed state, still holds a bit of
information that is either unknown (external
view) or incompressible (As internal view), and
thus is entropy, and can never go away (by the
2nd law of thermo.).
68
Simulating the Schroedinger Wave Equation

• A Perfectly Reversible Discrete Numerical
  Simulation Technique

69
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...

70
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...

71
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)
72
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.

73
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

74
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
?
75
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
• This is 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.

76
Phenomena Observed in Model

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

77
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