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Abstract In 1960, Stueckelberg showed that if the real-vector-space version of quantum theory is supplemented by a specific "superselection rule" limiting the set of allowed observables,

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Thesis title The ubd.:: An &'/6/a h-on- t;f �c, J -Vee-(4,y -tau Qu'<ff�,* 'TLu7

Library Use

Signatures removed.

The Ubit: An Exploration Of Real-Vector-Space Quantum

Theory

BY ANTONIYA A ALEKSANDROVA

WITH

WILLIAM K WOOTTERS, ADVISOR

A thesis submitted in partial fulfillment

of the requirements for the degree of

Bachelor of Arts with Honors

in Physics

WILLIAMS COLLEGE Williamstown, Massachusettes

June 5,2011

I would like to express my gratitude towards my advisor, Professor William K Wootters He was not only the inspiration and driving force behind this project, but also an extremely patient and supportive teacher I am also especially thankful to Professor Frederick Strauch for his invaluable feedback and to Professor Daniel Aalberts for his generosity and friendship I would have never been able to complete this work without the love and support of my family and my wonderful friends, to whom I am infinitely indebted

Abstract

In 1960, Stueckelberg showed that if the real-vector-space version of quantum theory is supplemented by a specific "superselection rule" limiting the set of allowed observables, then the theory is mathemati­cally equivalent to ordinary quantum mechanics Here we attempt to give an information-theoretic interpretation of this rule by proposing

a model in which all real quantum objects can interact with a uni­versal binary object called the ubit In addition, we use our model

to investigate whether the rule can be achieved through a generic dy­namics rather than being assumed Vie show that by allowing the ubit to rotate sufficiently fast, we can make the real-vector-space evo­lution operator arbitrarily close to a standard quantum mechanical opearator

Contents

List of Figures

1 Introduction

2 Background

2 1 Ordinary Quantum lVlechanics

2.2 Real-Vector-Space Quantum Theory

operators with rotation of the ubit ( IB2) 20

3.2.2 Commutativity with rotat.ions of t.he ubit implies an

infor-mationally inaccessible ubit (2=?3) 2 1

3.2.3 Informat.ionally inaccessible ubit leads to St.ueckelberg's

2 1 Precession of a proton in a magnetic field 5

3.1 The New Model: the ubit in a real-vector-space world 1 7

4.1 Study Model: The system consists of a rebit interacting with the ubit through a Stueckelbergian 81, The rotation

of the ubit at a rate w is represented by wJ The con­ nection between the system and the rest of the world is established through the Stueckelbergian of the ubit and the environment , 82, 28

4.2 Dependence o f t h e distance from t h e ideal on t h e rotation rate; dimension: 100, strength of interaction with envi­ ronment:l, time: 10 000 The green diamonds represent

the badness in the final state of the system evolved by: (a)

completely random Stueckelbergian 81 (b) 81 designed to generate precession of the Bloch vector representing the system's good part around the z-axis 32 4.3 Due to the bad part o f 81, t h e goodness in the rebit-ubit system decreases 33 4.4 Decay of the Bloch vector: We have taken our legal initial rebit-ubit state interacting with a 100-dimensional envi­ ronment with strength 1 and evolved it at three different rotation rates, keeping everything else the same 34

LIST OF FIGURES

4.5 Effect of the interaction strength 5 on the decay: the ubit is rotating at a rate of 450 and interact ing with 100- dimensional environment , 35 4.6 Effect of the dimension N of the environment on the decay: the ubit is rotating at a rate of 450 and interacting with

the environment 37

4.7 Decay of the length of the separate components of the Bloch vector: N = 100, w = 1 00,5 = 1 38

4.8 The Wobbling Effect : the size of the 'Ey- and z-component

of the Bloch vector oscillate irregularly in a tary way with each other The simulation was performed for a 100-dimensional environment interacting with a strength

complemen-of 0.1 with the ubit rotating at w = 1000 39 4.9 The Ring-down effect at the beginning of the evolution: exhibited for 1 00-dimensional environment interacting with

a strength of 0.01 with the ubit rotating at w = 50 40

4.11 The effect of badness (denoted by the green diamonds) 42 4.12 The effect of the enviornment 's size on the badness: w =

100, 5 = 1 42 4.13 Evolution without environment : N = 100, 5 = 1 , w = 100 44

4.14 Log-Log plot of the badness vs rotation rate of the ubit 45

4.15 Evolution of a random initial state of the rebit-ubit system with no 51: interaction strength 5 = 1

4.16 Comparison between the actual real-vector-space evolu­

t ion (blue and green) and the expansion approximation

46

(red and orange): N = 100, 5 = 1 51 4.17 Comparison between the actual real-vector-space evolu­ tion (blue and green) and the expansion approximation (red and orange) for completely random initial conditions:

N = 100, 5 = 1 , t = 1 000 52 4.1 8 Approximating t he badness of the rebit-ubit system: N =

100, 5 = 0.1 , t = 1 000 53

Introduction

In broad strokes, physical knowledge today can be classified in five major theories: classical mechanics, electromagnetism, thermodynamics and statistical mechan­ics, theory of relativity, and quantum mechanics While st.udying the first four

of these, a physicist can rely entirely on the real numbers In fact, the only time complex numbers appear is when we are looking for a mathematical shortcut such as computing the impedance in a multi-component AC circuit Since most

of what we observe in our surroundings is conveniently described by real num­bers, it makes sense that physical theories are defined over the real field However, this is not the case with quantum mechanics In fact, complex numbers are so intertwined in its theoretical framework, that even the fundamental equation de­scribing quantum mechanical behavior, the Schrodinger equation, contains the imaginary number i One is bound to wonder: Are complex numbers truly the most natural way of describing the quantum world? If so, \vhat principal aspect

of the universe's workings do they reflect?

These and similar questions have been preoccupying physicists for a long time The issue about the conceptual foundations of the theory has been raised from the very dawn of quantum mechanics It has been a topic of controversy among all of its notable founders and their contemporaries, including Bohr, Einstein and Schrbclinger After a century of active development in which its applications have become essential for many fields of modem research, quantum mechanics is still mostly a mathematical description rather than a physical account of the world

In recent years, many scientists interested in foundational physics have focused on

quantum information as a promising method of understanding the principles be­hind the quantum formalism [5] For example, in his article 'Quantum Mechanics

as Quantum Information (And Only A Little More) ,' Chrjstopher Fuchs argues that the structure of quantum theory can largely be accounted for in terms of information [4] , while John A Wheeler sums up his belief that reality is a mani­festation of information with the slogan "it from bit" [12] Many of the attempts

to reconstruct quantum mechanics from information principles, however, rely on the assumption of the complex number field

The question of the role of the complex field in quantum theory has also been a long-standing issue among the scientific community The existence of alternatives

to standard quantum mechanics, such as having the Hilbert space of quantum mechanics be real or quaternion, was recognized and developed [1] some years ago Ernst Stueckelberg, a notable Swiss theoretical physicist, was particularly interested in the real case and, together with associates, wrote extensively about

it from 1959 to 1962 [3,9, 10, 1 1] Stueckelberg showed that by imposing a special mathematical rule on the real Hilbert space (called a superselection rule) , one can make real quantum theory equivalent to standard quantum mechanics The ad­vantages of expressing quantum mechanics over the real field are multiple F irst,

as Dyson noted, the time reversal operator is antilinear in standard quantum theory, whereas it is linear in the real quantum theory [2] This is the case, for example, for the fundamental CPT symmetry of quantum field theory [7] Dyson

to real quantum mechanics Real-vector-space quantum mechanics is more con­

This is not the case for a finite-dimensional complex Hilbert space Myrheim also points out that over the real field it is possible to have unconventional canonical quantization based on the harmonic oscillator

Real-vector-space quantum mechanics could also offer simplifications in per­forming certain calculations since it is an alternative way of expressing physical conditions For example, \i\Tootters showed that entanglement looks very differ­ent in the real case compared to its complex quantum theory representation [14] While Stueckelberg proved the mathematical equivalence of real and standard

2

As a parallel, one might think about Heisenberg and Schrodinger's versions of quantum mechanics ·While Heisenberg first developed the theory, Schrodinger's interpretation stimulated physicists to think about the ,vave nature of objects Finally, real-vector-space quantum theory can potentially reveal a way of modify­ing and further developing quantum mechanics, conceivably for the development

of a theory of quantum gravity

vVith all this in mind, we set out to build on Stueckelberg's work and explore the world of real-vector-space quantum theory After providing the reader with the necessary background for comparing the complex and the real quantum the­ories in Chapter 2, we proceed to suggest an information-theoretic interpretation

of Stueckelberg's superselection rule in Chapter 3 This gives a physical meaning

to the otherwise purely mathematical treatment offered by Stueckelberg In par­ticular, we conclude that standard quantum mechanics is equivalent to a world

of real objects interacting with a unique binary object (the ubit) , in which the ubit is informationally inaccessible In Chapter 4, we investigate whether the generic dynamics of real-vector-space theory can give rise to the superselection rule, i.e ordinary quantum mechanics Once again we present our ubit model but we do not impose any rules on its behavior \Ve thus find out that if we allow the ubit to rotate infinitely fast, the evolution operators in complex and real-vector-space quantum mechanics will be identicaL Our results and directions for further investigation are summarized in Chapter 5

Chapter 2

Background

We begin with a brief overview of the ordinary complex-vector-space quantum mechanical tools At the heart of quantum mechanics is the idea that if for a given system we can assign a state vector I'll) that describes it, then the system behaves according to Schrodinger equation:

ai'll)

where H is the Hamiltonian of the system and is assumed to be a hermitian operator, that is, its matrix is equal to its hermitian conjugate HI = H Then the evolution of the state in time is given by U = e-iHt/1i so that:

(2.2)

Since H is hermitian, we have that U is a unitary operator Note that a unitary operator is one that has the property UI = U-1

For example, let us consider the simple case of a spin 1/2 particle, such as

a proton The general state of the spin of a proton can be expressed as a two­element state vector:

I'll) = (�) = a G) +,8 e) = alO) + 131 1) (2.3)

where a and 13 are complex numbers vVe shall call a quantum binary object such as a spin 1/2 particle a qubit Now let us see what happens when we place

4

the proton in a uniform magnetic field [6] We shall assume the field points in the z-direction and is thus given by B = Bok The proton then experiences a torque, J.L x B, which causes it to precess around the z-axis Here J.L denotes the magnetic dipole of the charged particle and is given by J.L = ,,(S, where the proportionality constant, ,,(, is the gyromagnetic ratio The situation is illustrated

in Figure 2.1

d�

Figure 2.1: Precession of a proton in a magnetic field

The energy associated with this torque contributes to the Hamiltonian of the system:

(2.4)

to solve Equation 2.1, we obtain the time-dependent expression for the proton's

state vector:

Iw(t)) = sin (()/2)e-i,"(Bot/2 (2.5)

The example of the proton illustrates the utility of state vectors and matrix Hamiltonians in describing a given system However, this process can become very cumbersome if we are dealing with a system with infinitely many energy eigen­states such as the quantum harmonic oscillator Then, as Schrodinger pointed out, it would be much more convenient to deal with wave functions instead of

2 Background

state vectors and operators instead of matrices For the purpose of our study,

we will assume from now on that we are dealing only with finite N-dimensional systems and we will thus choose to use state vectors instead of wave functions

Let us imagine that instead of a single qubit, we now have a composite system comprising two subsystems such as two qubits, A and B If the two quibits are independent, then we can assign separate quantum states to each of them, 11/)(11)) , I¢/B)) Then the state of the composite system is given by

be given by p(JlB) = ptA) p(B) To see this, note that we can easily find the probability that both an outcome a for qubit A and b for qubit B will occur by simply using the state vector:

p(AB)(a, b) = l(a(A) @ b(B) 1 'lj; (A) @ ¢(B)) 12

= 1 (a(A) I'I/;( A)) (b(B) I¢(B)) 12

Often times, however, we cannot assign a definite state vector to each of the quantum subsystems An example of this would be the following state, where {IO)R, Il)R} and {IO) Q, Il)Q} are basis vectors for the Hilbert spaces of Rand Q

respectively:

(2.8)

Since the Q basis vectors are linearly independent, IW(RQ)) cannot be a product state and we cannot assign a definite state to either subsystem R or Q We say that the two subsystems are entangled Entangled states have the property that the outcome from a measurement of system Q affects the state of system R

6

Another instance when we are unable to assign a definite state vector is if

we lack information about the preparation of the quantum system For example, suppose all we know about system C is that there is equal likelihood for it to be

in state 10)c or 11)c Then we say that C is in a mixed state

In the case of both entangled and mixed states, the way to describe the systems

is by defining a density matrix For a quantum system C cOllsisting of a mixture

of states I'l/J')') prepared with probability P�I the density matrix is given by:

p = "L P'i 1 l);'/) ('l/J')'I (2.9) ')'

Notice that this definition of the density matrix is perfectly capable of accommo­dating a pure state I'l/J) as well, since we can simply have p = I'�))('l/JI The density matrix of the composite system RQ given in Equation 2.8, where R and Q are entangled, is again p(RQ) = IW(RQ))(W(RQ)I We would like to be able to use the density matrix to describe the state of the subsystem Q:

(2.10)

This expression makes use of the partial trace of p(RQ) over the subsystem R Here

we give a definition of partial trace of a density matrix, which can be extended via linearity to an arbitrary matrix Suppose that {I kR)} is a complete orthonormal basis for system R Then we have:

n-(R)p(RQ) = "L (k(R)lp(RQ)lkR)

Any legitimate density matrix must satisfy the following three properties:

Pji = Pij'

• Its trace must be one: Tr(p) = 1 In other words, L Pii = 1

• It must be a positive semi-definite matrix This means that all of its eigen­values are non-negative

2 Background

The density matrix contains all the information necessary to compute the

probability of any outcome on any future measurement on a quantum system Q

Let 0 be an observable, and suppose t.hat it has discrete eigenvalues Oc< each

associated with eigenspaces Vc<' Let ITc< be the projection operator onto Vc<' By

definition, this means that IT; = ITc< and t.he eigenvalues of ITc< are 0 and 1

Also note that Lc< ITc< = I, since the projection operators ITc< map to orthogonal

subspaces Then the probability of measurement outcome 0: is given by:

p(o:) = Tr(pITc<) (2 12)

and typically the resulting density matrix becomes:

I ITopITc<

p = Tr(pITc<) (2 13)

In this thesis, by measurement we will understand a projection operator mea­

surement, also known as von Neumann measurement This assumption does not

pose any significant restrictions since b:y Naimark's Theorem we know that any

generalized measurement, i.e any positive operator valued measure (POVlVI),

can be obtained from a projection-type measurement on a larger system [8]

Vve can also compute the (ensemble average) expectation for an observable O

It is given by:

Following the general rule set by Equation 2.2, the unitary time evolution of the

density matrix over the interval from time 0 to time t is given by:

p(t) = U(t)p(O)(U(t))t Therefore, the Schrodinger equation for an isolated system becomes:

For a qubit system, there is an elegant way to visualize the set of density

matrices describing it Given the Pauli basis matrices:

8

we can write any density matrix for a qubit system in the following form:

(2 18)

where the real numbers ax ,ay and az are the components of a 3D space vector a, which we will call Bloch vector In fact, (a · a) ::; I , with equality when p is a pure state Therefore, the Bloch vectors for pure states form a sphere known as the Bloch sphere

To illustrate how this works, let us consider once again the example of a proton

in a magnetic field For simplicity, we shall denote <p = -"(Bot Then we can write Equation 2.5 up to an overall phase factor as:

is defined as:

1

S(p) = -TI:' (p log2 p) = ZAk log2 Ak

where /\k are the eigenvalues of p Let us consider, for example, the state given

in Equation 2.8 The density matrix for that state would be:

p(RQ) = I w(J�Q)) (W(RQ) 1

IOR1Q) (OR1Q I - 1 1ROQ) (OR1Q I - IOR1Q) ( 1ROQ I + 1 1ROQ) ( 1ROQ I

2

2 Background

y

Figure 2.2: The state of a quhit on the Bloch sphere

where the tensor product sign ® was omitted for neatness The density matrix

of the subsystem Q becomes:

10

In fact, it is very hard to have completely isolated systems Typically, a system is entangled with the environment, which leads to sharing of quantum information with the surroundings The loss of information to the environment

is known as decoher-ence Decoherence causes pure states to become more mixed and it increases the entropy of mixed states The effective decoherence can be seen by starting with a density matrix describing the system and the environment together and tracing out the environment

2.2 Real-Vector- Space Quantum Theory

We can also define quantum theory over the field of real numbers By itself, this

is a different mathematical and physical framework In the general real-vector­space quantum theory, we assume that pure states are simply vectors in ]RN

Together "vith mixed or entangled states, they can again be expressed through a density matrix However, we now insist that all density matrices have only real entries This means that instead of hermitian, matrices are simply symmetric

or, in other words, pT = p The other two of the three properties describing a legitimate density matrix remain unaltered: its trace still needs to be one and all

of its eigenvalues must be non-negative

As noted before, complex quantum theory requires the evolution of states to

be given by the unitary matrix U = e-iHt In the real vector space, unitary matrices are replaced by orthogonal ones Therefore, instead of U we shall have

o such that OT = 0-1 Also, the hermitian Hamiltonian would now be given by

a symmetric matrix but since we are not allowed to have i in our real quantum theory, we replace the entire combination (-iH) with an anti-symmetric matrix

A This means that A = -AT and 0 = eAt In this fashion, the time evolution

of the real-vector-space density matrix p becomes:

2 Background

While we keep all previously mentioned definitions from complex quantum mechanics unchanged in real-vector-space quantum theory, it turns out that the entanglement bounds are not fundamental properties but a result of the field we use In particular, if we call all real-vector-space binary objects r·ebits, we know that a reb it can be maximally entangled with an unlimited number of other rebits That is, monogamy no longer applies Also, a given state vector that, when interpreted in complex quantum mechanics is unentangled, can be mEL"Ximally entangled in real quantum theory To illustrate this we shall use the example given

Notice that both are maximally entangled pure states in the complex vector space Consider the following desnity matrix:

It turns out that w can be decomposed into a sum of product states (whose vec­tors have complex entries) and is thus unentangled in complex quantum theory However, in real-vector-space quantum theory, such decomposition is impossible and hence the states represented by w are entangled In addition, every decom­

Therefore, we call w maximally entangled

In the 1 960s, Ernst Stueckelberg investigated the connection between com­plex and real-vector-space quantum theory In particular, he was interested in expressing quantum mechanics over the field of reals To describe his results, we begin by observing that a single complex number contains both real and imag­inary parts Thus, if we want to encode the same amount of information with the reals, we would need two numbers Hence, the transformation from ordinary quantum mechanics to real-vector-space quantum theory should involve doubling

of the dimensions But how exactly do we perform this so that \ve preserve the useful properties of complex numbers? Note that just any transformation will not work For example, take the following map:

h( a + ib) = (� �) for a, b t lR

1 2

Then take two complex numbers, a + ib and c + id Note that (a + ib)(c + id) = (ac - bd) + i(ad + bc) but:

h(a + ib)h(c + id) = (� �) (� �)

= h((a + ib)(c + id))

I n other words, this linear transformation does not preserve multiplication, con­trary to ·what we wanted This problem is fixed by the following suggestion If

we have a complex number a + ib, to write it in the real vector space, we will form a 2 x 2 matrix: ( a -b b a ) (2.24)

For a larger matrix of complex numbers, we can apply the same transformation

to each of its complex components and obtain a real-valued matrix with twice the dimensions of the original complex matrix Notice that in the case of a density matrix such a transformation doubles the trace so we need to divide by two to normalize it This mapping f (a + ib) = (�� � b ) for a, b co IR is a homomorphism, that is the addition and multiplication of real matrices is just like that of complex numbers Let once again c + id be another complex number, which gets mapped

the multiplication is not preserved Therefore, a natural question that arises is how to limit the allowed real-vector-space matrices to exclude the ones that do not have any complex-vector-space equivalent Stueckelberg's solution to this problem was to force all measurements and transformations to commute with an

elberg's superselection rule and we shall call J Stueckelberg's matrix A simple form of J would be:

Inforrnation-Theoretic Approach

Although Stueckelberg's superselection rule provides an elegant mathematical connection between complex and real-vector-space quantum mechanics, it does not give any insight about the physical difference between these two theories In other words, it does not shed any light on the principles in nature that give rise

to quantum mechanics This chapter revisits Stueckelberg's ideas and suggests a

information-theoretic explanation

First of all , Stueckelberg's proposal for a transformation between complex and real quantum theory entails a doubling of the dimensions of states This reminds one of doing a tensor product with a binary object (i.e an object whose state space is two-dimensional) - in effect, the result is exactly a matrix with twice the original dimensions This observation leads us to suggest the following model Imagine that the world consists entirely of objects with real-vector-space states When these objects are binary, we shall call them rebits Now suppose that in addition, there exists a unique binary object called the universal bit or, for short, the ubit, as in Figure 3.1 a) Note that because there is no monogamy of entanglement in real-vector-space quantum mechanics, it is possible for each real­vector-space object in the world to be simultaneously entangled with the ubit

We will illustrate how our model works in the example of the qubit in Chapter 2

In our model a qubit is represented by a rebit together with the ubit When the qubit is placed in a magnetic field as described in Chapter 2, its phase ei<p

becomes essential for describing its precessing motion Suppose our qubit is in

the pure state I w) = � (10) + ei<P11)) Then its density matrix is:

(3.1) Applying the transformation given by Equation 2.24 to each element o f this ma­trix and recalling that ei<p = cos rp + i sin rp, in real-vector-space quantum mechan-

as a pro duct state between the rebit and the ubit Now vve can write p(rp) =

D(tp)p(O)D(rpf, where D(rp) is given by:

D(rp) = 10)(01 0 I2 + 1 1) (1 1 0 e'Ph (3.4)

Bloch sphere, all possible real states form just a solid circle in the xz-plane The rotation of the ubit can be seen as adding another circle, which through the interaction of the ubit with the real states, gives rise to the three-dimensional

illustrates this idea

(a) An auxilary binary object (ubit) entangled

with N-dimensional real-vector-space objects

gives rise to a 2N -dimensional world

I All States I I Reat States I

real part and a ubit part

Figure 3 1: The New Model: the ubit in a real-vector-space world

Next, we need to give a physical meaning of the superselection rule in our model To do so, we will consider what the allmved operations in this alternative world are that correspond to standard quantum mechanics The following theo­rem establishes this connection

Theorem 3 1 Suppose that all measurement, transformation and density matrix operators that commute with J are allowed In addition, assume that if

an operation is allowed, then a controlled version of it is allowed as well Then we claim that the following statements are equivalent in real-vector-space quantum mechanics:

1 (Stueckelberg's version) All allowed measurement operators E and trans­formation operators T must commute with J

rotations of the ubit

3 A rotation of the ubit can have no observable effect

Since Stueckelberg's version is equivalent to quantum mechanics, proving that the rest of the claims are in their essence the same as ( 1 ) will imply that they are enough to generate quantum mechanics out of real-vector-space quantum theory Notice that what statement (3) is actually saying that the only rule we need to impose is that it is impossible to extract information from the ubit This last claim gives us a physical interpretation of at least one of the fundamental aspects

of quantum theory, the appearance of complex numbers, in terms of information

Before we approach the actual proof of Theorem 3.1, we will prove the following two lemmas:

Lemma 3 2 Let M be any square matrix Then M can written as a sum

of a term that commutes with J and a term that anti-commutes with J These terms will be denoted M+ and M- respectively [7]

Proof Let M+ = � (M - JMJ) and M- = � (M + JMJ ) First note that:

Proof Using the notation and results of Lemma 3 2 , we see that:

So we proceed to the proof of Theorem 3 l

3 Information-Theoretic Approach

3 2 Proof o f Theorem 3 1

of all operators wit h rotat ion of t he ubit (1{::?2)

Without loss o f generality, consider a system o f a rebit A and the ubit U An operation on this system 'which results solely in a rotation of the ubit can be written in the following way:

J = fA �' ( � �1)

= fA 0 J2

20

Then we have that:

informationally inaccessible ubit (2=?3)

According to ( 2 ) , all E and T commute with eOJ Therefore:

Tr(EeOJ pe-OJ) = Tr(e-OJEeOJ p)

= n-(Ee-OJ eOJ p)

= n-(Ep) Since Tr(EeOJ pe-OJ) = n'(Ep) for all E and p, no effect of the rotation of the ubit can be observed .•

Here we will do a proof by contrapositive Suppose there exists a projection operator E (E2 = E) such that E does not commute with J We will show that

we will be able to detect a rotation of the ubit Let

1 [E - JEJ]

P = 2 n'(� [E - J EJ] ) (3.6)

Since E is a positive semi-definite matrix and is not the zero matrix, we know that Tl '(E2) is greater than zero Therefore, it is left to show that the second term on the right-hand side is non-negative Let 0 be an orthogonal matrix such that OEOT is diagonalized Then we have:

On the other hand, for the rotated final density matrix we have:

P2 = Tr(EJ PfJT)

= aTr(EJEpEfT)

= aTr( -EJE(E - JEJ) EJ)

= -aTr(EJ EJ E) + aTr(EJ EJ EJ EJ)

Now let us take a look at the difference of the two probabilities:

PI - P2 = aTr(E3) - aTr(EJ EJ EJ EJ)

To show that this difference is not zero, we need to show that E and J EJT do not project on the same subspace We \vill prove this by contradiction Suppose they do Then:

JEJT = E

� [J, Ej = 0 which is a contradiction with our initial assumption that J and E do not com­mute Therefore, E and J EJT do not project on the same subspace Let E project on a d-dimensional subspace anel:

3 Information-Theoretic Approach

where IUj) = JEJ7' lv:j) We note that for any normalized I'uj) we have ('uj IEI'Lij) �

1 Moreover, due to the fact that E and J EJ7' do not project on the same sub­space, there is at least one IU:j) such that (uj IEI'Lij) < 1 Therefore:

or, comparing this with Equation 3.8, we end up with:

(3 1 1 )

To do so, we shall recall that we can do a "controlled T," which we will denote

by C We pick C to be such that:

A s we have noted earlier, every matrix ca.n b e written as a sum o f its good and bad parts Since the good part of the rotation operator does not contribute significantly to our argument, we are just going to examine the effects of its bad part That is, we shall assume that T is entirely bad, or:

TJ = -JT

24

Let F be a good measurement operator and p be a good density matrix so that:

E = CFCT (T = �C [( 10) + 1 1)) ( (01 + ( 1 1) ® p] CT

Using Equation 3 12, Equation 3.14 becomes:

(3 13) (3 14)

B and the 'ubit, then no action on A can have an observable effect on B

This statement is a version o f "no information can b e transmitted instanta­neously over a distance." It is taking our information-theoretic interpretation a step further