Classical and Quantum Info Theory

Lecture on classical and quantum information theory
Krzysztof Pomorski
University of Warsaw
kdvpomorski@gmail.com
30 marca 2017
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 1 / 66

Zakres wykładu:
Omówione zastaną wybrane zagadnienia z klasycznej i kwantowej teorii
informacji. Potwierdzona zostanie teza Landauera. Wskazane zostaną
procesy utraty informacji w trakcie przekazywania danych na odległość.
Nakreślone zostanie odwołanie do termodynamiki. Wskazana zostanie
między innymi teza Żurka o nieklonowalności kubitu.

1 Motivation
2 Interlink between information and physics
Maxwell’s demon
The four laws of thermodynamics
Thermodynamics of big systems
3 Classical information theory
Shannon theory
Classical measurement
4 Quantum information theory
Analogy between QM and Statistical Physics
Quantum measurement
Qubit as Bloch sphere
New properties from quantum mechanics
Interaction of 2 quantum systems
5 Classical vs quantum information theory

Motivation
1. China has just launched the world’s ﬁrst quantum communications
satellite [600kg]. The satellite is both an extreme test of the weird
properties of quantum mechanics, and a technology tested for what could
be the start of a global, unhackable communications network.
2. Quantum Cryptography is like computing all over again. We cannot
possibly tell what the implications may be. Andrew Hilton, director of CSFI
3. Quantum computer going beyond classical computer.
4. Construction of supersensitive devices.

Information and physics [4]

Chaos vs order in classical systems [12]
Detection of chaos in simple classical systems.

Maxwell’s demon
Maxwell’s demon is the name given to a thought experiment designed to
question the possibility of violating the second law of thermodynamics. It
was formulated and named after the Scottish physicist James Clerk
Maxwell in 1867.
Maxwell’s demon demonstration turns information into energy [if we
accomodate all gas particle on one side thanks to information on position
of particles].

Small quantum systems
The Ohm law or Heat Flow law does not work properly in small systems
since electron or phonon ﬂow is not fully deterministic [it works on
average].

The four laws of thermodynamics
Zeroth law of thermodynamics: If two systems are in thermal
equilibrium with a third system, they are in thermal equilibrium with each
other. This law helps deﬁne the notion of temperature.
First law of thermodynamics: When energy passes, as work, as heat, or
with matter, into or out from a system, the system’s internal energy
changes in accord with the law of conservation of energy.
Second law of thermodynamics: In a natural thermodynamic process,
the sum of the entropies of the interacting thermodynamic systems
increases.
Third law of thermodynamics: The entropy of a system approaches a
constant value as the temperature approaches absolute zero.With the
exception of non-crystalline solids (glasses) the entropy of a system at
absolute zero is typically close to zero, and is equal to the logarithm of the
product of the quantum ground states.

Thermodynamics of big systems [4]

Gibbs entropy [11]
In statistical thermodynamics, entropy S is a measure of the number
of microscopic configurations Ω that a thermodynamic system can
have when in a state as specified by some macroscopic variables.
Specifically, assuming that each of the microscopic configurations is
equally probable, the entropy of the system is the natural logarithm of that
number of configurations, multiplied by the Boltzmann constant kb. Let us
consider the system with W degeneracies for given energy. For fixed
internal energy U, volume V, number of particles N,
S = −kb
i
pi log(pi ) = −kb
i
(1/wi )log(1/wi ) = kblogW (U, V , N)
(1)
We can view W = Ω as a measure of our lack of knowledge about
a system [microcanonical ensemble]. 3 type of ensembles are given as:
(1) The Microcanonical ensemble is an isolated system.
(2) The Canonical ensemble is a system in contact with a heat bath.
(3) The Grand Canonical ensemble is a system in contact with a heat and
particle bath.

Schematic representation of communication channel
Rysunek: From Shannon ’s A Mathematical Theory of Communication, page 3.

The Shannon information content h(x) of an outcome x is deﬁned to be
h(x) = log2(
1
P(x)
). (2)
It is measured in bits. [The word ’bit’ is also used to denote a variable
whose value is 0 or 1].
The entropy of an ensemble X is deﬁned to be the average Shannon
information content of an outcome:
H(x) =
1
P(x)
log2(
1
P(x)
), (3)
with the convention for P(x)=0 that 0 × log(1/0) = 0, since
limΘ→0+ Θlog(1/Theta) = 0.

Deﬁnition of channel capacity C
The capacity C of a discrete channel is given by
C = LimT→∞
logN(T)
T
, (4)
where N(T) is the number of allowed signals of duration T.

The ﬁrst theorem deals with communication over a
noiseless channel.
Let a source have entropy H(bits per symbol) and a channel have a
capacity C (bits per transmit at the average rate C/H - symbols per
second over the channel where is arbitrarily small. It is not possible to
transmit at an average rate greater than C/H.

Shannon-Hartley theorem
It states that the channel capacity C, meaning the theoretical
tightest upper bound on the information rate of data that can be
communicated at an arbitrarily low error rate using an average
received signal power S through an analog communication channel
subject to additive white Gaussian noise of power N:
C = B log2 1 +
S
N
(5)
where: C - the channel capacity in bits per second, a theoretical upper
bound on the net bit rate (information rate, sometimes denoted I)
excluding error-correction codes; B - the bandwidth of the channel in hertz
(passband bandwidth in case of a bandpass signal); S - the average
received signal power over the bandwidth (in case of a carrier-modulated
passband transmission, often denoted C), measured in watts (or volts
squared); N - the average power of the noise and interference over the
bandwidth, measured in watts (or volts squared).

Classical channel with white noise
The capacity of a channel of band W perturbed by white thermal noise
power N when the average transmitter power is limited to P is given by
C = W × log(P + S/N) (6)

Landauer principle: .Each time a single bit of information is erased it the
amount of energy dissipated to environment is kBTln2. where T is the
temperature of enviroment and kB is Boltzmann constant.

Derivation of Landauer principle [4]

Classical measurement-example
Imagine two wooden balls of Radius R1 and R2 in space [as far from
Pluton ”planet”] that are charged with Q1 and Q2 where Q1 and Q2 have
opposite signs and that they have bounded state. They are analogical to
Newton gravitational problem of 2 bodies. We can photograph two moving
balls by shining radiation on this system. In such case we perturbed the
already existing [because of photon pressure] state by changing balls
trajectory, moving some of charge to vacuum and so on. It is impossible to
copy dynamical behavior of the system without changing it by small
perturbation!!!! In very real sense it is example of classical
non-cloning theorem since we cannot copy the exact classical state
[only with certain approximation]. It is desirable to refer to [17].

Analogy between Quantum Mechanics and Statistical
Physics
Superfluid liquid helium flowing out of container against gravitational field.

Hamiltonian and Lagrangian formalism
Hamiltonian equations of motion:
d
dt
pk = −
dH
dqk
,
d
dt
qk = −
dH
dpk
.L =
k
d
dt
(q)p − H (7)
Lagrangian equations of motions:
dL
dqk
=
d
dt
dL
d(dqk
dt )
. (8)
H = Ek + Ep =
1
2
m(
d
dt
y)2
+ mgy, L = Ek − Ep =
1
2
m(
d
dt
y)2
− mgy. (9)

Moving from classical to quantum mechanics
We have given equations of motion in Hamiltonian formalism. Suppose
that f (p, q, t) is a function on the manifold. Then
d
dt
f (p, q, t) =
∂f
∂q
dq
dt
+
∂f
∂p
dp
dt
+
∂f
∂t
. Further, one may take p = p(t) and
q = q(t) to be solutions to Hamilton’s equations; that is,
˙q = ∂H
∂p = {q, H}



˙q = ∂H
∂p = {q, H}
˙p = −∂H
∂q = {p, H}
Then
d
dt
f (p, q, t) =
∂f
∂q
∂H
∂p
−
∂f
∂p
∂H
∂q
+
∂f
∂t
= {f , H} +
∂f
∂t
.
Quantization is by replacement Poisson bracket {A, H} with
commutator[A, H] = AH − HA. All observables in Quantum Mechanics
as x, p or any other are operators and they do not need to commute and
they cannot be fully determined at the same time [as it is in classical
mechanics] in general!!! [For example ˆx = x and ˆpx = i
d
dx ] Since
[ˆx, ˆp] = i I they do not commute and cannot be measured at the same
time. For non-commuting variables x and p we have∆x∆p >= /2.

Quantum harmonic oscillator
(H0(x) + V (x))ψ(x) = (
1
2m
ˆp2
+ ˆV (x))ψ(x) = Eψ(x) (10)
(−
2
2m
d2
dx2
+
1
2
kx2
)ψ(x) = Eψ(x) (11)
Here ψ(x) is eigenfunction and E is eigenvalue.

Quantum measurement-example
Particle is at certain measurement with output A1 with certain probability.
Once the measurement says A1 value the whole wavefunctions collapses
and particle in only this state with probability 1.

Qubit and quantum-mechanics
Rysunek: Qubit in hydrogen atom.
ˆH = ( ˆH0 + V (x))|ψ >= E|ψ > (12)
|ψ(x) >=






ψ1(x)
ψ2(x)
...
ψm(x)






(13)

|ψ >= cos(Θ/2)|0 > +sin(Θ/2)eiϕ
|1 >, (14)
|ψ >= cos(Θ/2)
0
1
+ sin(Θ/2)eiϕ 1
0
, (15)
Bloch sphere representation of qubit, where Θ is the ’latitude’ and ϕ the
’longitude’ angles of ϕ on the Bloch sphere.

Physical implementation of qubit
Rysunek: Josephson junction as system of two coupling superconductors.

T1 is the average time that the system takes for its excited state |1 > to
decay to the ground state |0 >. T2 represents the average time over which
the qubit energy-level diﬀerence does not vary [10].

Quantum interference
Electron [particle] behaves as wave !!!

Quantum parallelism
Rysunek: Photon distribution in system.

Quantum entanglement
|ψ >=
1
2
(|0 > |1 > −|1 > |0 >) (16)
Entanglement is the strange phenomenon in which two quantum particles
become so deeply linked that they share the same existence. When this
happens, a measurement on one particle immediately influences the other,
regardless of the distance between them. Measurement with 1 in first qubit
gives state
(|1 >< 1|×I)|ψ >= (|1 >< 1|×I)
1
2
(|0 > |1 > −|1 > |0 >) = −
1
2
|1 > |0 >
(17)
while the measurement of 0 in first qubit gives the state
(|0 >< 0|×I)|ψ >= (|0 >< 0|×I)
1
2
(|0 > |1 > −|1 > |0 >) = +
1
2
|0 > |1 > .
(18)

Cooper pair splitter

Interaction of two quantum systems A and B [as qubits]
ˆH = ˆHa + ˆHb + ˆHab = Ha(2times2) × I2by2 + I2by2 × Hb(2times2) + Hab(4times4)
(19)
Here × denotes tensor product and I is identity matrix. The eigenstate of
two isolated (non-interacting systems ˆHab = 0) A and B is given as
|ψ >= |ψA > |ψB >=





ψ1A
ψ2A
ψ1B
ψ2B





. (20)
Consequently we have
(Ha + Hb)|ψ >=





HA(1, 1) HA(1, 2) 0 0
HA(2, 1) HA(2, 2) 0 0
0 0 HB(1, 1) HB(1, 2)
0 0 HB(1, 1) HB(1, 2)





= E|ψ > .
(21)

Idea of quantum computer
The DiVincenzo criteria is a list of conditions that are necessary for
constructing a quantum computer proposed by the theoretical physicist
David P. DiVincenzo in his 2000 paper ”The Physical Implementation of
Quantum Computation”. Quantum computation was first proposed by
Richard Feynman (1982) as a means to efficiently simulate quantum
systems. There have been many proposals of how to construct a quantum
computer, all of which have varying degrees of success against the
different challenges of constructing quantum devices. Some of these
proposals involve using superconducting qubits, trapped ions, liquid and
solid state nuclear magnetic resonance or optical cluster states all of which
have remarkable prospects, however, they all have issues that prevent
practical implementation. The DiVincenzo criteria are a list of conditions
that are necessary for constructing the quantum computer as proposed by
Feynman.

Idea of quantum computer and DiVincenzo criteria
In order to construct a quantum computer the following conditions must
be met by the experimental setup. The first five are necessary for quantum
computation and the remaining two are necessary for quantum
communication.
1. A scalable physical system with well characterised qubits.
2. The ability to initialise the state of the qubits to a simple fiducial state.
3. Long relevant decoherence times.
4. A universal set of quantum gates.
5. A qubit-specific measurement capability.
6. The ability to interconvert stationary and flying qubits.
7. The ability to faithfully transmit flying qubits between specified
locations.

Classical annealing

Quantum annealing

Quantum memory

Fidelity

Quantum memory performance

Physical system implementing quantum annealing

Entanglement in Josephson junction system

Unitary quantum gates
The diagram representing the action of a unitary matrix U corresponding
to a quantum gate on a qubit in a state UU† = 1.

CNOT gate

Basic quantum gates

Zurek no-cloning reasoning

Quantum teleportation
Quantum reexportation is a process by which quantum information (e.g.
the exact state of an atom or photon) can be transmitted (exactly, in
principle) from one location to another, with the help of classical
communication and previously shared quantum entanglement between the
sending and receiving location.

Example of weak measurement

Concept of density matrix
The density matrix is generalization of description given by Schrodinger
equation and formally is deﬁned as the outer product of the wavefunction
and its conjugate so it is matrix of the form as ρ(t) = |ψ(t) >< ψ(t)|. We
have H|ψ(t) >= E|ψ(t) > and < ψ(t)|H† =< ψ(t)|E. Liouville-Von
Neumann equation describes the dynamics of density matrix with time:
d
dt
ρ =
i
[ρ, H]. (22)
Average of observable A is given as
< A >= tr(Aρ). (23)
In case of thermal ensemble in equilibrium we have density matrix given as
Z = tr(e−H/(kT))-partition function, ρequilibrium = e−En/kT /Z
[thermodynamical ensemble].
ρ2
=
1 for pure state
< 1 for mixed state
(24)

Tensor product of matrices
a11 a12
a21 a22
×
b11 b12
b21 b22
=
=






a11
b11 b12
b21 b22
a12
b11 b12
b21 b22
a21
b11 b12
b21 b22
a22
b11 b12
b21 b22







Quantum Entropy (Von Neumann entropy)
Deﬁnition of Entanglement in density matrix picture
Entangled state is when its density matrix cannot be written as tensor
product of two density matrices that is ρ = ρA × ρB.
Quantum entropy is given by formula
H(ρ) = Tr[ρlog(1/ρ)] = −Tr[ρlog(ρ)]. (25)
If ρ is the joint state of two quantum systems A and B then the quantum
mutual information is
I(A, B) = H(ρA) + H(ρB) − H(ρ). (26)

QM equation of motion with presence of dissipation
Decoherence and equations of motion.

QM equation of motion with use of Wigner functions
W (x, p) =
1
π
+∞
−∞
ψ†
(x + y)ψ(x − y)e
2iyp
dy (27)
Equations of motion for Wigner function.

Quantum information theory
Quantum information theory deals with four main topics:
(1) Transmission of classical information over quantum channels.
(2) The tradeoﬀ between acquisition of information about a quantum state
and disturbance of the state (brieﬂy included in quantum cryptography).
(3) Quantifying quantum entanglement.
(4) Transmission of quantum information over quantum channels.

Properties of the Isolated Quantum System with Finite Volume and a
Finite Number of Particles [7] 1) These quantum systems evolve for reversible
equations of motion (Schrodinger’s equation)
2) Poincare’s theorem is also accurate for these systems.
3) For quantum systems, it is also possible to define the entropy of ensemble.
Entropy is a measure of uncertainty about the state of a system. A pure state
provides a maximally complete description of a quantum system. Therefore, any
pure state entropy is zero by definition. For the mixed-state case, the system
corresponds to a set of pure states. Therefore, entropy already exceeds zero.
Assume that the probability of a pure state is near 1. This mixed state is almost
pure and its entropy is almost zero. On the other hand, when all pure states of
the mixed state have equivalent probabilities, entropy reaches a maximum.
4) During the evolution of a quantum system, the pure state can evolve to a pure
state only. In the mixed state, the probabilities of pure states also remain
unchanged. Thus, the entropy of ensemble does not change during the evolution
of a quantum system.
5) We can represent a large quantum system by a small number of parameters
named macroscopic parameters. A large set of pure states defined by microscopic
parameters corresponds to this mixed macroscopic state. The entropy of a
macroscopic state can be calculated based on this pure set. We define this
entropy as macroscopic entropy. In contrast with the entropy of ensemble, the
macroscopic entropy should not be conserved during the evolution of a qsystem.

6) A quantum system will not be considered to be an isolated system due
to its interaction with the measuring device. Its initial pure state evolves to
a mixed state and its microscopic entropy increases. This evolution cannot
be reversed by inversion of the measured system as inversion of the
measuring device is also necessary.
Poincare recurrence theorem [ from Wikipedia ]
Any dynamical system defined by an ordinary differential equation
determines a flow map f t mapping phase space on itself. The system is
said to be volume-preserving if the volume of a set in phase space is
invariant under the flow. For instance, all Hamiltonian systems are
volume-preserving because of Liouville’s theorem. The theorem is then: If a
flow preserves volume and has only bounded orbits, then for each open set
there exist orbits that intersect the set infinitely often.

Quantum vs classical information
Quantum information diﬀers strongly from classical information,
epitomized by the bit, in many striking and unfamiliar ways. Among these
are the following:
A unit of quantum information is the qubit. Unlike classical digital states
(which are discrete), a qubit is continuous-valued, describable by a
direction on the Bloch sphere. Despite being continuously valued in this
way, a qubit is the smallest possible unit of quantum information. The
reason for this indivisibility is due to the Heisenberg uncertainty principle:
despite the qubit state being continuously-valued, it is impossible to
measure the value precisely. A qubit cannot be (wholly) converted into
classical bits; that is, it cannot be ’read’. This is the no-teleportation
theorem. Despite the awkwardly-named no-teleportation theorem, qubits
can be moved from one physical particle to another, by means of quantum
teleportation. That is, qubits can be transported, independently of the
underlying physical particle. An arbitrary qubit can neither be copied, nor
destroyed. This is the content of the no cloning theorem and the
no-deleting theorem.

Although a single qubit can be transported from place to place (e.g.
via quantum teleportation), it cannot be delivered to multiple
recipients; this is the no-broadcast theorem, and is essentially
implied by the no-cloning theorem. Qubits can be changed, by applying
linear transformations or quantum gates to them, to alter their state.
Classical bits may be combined with and extracted from configurations of
multiple qubits, through the use of quantum gates. That is, two or more
qubits can be arranged in such a way as to convey classical bits. The
simplest such configuration is the Bell state, which consists of two qubits
and four classical bits (i.e. requires two qubits and four classical bits to
fully describe). Quantum information can be moved about, in a quantum
channel, analogous to the concept of a classical communications channel.
Quantum messages have a finite size, measured in qubits; quantum
channels have a finite channel capacity, measured in qubits per second.
Multiple qubits can be used to carry classical bits.

Although n qubits can carry more than n classical bits of
information, the greatest amount of classical information that can
be retrieved is n. This is Holevo’s theorem. Quantum information,
and changes in quantum information, can be quantitatively
measured by using an analogue of Shannon entropy, called the von
Neumann entropy.

Many of the same entropy measures in classical information theory can
also be generalized to the quantum case, such as Holevo entropy and the
conditional quantum entropy. Quantum algorithms have a diﬀerent
computational complexity than classical algorithms. The most famous
example of this is Shor’s factoring algorithm, which is not known to have a
polynomial time classical algorithm, but does have a polynomial time
quantum algorithm. Other examples include Grover’s search algorithm,
where the quantum algorithm gives a quadratic speed-up over the best
possible classical algorithm. Quantum key distribution allows
unconditionally secure transmission of classical information, unlike classical
encryption, which can always be broken in principle, if not in practice.
(Note that certain subtle points are hotly debated).

Holevo bound
Initially the sender, Alice, holds a long classical message. She encodes
letter i (which appears with probability pi ) of this message into a pure
state which, during the transmission, is turned into a possibly mixed
quantum state q i due to the incomplete knowledge of the environment or
of Eve’s actions. These quantum states are then passed on to the receiver,
Bob, who then has the task to infer Alice’s classical message from these
quantum states. The upper bound for the capacity for such a transmission,
i.e. the information I that Bob can obtain about Alice’ s message per sent
quantum state, is known as the Holevo bound
I <= S(ρ) −
i
pi S(ρi ) (28)

References
[1]. http://www.ieee.ca/millennium/radio/radio_differences.html
[2]. Jakob Foerster, Lecture 1 of the Course on Information Theory, Pattern
Recognition, and Neural Networks. Produced by: David Mac Kay (University of
Cambridge) https://www.youtube.com/watch?v=BCiZc0n6COY
[2]. Elementary gates for quantum computation, Adriano Barenco et al. ,
https://arxiv.org/abs/quant-ph/9503016
[3]. Entanglement in a Quantum Annealing Processor, PRX 4, 2014
[4]. Presentation: The physics of information: from MaxwellŹs demon to
Landauer by Eric Lutz from University of Erlangen-Nrnberg
[5]. Physics of Information F. Alexander Bais J. Doyne Farmer
http://samoa.santafe.edu/media/workingpapers/07-08-029.pdf.
[6]. https://ru.coursera.org/learn/quantum-optics-single-photon/
lecture/eo9Ym/7-3-one-photon-polarization-as-a-qubit
[7]. Basic Paradoxes of Statistical Classical Physics and Quantum Mechanics by
Oleg Kupervasser.
[8]. http://web.physics.ucsb.edu/~quniverse/dhqm-exprob.html
[9]. Quantum memories in atomic ensembles-G.Braunbeck

[10]. Superconducting circuits and quantum information-Physics today, J.You and
F.Nori, 2005
[11]. Wikipedia
[12]. Deterministic chaos-Shuster
[13]. Landauer and noise
[14]. Anne Hillebrand, PhD thesis: Quantum Protocols involving Multiparticle
Entanglement and their Representations in the zx-calculus, 2011
[15]. Lecture on statistical physics,
http://www.physics.udel.edu/~glyde/PHYS813/Lectures/chapter_6.pdf
[16]. A single quantum cannot be cloned, W.K.Wootters, W.H.Zurek, Nature,
Vol.299, 1982
[17]. Physical Review Letters, Vol. 88., No.21.,2002, Classical No-Cloning
Theorem, A. Daﬀertshofer et al.
[18]. Teleportation breakthrough made, Paul Rincon, 2004
http://news.bbc.co.uk/2/hi/science/nature/3811785.stm
[19]. Quantum information theory and Holevo bound
https://www.cs.cmu.edu/~odonnell/quantum15/lecture18.pdf

Classical and Quantum Info Theory

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