Geometric Phases In Quantum Systems Of Pure And Mixed State

(1)

Geometric phases in quantum systems of pure and mixed state

Geometriska faser in rena och blandade kvantmekaniska system

By Miran Haider

Faculty of health and science

Master’s thesis in Engineering physics, 30ECTS Supervisor: Prof. Fuchs J¨urgen

Examinator: Prof. Johansson Lars Date: February 23, 2017

(2)

Abstract

This thesis focuses on the geometric phase in pure and mixed quantum states.

For the case of a pure quantum state, Berry’s adiabatic approach (4.1.10) and Aharonov & Anandan’s non-adiabatic generalization of Berry’s approach (4.2.8) are included in this work. Mixed quantum state involves Uhlmanns approach (5.1.42), which is used extensively in Section 7 and Sj¨oqvist’s et al. approach (5.2.22), used extensively in Section 6. Sj¨oqvist’s approach states that the Uhlmann phase is an observable and provides the experimental groundwork using an interferometer. The geometric phase for a spin-¹₂ system is given by (5.2.47) γ_GrΓs “ ´ arctan“r tan `^Ω₂˘‰ , which was proven, by Du et al.[45] to reproduce experimental data (Figure 19) on page 56.

The Uhlmann phase can be used to observe the behaviour of topological kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain and the SSU-model. It is found that the Uhlmann phase is split into two regimes with the dividing parameter being the temperature. This temperature is called the critical temperature, Tc, and is given by Tc “ ¹

lnp^2`^?³q. If the temperature is below the critical temperature, the Uhlmann phase yields π and if the temperature is above the critical temperature, the Uhlmann phase yields zero.

Detta examensarbete behandlar geometriska faser i rena och blandade kvanttillst˚and. I rena kvanttillst˚and finner man Berrys adiabatiska behandling av den geometriska fasen (4.1.10) och Aharonov & Anandan icke-adiabatiska generalis- ering av Berry fasen (4.2.8). I det blandade kvanttillst˚anden har Uhlmann introducerat en förlängning av den geometriska fasen som sträcker sig till det blandade kvanttillst˚anden (5.1.42), detta finner man i sektion 7. Senare har Sjöqvist et al. introducerat ett alternativ till att angripa geometriska faser (5.2.22) som beskrivs i sektion 6. Sjöqvist konstaterade att Uhlmannfasen är observerbar, i kvantmekanisk mening, och presenterade ett experimentelt upplägg där han visade just detta med hjälp av en interferometer. Den geometriska fasen för ett spin-¹₂ system ges av (5.2.47) γGrΓs “ ´ arctan“r tan `^Ω₂˘‰ , vilket senare bevisades av Du et al.[45] där de experimentella mätvärdena stämde överens med dem teoertiska (se figur 19 p˚a sidan 56).

Uhlmannfasen kan även användas för att observera topologiska ”kink”-lösningar.

Detta testades f¨or 3 olika modeller; Creutz stege formationen, Majorana kedjan och SSU modellen. Det visade sig att Uhlmannfasen delades up i tv˚a regioner och var starkt beroende p˚a temperaturen. Denna temperaturen kallades f¨or den kritiska temperaturen Tc, och ges av Tc “ ¹

lnp^2`^?³q. Om temperaturen ligger under eller ¨over den kritiska temperaturen f˚ar man att Uhlmannfasen ger π eller 0 respective.

(3)

Acknowledgements

I would like to extend my heartfelt gratitude and appreciation to my supervisor, Prof. Fuchs J¨urgen, for constantly being there when needed and has, on multiple occasions, guided me through several barriers. Without his persistent help and guidance, this work would not have been possible.

I would also like to thank my family and friends for their constant love and support.

(4)

List of Figures

1 [4] The Bloch sphere, a geometrical representation of the two- dimensional Qudit system called the Qubit. . . 3 2 [7] The transport of a vector along the closed curve C on a 2-sphere. 12 3 [7] A spin-¹₂ particle moving in a adiabatically, rotating, θ, exter-

nally applied magnetic field B with angular velocity ω, emulating Equation (4.1.16) . . . 16 4 [7] Magnetic field, Equation (4.1.16), following the path C, ex-

pressed in parameter space . . . 17 5 [7] The path C in H being projected onto the projective state

space C¹ on PpHq. Note that the path C need not necessarily be closed for the path C¹ to be closed. . . 19 6 Comparison [33] between the Berry (Equation (4.1.10)) and Uhlmann

(Equation (5.1.42)) aproaches in obtaining the geometric phase. . 27 7 [12] A conventional Mach-Zehnder ineterferometer with two beam-

splitters and two mirrors . . . 29 8 [?, ?] Sj¨oqvist’s interferometry model BS1 and BS2 are beam

splitters, M1 and M2 are Hadamard mirrors, U is a unitary operator acting on the internal states of the photons and χ is an operator that shifts the phase. . . 29 9 Nucleus axis of rotation precessing around the magnetic field. . . 41 10 Energy levels splitting of an fermion due to an externally applied

magnetic field, known as the Zeeman splitting. . . 42 11 [55] An α pulse transforms the net magnetization, M0, into an-

other oriantation, M_x,y, by a degree α. A 90 degree pulse would shift the net magnetization entierly from one axis to another. B is the externally applied magnetic field. . . 43 12 [55] A spin ¹₂ particle absorbing RF radiation with an energy

corresponding to a spin-flip from . . . 44 13 A pictorial presentation of the JJ-coupling effect on respective

proton . . . 46 14 [55] General/conventional CW spectrometer. Sample solution is

placed in a rotating glass tube oriented between magnetic poles.

Radio frequency (RF) is emitted from the RF antenna, enclosing the glass tube and a receiver to detect emitted radiation from the sample. The receiver transmits the data to a control console which displays the result on a screen. . . 47 15 Spin echo method . . . 52 16 [57] A refocusing of spin moments, yielding a singal, echo, when

no pulse was applied. a, b, c, d, e, f . . . 52 17 [45] Quantum network describing the experiment where the top

line represents the auxiliary qubit or particle of spin-1{2 while the lower line represents the qubit, spin 1{2 prticle, that undergoes a cyclic evolution induced by the unitary operator U . . . 53 18 The solid angle Ω is subtended by the cyclic path ABCDA. The

solid angle can be varied by changing θ, the angle of inclination between x, y plane and the ABC plane. . . 54

(7)

19 Experimental data [45] (represented by circle and squares) versus theoretical predictions (represented by solid lines and Equation (5.2.22)). The geometric phase ,γ, is presented versus purity, r, for three different solid angles, Ω. . . 56 20 A sample in the presence of an externally, perpendicularly, ap-

plied magnetic field. The electrons on the sample are effected by this magnetic field, causing electrons to orbit. Electrons close ot the edges experience Edge states, effectively contributing to current along the egdes, while electrons in the middle contiue to orbit. . . 58 21 The Creutz ladder formation. Particles can hop both diagonally

and vertically. . . 61 22 [33] The Topological Uhlmann phase for the Creutz-ladder (a),

the Majorana-chain model (b) and the SSH-model(c). The topological phase is equal to π inside the green volume and zero outside. The FPB (Flat Band Points) are indicated by arrows. . . . 62 23 [34] (a): non-trivial topology and (b): trivial-topology . . . 62 24 Real and imaginary part of fn (7.5.9) is plotted against the pa-

rameter n. In the figuren n appears to be continuous, however, we are only interested in the integer numbers. Only possible integer value, for which (7.5.9) is real, is given for n “ 0 . . . 65 25 Real and imaginary part of fn (7.5.9), presented in figure (24), is

plotted on top of each other. Again, only possible integer value, for which (7.5.9) is real, is given for n “ 0 . . . 65 26 Top chain showing trivial phase with paired Majorana fermions

(in blue) located on the same sites on a physical lattice. Bot- tom chain showing non-trivial topology (topological phase) with bound pairs located at neighboring sites resting as unpaired Ma- joranas at each ends, represented by red coloured balls . . . 67 27 The Su-Schrieffer-Heeger model . . . 68

(8)

1 Introduction

The purpose of this work is to introduce the concept of geometric phase and to describe different variants for quantum systems which are in a pure or mixed state. This work is divided into 5 parts. A preliminary section is intended to prepare the reader for the mathematical concepts and reasoning implemented in the upcoming sections, a theoretical section which introduces the concept of geometric phase, a section discussing experiments in which the geometric phase has been experimentally demonstrated, a more abstract section where the relations of geometric phases are extended to topological notions and their usage in describing behaviours of certain insulating materials is discussed, and finally a section giving summary and mentioning further studies.

When a quantum system (see Sections 2.1 and 2.2) undergoes a cyclic unitary evolution, the state aquires a phase. This phase consists of two parts; a dynamic phase which depends on the Hamiltonian, and a geometric phase which depends only on the path of the evolution that the system takes in the projective Hilbert space [9], [10], [16]. For a qudit system (see Section 2.1), the qubit in particular, the projective Hilbert space is given by a sphere (Bloch Sphere, See Section 2.1.1) where the geometric phase depends on the geodesical solid angle subtended at the center of the sphere by a path by state vector. The concept of geometric phase was first introduced in an adiabtic context [9], and later a non-adiabatic generalized theory was proposed by Aharonov et al. [16].

The adiabatic approach requires the state vector to be parallel transported adiabatically to ensure that the system always remains in one of the eigenstates of the instantaneous Hamiltonian during the evolution.

The system in the non-adiabatic approach, is allowed to change abruptly.

This implies that the system reverts to its initial state through intermediate states. The dynamical phase is eliminated from the total phase by various methods in order to be able to experimentally measure the geometric phase. One possibility to eliminate the dynamical phase is by using the Nuclear Magnetic Resonance (NMR) technique called spin-echo [57] (see Section 6.2.4).

In 1986, Uhlmann approached the concept of mixed state geometric phase [28] using the notion of amplitude (5.1.42). Here, Uhlmann has taken a large system in a pure state and a subsystem in a mixed state. Uhlmann then obtained the unitary evolution for the subsystem which was transported in a maximally parallel fashion.

Recently, Sj¨oqvist et al. [19] provided additional insight into the nature of geometric phases by providing a concept different to the one given by Uhlmann (see Section 5.2), which is adapted to experimental use. In a quantum interferometer, the system undergoes multiple unitary evolutions, for which the probability of discovering the system in one of its eigenstates behaves like a oscillatory function. This oscilattory function, yielding a oscillation pattern of probability, resembles an optical interference pattern. Now a shift in this interference pattern is a function of the geometric phase acquired by the quantum system undergoing unitary evolution as well as the purity of the system (see Equation (5.2.30)) [19]. The geometric phase can then be directly measured from this shift in the interference pattern.

The mixed state geometric phase has been experimentally observed by by Du et al. [45]. Du et al. obtained the geometric phase by measuring the relative phase change of an auxiliary spin (Section 6).

(9)

Since the discovery of the geometric phase by Berry [10], it has played an essential role in the study of many quantum mechanical phenomena. An example of this is the characterization of the transversal conductivity σxx in the quantum Hall effect, by means of the integral of the Berry curvature over the two-dimensional Brillouin zone, and its relation to the Chern number. Using the TKNN formula [46], one can show that the Hall conductivity is related to the Chern number (a topological invariant) by a constant e²{h [27].

Section 7 focuses on the topological aspects of the Uhlmann phase for the case of one-dimensional fermionic systems. The notion of topology and topological protection are explained in Section 7.2 where the latter is related to insulators and superconductors. Mathematically, the situation is quite similar to what happens in the work by J.M. Kosterlitz and D.J. Thouless, receivers of the 2016 Nobel prize in physics, regarding topological phase transitions [48].

With the developments on fault-tolerant methods, it is known that, in principle, operators of a quantum computer can actively intervene to stabilize the device from errors in a noisy environment [52]. That is, controlled encoded quantum information can protected from errors due to its interaction with the environment. However, in the long term, it is favourable to have an intrinsic fault-tolerance hardware within the device. Recently, it has been proposed that fault-tolerant (intrinsic) quantum computing can be performed with the aid of a geometric phase [50] [51].

(10)

2 Single quantum system

2.1 Preliminaries

Definition 2.1.1. A Qudit system is a quantum system of d linear independent states described by the d dimensional Hilbert space C^d

2.1.1 The Qubit example and the Bloch Sphere

Definition 2.1.2. A binary relation on a set A is said to be an equivalence relation if and only if it is, reflexive, symmetric and transitive.

That is, for @a, b, c P A ,the relation is reflexive if a „ a, symmetric if a „ b iff b „ a, and transitive if a „ b and b „ c then a „ c.

The Projective Hilbert space H of a complex Hilbert space H is the set of equivalence classes (a class which contains a notion of equivalence formalized as equivalence relation given by Definition 2.1.2) of vectors |vy P H ´ t0u for the relation „ given by v „ w when v “ λw where λ P C ´ t0u. The equivalence classes for the relation „ is called rays which is the usual construction of projectivication applied to a complex Hilbert space. That is, for λ P C ´ t0u, P pλv, wq “ P pv, λwq “ P pv, wq since |vy is the same state as |wy. Thus, a ray is a set of all vectors describing the same state

R_w“ t|vy P H|Dλ P C : |vy “ λ|wyu. (2.1.1) The block sphere is a geometrical representation of the pure state case of the Qubit, the restricted Qudit system, for which one deals with the two-dimensional complex Hilbert space C². The projectivication of the two-dimensional complex Hilbert space is the complex projective line P pC²q “ CP¹, which is known as the Bloch sphere, see figure (1).

Figure 1: [4] The Bloch sphere, a geometrical representation of the two- dimensional Qudit system called the Qubit.

(11)

By requiring that the pure state |ψy be normalized, given any orthonormal basis (t|0y, |1yu for the case of the qubit), it can be represented as

|ψy “ cosˆ θ 2

˙

|0y ` e^iφsinˆ θ 2

˙

|1y. (2.1.2)

Then the density operator is given by

ρ “ |ψyxψ| “ 1 2

´

I₂` ~b ¨ σ

¯

, (2.1.3)

where

σ “ pσx, σ_y, σ_zq (2.1.4) are the Pauli matrices, I₂ is the two-dimensional unite matrix and the Bloch vector, ~b, is given by

~b “ psinpθq cospφq, sinpθq sinpφq, cospθqq (2.1.5) with θ P r0, πs and φ P r0, 2πs.

2.2 Density operator

Before discussing the density operator and density matrix some terms need to be defined. Consider all of space to be the unbounded set S P R³. Then a Quantum System is a bounded subset of space, S¹Ď S

Following the definition of a quantum system, then it is a subset of space for which quantum mechanical studies are being acted upon. Thus, it is a system of a size ”suitable” for experiments for which all the outcomes in S¹ reflects the behaviour of all the outcomes in S under similar conditions.

Definition 2.2.1. The state of a quantum system is called ”pure”, or pure Quantum State, if the system can be completely described by a single state- vector |Ψy P H [1].

Pure quantum state is synonymous with the term, proposed by J. Von Neu- mann, Pure ensemble: collection of physical systems for which each member can be characterized by the same state vector, say |ψy P H.

Definition 2.2.2. A mixed Quantum State is a statistical Ensemble of pure Quantum States.

That is; a Convex Combination of ”pure” Quantum states. Suppose one performs a measurement on a mixed ensemble of some observable A, that has a representation on the Hilbert space H of a system. Then, the Ensemble average of A is given by;

(12)

xxAyy ”

dimpHq

ÿ

α

ραxψα|A|ψαy

“ lim

R¹Ñ8 dimpHq

ÿ

α R¹

ÿ

a¹

ραxψα|A|a¹yxa¹|ψαy

“ lim

R¹Ñ8 dimpHq

ÿ

α R¹

ÿ

a¹

ρα|xa¹|ψαy|²a¹, (2.2.1)

where the Closure relation (equation (A.2.15)) has been inserted in the first line of equation (2.2.1) and that |a¹y is an eigenket of A. Note that the index α has been chosen to further remind us that |ψαy need not necessarily be orthogo- nal. Further note the probabilistic nature of Equation (2.2.1), for which two probability factors appear.

• ρα; Probability factor in an ensemble of a state characterized by |ψαy.

That is, ρα is the probability factor for which the ensemble exist in the state |ψαy

• |a¹x|ψαy|²; Quantum mechanical probability for the state |ψαy to be found in an eigenstate of A, |a¹y

This motivates one to reconsider (rather re-express) the ensemble average, equation (2.2.1), using the orthonormal basis, t|b¹y, |b²yu

xxAyy “ lim

R¹,R²Ñ8 dimpHq

ÿ

α

ρα R¹

ÿ

b¹ R²

ÿ

b²

xψα|b¹yxb¹|A|b²yxb²|ψαy

“ lim

R¹,R²Ñ8 R¹

ÿ

b¹ R²

ÿ

b²

¨

˝

dimpHq

ÿ

α

ραxb²|ψαyxψα|b¹y

˛

‚xb¹|A|b²y. (2.2.2)

Where the bracket (the α-sum) is a collection of pure ensembles. This, further, motivates one to define the density matrix and the density operator, for which;

ρ “

dimpHq

ÿ

α

p_α|ψαyxψα|, (2.2.3)

is defined as the density operator. One can express the density operator, ρ, as a matrix, after a choice of basis;

xb²|ρ|b¹y “

dimpHq

ÿ

α

pαxb²|ψαyxψα|b¹y. (2.2.4) Note that equation 2.2.2 can be rewritten as xxAyy “ trpρAq, certain conditions have to be fulfilled in order for an matrix to be called a density operator.

Definition 2.2.3. An operator, Q, is a Density operator if and only if

(13)

• Trace; T rpQq “ 1.

• Hermitian; Q “ Q^:.

• Positive definite: xφ|Q|φy ě 0, @|φy P H.

Now, one is ready to defining Pure states using the density operator.

Definition 2.2.4. A pure ensemble is defined as; for @α P Ně0, D!α “ α¹ such that ρα¹ “ 1 and ρα“ 0, @α P Nztα¹u.

ρ “ |ψα¹yxψα¹|. (2.2.5)

Definition 2.2.5. A mixed ensemble is an ensemble which is not pure.

After the definition of a pure density operator, one could argue that a density operator is pure if and only if the trace, T rpρ²q “ 1 is equal to 1. This is immediately evident considering that the density operator of an pure ensemble is idempotent. However, this can be proven more elegantly.

Proposition 2.1. A density operator, ρ, is pure if and only if T rpρ²q “ 1.

Proof. Assume that ρ corresponds to the density operator of a mixed ensemble.

Then ρ “ř

αρα|ψαyxψα| and

T rpρ²q “ÿ

b¹

ÿ

b²

xb¹|

˜ ÿ

i

ÿ

j

ρ_iρ_j|ψiyxψi|b²yxb²|ψjyxψj|

¸

|b¹y

“ÿ

i

ÿ

j

ρiρjxψi|ψjyxψj|

˜ ÿ

b¹

|b¹yxb¹|

¸

|ψiy

“ÿ

i

ÿ

j

ρiρj|xψi|ψjy|²

ďÿ

i

ρi

ÿ

j

ρj“ 1. (2.2.6)

Since, for a mixed ensemble, 0 ď ρi ă 1 @i P N, therefore ρ²i ă ρi. This implies that the squared density operator of a mixed ensemble cannot have a trace equal to 1. Thus a mixed ensemble cannot be pure.

2.3 Time-Evolution of the density operator

The time-evolution of the density operator, ρptq, can be predicted using the Schr¨odinger equation. Say density operator at a given time is given by;

ρptq “

dimpHq

ÿ

α

pα|ψαptqyxψαptq|. (2.3.1) Then inserting equation (2.3.1) into the Schr¨odinger equation, yields;

(14)

Bρptq Bt “

dimpHq

ÿ

α

ρα

„ˆB

Bt|ψαptqy

˙

xψαptq| ` |ψαptqy ˆB

Btxψαptq|

˙

“ 1 i~

dimpHq

ÿ

α

ραrpH|ψαptqyq xψαptq| ´ |ψαptqy pxψαptq|Hqs

“ 1

i~rHρptq ´ ρptqHs

“ 1

i~rH, ρptqs. (2.3.2)

The general solution (provided that the Hamiltonian is time-independent) to the differential equation given by equation (2.3.2) is given by

ρptq “ U pt, t0qρpt0qU^:pt, t0q. (2.3.3) Where the time-shift operator, U ,is given by equation (A.2.10).

2.4 Spin operators

For a fermionic system of spin-¹₂, the spin operators are given by

I_x“1 2

„0 1 1 0



“ 1

2σ_x, (2.4.1)

Iy “1 2

„0 ´i i 0



“1

2σy, (2.4.2)

Iz“1 2

„1 0 0 ´1



“ 1

2σz, (2.4.3)

rIx, Iys “ iIz, (2.4.4)

where ~ has been suppressed Do not confuse the operators for the identity operator, if both are included, a notification will be given. The eigenkets are represented by

|αy “„1 0



, (2.4.5)

|βy “„0 1



. (2.4.6)

Thus, applying the spin operators onto the eigenkets yields

Ix|αy “ 1

2|βy, Ix|βy “ 1

2|αy, (2.4.7) I_y|αy “ i

2|βy, I_y|βy “ ´i

2|αy, (2.4.8) Iz|αy “ 1

2|αy, Iz|βy “ ´1

2|βy. (2.4.9)

(15)

3 Multiple quantum systems

3.1 Entanglement and Separability for mixed states

A mentioned previously, the notion of entanglement and separability must be defined.

Definition 3.1.1. A mixed state is called separable if and only if the density operator, ρ_Y

jA_j, corresponding to the systemÂn

jPNą0H_A_j can be expressed as ρ_Y_j_A_j “

j

ÿ

i n

â

j

ρiρ_i,A_j. (3.1.1)

Then Entanglement is defined as:

Definition 3.1.2. An Entangled state is a state that is not separable.

In other words, the state of a composite quantum system (more on this in the next section and [14]) is called entangled (or not separable) if and only if it cannot be represented as a tensor product of states of its subsystems.

3.2 Composite quantum systems

While entanglement is not the subject of this thesis, it will be mentioned briefly since certain concepts are adopted in the discussion of the bipartite quantum system. If n P N subsystems, Ťn

iPNą0A_i, are related to each other in the sense that they have, at some point, interacted with one another then it is generally impossible to assign a single state vector to either of the subsystems. This principle is known as non-separability. For the case of the bipartite quantum composite system, separability will be discussed in terms of pure and mixed states.

The state of a quantum system is described in terms of a Hilbert space H.

Consider n subsystems tHA_iuⁿ_i“1;

Definition 3.2.1. A composite system of the subsystems tHA_iuⁿ_i“1is described as the tensor product Hilbert-space;

H^Ť

iA_i “â

i

HAi. (3.2.1)

3.3 Restricted case: Pure Bipartite Quantum composite System (Pure Bipartite qudit state)

The state vector for a pure, separable quantum system consisting of ”n”,

”d “ dimpHq”-dimensional subsystems is defined as |ψy P Ân iPNą0H^d_A

i. The bipartite quantum system consists of 2 subsystems. Then the d-dimensional Hilbert space, H^d, of the Bipartite system, A Y B, is given by HA,B“ H^d_Ab H_B^d (due to the separability condition). That is if t|µyAu is an orthonormal basis for H^d_A and t|γyBu is an orthonormal basis for H_B^d, then t|µyAb |γyBu is an orthonormal basis for H^d_Ab H^d_B. Thus an arbitrary pure state of H^d_Ab H^d_B can

(16)

be expanded (this is indeed possible due to the separability of the two Hilbert spaces for system A Y B which, in turn, implies that the the systems A Y B are not entangled ) as;

|ψyA,B“

dimpHAq

ÿ

i

dimpHBq

ÿ

j

a_i,j | µiyAb | γjyB (3.3.1) Whereř

i

ř

j|ai,j|² “ 1, ai,j P C. Roughly speaking, in this case, a pure state is separable if, in equation p3.3.1q, ai,j “ bicj, such that |ψyA,B“ |ψyAb

|ψyB. Note that it is possible to express equation (3.3.1) as a single sum by denoting |˜γyB “ ř

jai,j|γyB but at the expense of |˜γyB not, in general, be orthonormal. Say one is only interested in subsystem A, this is achieved by the observable MAbIB, where MAis an self-adjoint operator acting on subsystem A and IB, is the identity operator acting on subsystem B. The expectation value of an observable MAb IB is given by;

xMAy “A,Bxψ|MAb IB|ψyA,B

“

˜ ÿ

k

ÿ

l

a^˚_k,lAxµk| bBxγl|

¸

pMAb IBq

˜ ÿ

i

ÿ

j

ai,j|µiyAb |γjyB

¸

“ÿ

k

ÿ

l

ÿ

i

ÿ

j

a^˚_k,la_i,jrAxµk|MA|µiyAbBxγl|IB|γjyBs

“ÿ

k

ÿ

i

ÿ

j

a^˚_k,ja_{i,j A}xµk|MA|µiyAδ_l,j

“ T r pMAρ_Aq , (3.3.2)

where ρ_A “ T rBpρ_A,Bq “ T rBp|ψyA,B A,Bxψ|q ” ř

k

ř

i

ř

ja^˚_k,jai,j|µkyAAxµi|, for which the total density operator for the entire system, A Y B is denoted as ρ_A,B. That is, the pure density operator is given by

ρ_A,B“ |ψyA,Bb |ψyA,B“ |ψyA,B A,Bxψ| (3.3.3) Then the reduced density operator, ρ_A(ρ_B), for subsystem A is obtained by taking the partial trace, T r_B (T r_A), over subsystem B (A). The reduced density operator, ρ_A(ρ_B), is self-adjoint, positive definite and its trace is equal to 1.

3.4 Purification

Generally, any result in a mixed quantum state can be viewed as the reduced state of a pure state in a larger dimensional Hilbert space. This is the notion of purification and has already been touched upon in the previous section. A formal definition shall be given now [44].

Theorem 3.1. Let ρ be the density operator acting on the Hilbert space H_Aⁿ. Then there exists a Hilbert space Hⁿ_B and a pure state |ψyA,BP Hⁿ_Ab H_Bⁿ such that the partial trace of |ψyA,B A,Bxψ| with respect to Hⁿ_B is given by

tr_Bp|ψyA,B A,Bxψ|q “ ρA, (3.4.1) where |ψy is called the purification of ρ

(17)

Proof. A positive semi definite density matrix, ρ, is given by ρ “řdimpHnq i pi|iyxi|

for some orthonormal basis tiu. Furthermore, let Hⁿ_B be another copy of Hⁿ_A with the orthonormal basis ti¹u. Define |ψyA,BP Hⁿ_Ab Hⁿ_B by

|ψyA,B“

dimpHq

ÿ

i

?pi|iyAb |i¹yB. (3.4.2)

Then, using the partial trace, one obtains

trBp|ψyA,B A,Bxψ|q “ pI b trq

«˜

ÿ

i

?p_i|iyAb |i¹yB

¸ ˜ ÿ

j

?p_{j A}xj| bBxj¹|

¸ff

“

pI b trq

« ÿ

i

ÿ

j

?pipj|iyAAxj| b |i¹yB Bxj¹| ff

“ ÿ

i

ÿ

j

?pipj|iyAAxj| tr`

|i¹yB Bxj¹|˘

“ ÿ

i

ÿ

j

?p_ip_j|iyAAxj|δij “ ρA. (3.4.3)

This proves the claim.

The notion of purification, in the sense of the definition given by Uhlmann and Sj¨oqvist, will be discussed and further elaborated in the upcoming sections.

3.5 Tensor product spaces

The wave-function, |ψy in product basis is expressed as the direct products of wave-functions for corresponding spin.

|ψy “

N

â

k

|miy. (3.5.1)

That is, the direct product of two matrices A and B yields

A b B “„A₁₁ A₁₂ A21 A22



b„B₁₁ B₁₂ B21 B22



“„A₁₁B A₁₂B A21B A22B



. (3.5.2)

Thus the wave function |ψy in product-basis of a two-spin system is (using the eigenkets as described in Equation (2.4.5 and 2.4.6))

(18)

|ψ1y “ |αy b |αy “„1 0

 b„1

0



“

»

—

— – 1 0 0 0 fi ffi ffi fl

, (3.5.3)

|ψ2y “ |αy b |βy “„1 0

 b„0

1



“

»

—

, (3.5.4)

|ψ3y “ |βy b |αy “„0 1

 b„1

0



“

»

—

, (3.5.5)

|ψ4y “ |βy b |βy “„0 1

 b„0

1



“

»

—

. (3.5.6)

Consider the sum of the operator I¹_z and I²_z(where I_zis given by Equation (2.4.3))for a two-spin system (where the superscript 1 stands for first spin and 2 for the second spin). However, the sum I¹_z` I²_z yields a 2-by-2 matrix which is incorrect since it is a sum of one-spin operators. Formally, the operators of a two-spin system can be calculated from the direct product of the one-spin operators with the identity operators. That is, if one denotes one-spin operators as¹I, then (also for the sake of avoiding misunderstanding, the identity operator shall be denoted E)

2I¹_z“¹I¹_zb E², (3.5.7)

2I²_z“ E²b¹I²_z. (3.5.8) Thus, the sum of operators for a two-spin system is then correctly given by (using the eigenkets described in Equation (2.4.5) and 2.4.6))

2I¹_z`²I²_z“¹I¹_zb E2` E2b¹I²_z“

»

—

— –

1 0 0 0

0 0 0 0

0 0 0 ´1 fi ffi ffi fl

. (3.5.9)

Thus, for a two-spin system, the operator algebra in direct product spaces is given by:

pA b Bq p|αy b |βyq “ A|αy b B|βy. (3.5.10)

(19)

4 Geometrically induced phases for pure states

The phenomenon of a geometric phase, mathematically known as holonomy (transport of a vector along a closed curve on a manifold, in particular: transport of a state vector within the space of states), arises due to parallel transport of a vector on a curved area, that is, when the system undergoes a loop, see figure 2.

Figure 2: [7] The transport of a vector along the closed curve C on a 2-sphere.

Classically, it is impossible to state whether or not the system has under- gone an evolution in quantum mechanics, the system retains some information about its motion in the form of a phase factor of the state vector. In 1984 M.V. Berry addressed the issue of unitary cyclic evolution under the action of a time-dependent Hamiltonian in a quantum system. Supposedly, the process changes adiabatically, which implies that the time scale of the changing Hamil- tonian is much larger than the time scale of the system. It was assumed that the quantum system for a cyclic Hamiltonian would only acquire a dynamical phase (Since it comes from the dynamics of the system), deprived of any physical meaning, which could be eliminated by using a suitable gauge-transformation of the from |ψy Ñ e^iφ|ψy. However, M.V. Berry discovered that there is an additional phase, beside the dynamical phase, which is purely geometrical and depends only on the path that |ψy describes in the parameter space.

(20)

4.1 Adiabatic Geometric phase (Berry’s phase)

Adiabatic Process [49] involves a slow perturbation acting on the systems eigenstate. As such, it is required that the parameter ~R of the Hamiltonian Hp ~Rq changes slowly for the adiabatic theorem to be in affect. This implies that the system will remain in an eigenstate of Hp ~Rptq; tq at any time t, provided that the system initially is in an eigenstate of H. Assuming that the evolution is cyclic ( ~Rp0q “ ~RpT qq for some period T ), then the Hamiltonian takes on its original and final form at T , resulting in the system returning to its initial state. The state has been transported around a loop, according to C : t P r0, T s ÞÑ |ψy, in parameter space with |ψy describing the instantaneous state of the system, equivalent to the eigenstate |n; ~Rptqy of the instantaneous Hamiltonian:

Hp ~Rptq; tq|n; ~Rptqy “ Enptq|n; ~Rptqy, (4.1.1) where E_n denotes the n:th energy eigenstate and |n; ~Rptqy is assumed to be normalized. The Schr¨odinger equation for the evolution of the system’s state vector |ψy is given by

i~d

dt|ψy “ Hp ~Rptq; tq|ψy. (4.1.2) Let the initial state |ψp0qy “ |n; ~Rp0qy be an eigenstate of Hp ~Rp0q; 0q at time t “ 0. Assuming that the solution due to adiabatic evolution, reads;

|ψptqy “ e^iΦptq|n; ~Rptqy. (4.1.3) The phase Φptq is determined by inserting equation (4.1.3) into equation (4.1.2):

´i~

„ idΦ

dte^iΦptq|n; ~Rptqy ` e^iΦptqd

dt|n; ~Rptqy



“ e^iΦptqE_np ~Rptq; tq|n; ~Rptqy.

(4.1.4) By multiplying equation (4.1.4) with the dual of |ψy from the left and uti- lizing the orthogonality property of the eigenstate, one is left with

dΦ dt “ ´1

~Enp ~Rptq; tq ` ixn; ~Rptq|B

Bt|n; ~Rptqy. (4.1.5) Equation (4.1.5) can be integrated with respect to time t P r0, T s, leaving you with

Φptq “ ´1

~ żT

0

Enp ~Rptq; tqdt ` i żT

0

xn; ~Rptq|B

Bt|n; ~Rptqydt. (4.1.6) The dynamical phase is the first integral in equation (4.1.6). It can be written as

(21)

θnptq “ ´1

~ żT

0

xn; ~Rptq|Hp ~Rptq; tq|n; ~Rptqydt. (4.1.7) For the second integral in equation (4.1.6), the time dependence has been assumed to be implicit, that is _Bt^B|n; ~Rptqy “ ∇R~|n; ~Rptqy^{d ~}_dt^R. Inserting this expression yields

Φptq “ ´1

~ żT

0

Enp ~Rptq; tqdt ` i żRpT q

Rp0q

xn; ~Rptq|∇R~|n; ~Rptqyd ~R. (4.1.8) If one now considers a closed path, then Rp0q “ RpT q, and

Φptq “ ´1

~ żT

0

Enp ~Rptq; tqdt ` i

¿

C

d ~Rxn; ~Rptq|∇R~|n; ~Rptqy. (4.1.9)

That is, the Berry phase is then given by

γnrCs “i

¿

C

d ~Rxn; ~Rptq|∇R~|n; ~Rptqy (4.1.10)

“

¿

C

A_Bp ~Rq ¨ d ~R, (4.1.11)

where A_B “ ixn; ~Rptq|∇R~|n; ~Rptqy is called the Berry Connection. The Berry phase is independent of time and only depends on the path C in parameter space. It is also worth noting that the Berry phase, γnrCs, is real, which can be easily shown using the normalization condition xn; ~Rptq|n; ~Rptqy “ 1, which gives

∇R~xn|ny “ xn|∇R~ny^˚` xn|∇R~ny “ 0, (4.1.12) implying that the real part is identically zero and Equation (4.1.10) is real.

4.1.1 Gauge invariance of the Berry phase

By choosing a different phase for the eigenvectors |nptqy, one can show that the geometric phase, γn, remains invariant, otherwise the quantity would not be physical since the choice of phase for the eigenvectors are arbitrary at each instant of time. That is, make the following gauge transformation

|nptqy ÞÑ |n¹ptqy “ e^iαptq|nptqy. (4.1.13) Then, the geometric phase is expressed as

(22)

γ_n¹1“i ż

dtxn¹ptq|d

dt|n¹ptqy “ γn´ żτ

0

dtαptq, (4.1.14) where αpτ q ´ αp0q “ 2πn (since initial and final eigenstate can only differ by a integer phase of 2π due to choosing single-valued eigenvalue basis). Thus γ_n¹1 “ γn mod 2π.

4.1.2 Informal description of the Chern number

An informal description of the Chern number is given here. Consider an adiabatic transport of an eigenstate around a small loop in the parameter space.

When a loop is completed, the particle will be in the same eigenstates as it started, up to a possible multiplication of a phase factor, say eîφ¹. By Stokes theorem, φ1should be a function describing the area inside the loop. Now, adiabatically transporting the eigenstate in the opposite direction around the loop yields a phase factor e^íφ² where φ2 is a function describing the area outside of the loop. However, the phase should be the same regardless of direction. Thus eîφ¹“ e^íφ² or eîpφ¹^`φ²^q“ 1. By setting φ “ φ1` φ2, one ends up with eîφ “ 1 which implies that φ must be an integer multiple of 2π. This integer is called the Chern number [40].

(23)

4.1.3 Example, spin 1/2 in an adiabatically rotating magnetic field Consider this particular example of a spin-¹₂ particle moving in a externally applied, adiabatically rotating magnetic field ~Bptq under angle θ around the z-axis, see figure 3.

Figure 3: [7] A spin-¹₂ particle moving in a adiabatically, rotating, θ, externally applied magnetic field B with angular velocity ω, emulating Equation (4.1.16)

The magnetic field is given by

Bptq “ | ~~ Bptq|

» –

cospθq sinpωtq sinpθq sinpωtq

cospθq.

fi

fl (4.1.15)

As the magnetic field varies, the spin´¹₂ particle follows the direction of the magnetic field in the sense that the eigenstate of Hp0q goes to the eigenstate of Hptq at later times ,t. The interaction Hamiltonian, in rest frame, for this system is given by

Hptq “ µ| ~~ Bptq|~σ “ µ| ~Bptq|

„ cospθq e^´iωtsinpθq e^iωtsinpθq ´ cospθq



, (4.1.16)

where the constant µ “ _2m^e ~. The eigenvalue equation

Hptq|nptqy “ En|nptqy, (4.1.17) is solved by the normalized eigenstates of Hptq

|n_`ptqy “

„ cos`_θ

2

˘ e^iωtsin`_θ

2

˘



, (4.1.18)

|n_´ptqy “

„ ´ sin`_θ

2

˘ e^iωtcos`_θ

2

˘



, (4.1.19)

with the corresponding eigenvalues

E_˘ “ ˘µ| ~Bptq|. (4.1.20)

(24)

Furthermore, one could argue that one could indeed interpret the eigenstates

|n˘y as spin up and spin down (for ` and ´ respectively) along the direction of the magnetic field.

The allowed values for the Hamiltonian, H “ Hpθ, φptq “ ωt, rptq “ | ~Bptq|; tq is identical to the parameter-space given by S². That is, the magnetic field ~Bptq traces our the curve; C : rptq “ r, θptq “ θ, φ P r0, 2πs, see figure 4.

Figure 4: [7] Magnetic field, Equation (4.1.16), following the path C, expressed in parameter space

Thus, the gradient spanned by the magnetic field is then given by

∇|n˘ptqy “ B

Br|n˘ptqyˆr `1 r

B

Bθ|n˘ptqyˆθ ` 1 r sinpθq

B

Bφ|n˘ptqy ˆφ. (4.1.21) Inserting Equation (4.1.18 & 4.1.19) into Equation 4.1.21 yields

∇|n`ptqy “ 1 r

„ ´¹₂sin`_θ

2

˘

1

2e^iωtcos`_θ

2

˘



θ `ˆ 1 r sinpθq

„ 0

ie^iωtsin`_θ

2

˘



φ,ˆ (4.1.22)

∇|n_´ptqy “ 1 r

„ ´¹₂cos`_θ

2

˘

´¹₂e^iωtsin`_θ

2

˘



θ `ˆ 1 r sinpθq

„ 0

ie^iωtcos`_θ

2

˘



φ.ˆ (4.1.23)

Multiplying by the corresponding bar vector from the left yields

xn_`|∇|n_`y “ isin²`_θ

2

˘ r sinpθq

θˆ (4.1.24)

xn_´|∇|n_´y “ icos²`_θ

2

˘ r sinpθq

θ,ˆ (4.1.25)

whereby integration along the path C yields the expression for berry phase (4.1.10)

(25)

γ_n,˘rCs “

¿

C

xn˘|∇|n˘yr sinpθqdθdφ “ iπp1 ¯ cospθqq, (4.1.26)

with the dynamical phase (4.1.7) given by

θnrT s “ ´1

~ żT

0

E_˘ptqdt “ ¯µ

~| ~Bptq|T. (4.1.27) Thus, the total state from initial state to final state (after on cycle) is then given by

|n˘ptqy “ e´iπp1¯cospθqqe^¯^µ^~^{| ~}^Bptq|T|n˘p0qy. (4.1.28) Note that the dynamical phase only depends on the period, T , of the rotation while the geometrical phase depends on the geometry of the problem.

(26)

4.2 Non-adiabatic Geometric phase (Aharonov-Anandan phase)

A generalization of the Berry phase was proposed in 1987 by Aharonov & Anan- dan [7]. The consideration of a closed path, unrestricted by the adiabatic theorem, allows one to bypass the notion of parameter space when describing the evolution of the Hamiltonian. When doing so, it is important to work with the projective state space in which the closed curve is traced by the system. The importance of this generalization is due to the fact that in real processes, the adiabatic condition is never exactly fulfilled.

Let us consider the projective state space, P, which is by definition, the set of equivalence classes of all state vectors of the Hilbert space, H, with respect to the equivalence relation (for the sake of convenience, assume that |ψy is normalized)

|ψ¹y „ |ψy iff |ψ¹y “ e^iφ|ψy, (4.2.1) where φ is a real number. The projection operator, Π, from the Hilbert space to the projective Hilbert space is defined as

Π : H Ñ PpHq,

|ψy ÞÑ |ψyxψ|. (4.2.2)

This implies that every ray is mapped to a point in P. That is, the pure state density operator (2.2.5) corresponds to the projective state space due to the loss of phase information. If dimpHq “ n then the projective Hilbert space PpHq is a manifold of dimension dimpPpHqq “ n ´ 1, see figure (5).

Figure 5: [7] The path C in H being projected onto the projective state space C¹ on PpHq. Note that the path C need not necessarily be closed for the path C¹ to be closed.

(27)

Now, consider a cyclic evolution of a state vector, |ψptqy of period T , in the projective state space. The unitary evolution

|ψp0qy ÞÑ |ψptqy “U ptq|ψp0qy (4.2.3) of the state vector produces the path C : r0, T s Ñ H, which is projected on the projective Hilbert space P pHq via the map, Π, as ΠpCq “ C¹. However, there are infinitely many paths in H that project to the same path in PpHq. That is, if |ψy describes the path C and | ˜ψy “ e^{if ptq}|ψy describes the path ˜C, then for any arbitrary real function, f ptq, they define the same path C¹ in PpHq under the projection operator π;

e^{if ptq}|ψy ÞÑ e^{if ptq}|ψyxψ|e^{´if ptq}“ |ψyxψ|. (4.2.4) Additionally, the evolution is cyclic if and only if the path in the projective Hilbert space, C¹, is closed (note that the path in the Hilbert space, C, need not necessarily be closed for the path in C¹ to be closed), that is

ΦrCs “ argtxψp0q|ψpT qyu, (4.2.6)

which signifies the phase shift of the system. Aharonov & Anandan [16] showed that the geometric phase can be obtained by subtracting the dynamical phase (reminiscent of Equation (4.1.7)):

θrCs “ ´ żT

0

xψptq|H|ψptqydt

“ ´i żT

0

xψp0q|U^:ptqdU ptq

dt |ψp0qydt, (4.2.7)

From the expansion for the total phase (Equation (4.2.6)), it then follows that the geometric phase is then given by

γrC¹s “ argtxψp0q|ψpT qyu ` i żT

0

xψp0q|U^:ptqdU ptq

dt |ψp0qydt. (4.2.8) This is a functional of C¹alone. The Aharonov & Anandan (AA) geometric phase (4.2.8) can be applied on a cyclic or adiabatic Hamiltonian [18] since the geometric phase depends only on the cyclic evolution of the system itself.

Thus, in the adiabatic limit, the AA-phase tends to the Berry phase. The arbitrariness of f ptq allows one to completely remove the dynamical phase, that is, equation (4.2.8) is a gauge-invariant expression for the geometric phase of a

(28)

pure state. By choosing f ptq “ ¹

~

şt

0xψpt¹q|Hpt¹q|ψpt¹qydt, one can impose the condition xψp0q|U^:ptq^{dU ptq}_dt |ψp0qy “ 0, @ψ P H, known as parallel transport.

The validity of the parallel transport condition can be easily proven, using the gauge transformation |ψ¹ptqy “ e^{if ptq}|ψptqy, whereby

xψ¹ptq|d

dt|ψ¹ptqy “ xψp0q|U^:ptqe^{´if ptq}d

dtU ptqe^{if ptq}|ψp0qy (4.2.9)

“xψp0q|U^:ptqe^{´if ptq}ˆ dU ptq

dt e^{if ptq}|ψp0q ` idf ptq

dt e^{if ptq}U ptq|ψp0qy

˙

(4.2.10)

“xψp0q|U^:ptqdU ptq

dt |ψp0qy ` idf ptq

dt (4.2.11)

dt |ψp0qy ` id dt

ˆ 1

~ żt

0

xψpt¹q|Hpt¹q|ψpt¹qydt

˙

(4.2.12)

dt |ψp0qy ` i

~xψptq|Hptq|ψptqy (4.2.13)

“xψp0q|u^:ptqdU ptq

dt |ψp0qy ´ ψp0q|U^:ptqdU ptq

dt |ψp0qy “ 0. (4.2.14) Here the orthogonality condition has been used in Equation (4.2.11) and the fundamental theorem of calculus in Equation (4.2.12). Moreover, from Equation (4.2.11), one could easily deduce the expression of f ptq that is needed in order to fulfil the condition of parallel transport. Thus, provided that parallel transport is ensured, the non-adiabatic Geometric phase, formulated by Aharonov et al.

is given by Equation (4.2.6). The notion of parallel transport will be discussed more thoroughly in Section 5.1 regarding mixed geometric phase given by A.

Uhlmann.

(29)

5 Geometrically induced phases for mixed states

For a pure state, Equation (4.2.6) gives the total phase (or the geometric phase provided parallel transport is ensured) gives the total phase change during an evolution. However, for a mixed state, this fails. The first problem lies in iden- tifying the change of the mixed state density operator ρ. States in of identically prepared quantum systems exhibits an uncertainty, hence the reason why they are described as a micture of pure states.

Uhlmann, [28] [29], was among the first to develope a theory for the geometric phase for parallel transported mixed states. This was approached using the notion of amplitudes and purification (Section (5.1.1)), which is then used, extensively, in Section 7.

Another approach was given by Sj¨oqvist et al. [19] where they later showed that, using a conventional Mach-Zehnder interferometer, it is possible to measure this phase using an interferometer by observing the intensity of the output signal. It turns out that the Uhlmann phase is, indeed, an observable and is later experimentally obtained in Section 6.

5.1 A. Uhlmann’s concept of mixed geometric phase

5.1.1 Parallel amplitude

The notion of parallel amplitudes is essential, for the Uhlmann case, to be able to define a parallel transport condition. Consider the amplitudes w1and w2for given states ρ1 and ρ2 respectively. The two amplitudes w1 and w2 are said to be parallel if they minimize the Hilbert space distance in Hw, that is w1 is parallel to w₂ (w₁k w2) if

k w₁´ w2k²“ min

wr₁,wr₂kwr₁´wr₂k², (5.1.1) where the minimum is taken over allwr1 andwr2which satisfies

ρ1“wr1wr^:₁, (5.1.2) ρ2“wr2wr^:₂. (5.1.3) One can rewrite the minimum condition to obtain properties of the amplitudes:

min

wr1,wr2

kwr₁´wr₂k²“ min

wr1,wr2

tr“

pwr₁´wr₂q^:pwr₁´wr₂q‰

(5.1.4)

“ min

wr₁,wr₂tr

”

wr^:₁wr₁`wr^:₂wr₂´wr^:₁wr₂´wr^:₂wr₁ ı

(5.1.5)

“ tr pρ1q ` tr pρ2q ´ max

wr₁,wr₂tr

”

wr^:₁wr2`wr^:₂wr1

ı

(5.1.6)

“2 ´ 2 max

wr1,wr2

Re

” tr

´ wr^:₁wr2

¯ı

, (5.1.7)

where in the first line the explicit expression for the Hilbert-Schmidt norm has been used (Definition (A.1.9)). Given the condition of maximum, for Repxq ď

(30)

|x|, it is clear that ˜w₁ and ˜w₂ are chosen such that ˜w₁^:w˜₂ is self-adjoint and positive definite.

A more explicit expression can be achieved by using the polar decomposition theorem. This theorem states that any operator A can be decomposed into A “ |A|UA, where |A| “a

A^:A and U_A is a unitary operator. Now for any unitary operator U , one has

Re rtr pAU qs ď | tr pAU q | “ | tr´a

|A|a

|A|U UA

¯

| ď

c

ptr p|A|qq

” tr´

U^:_AU^:|AU UA|

¯ı

“ tr |A|, (5.1.8) where in the second line, the Cauchy-Schwartz inequality is used. Equlity is obtained when U “ U^:_A, leaving us with

maxU Re rtr pAU qs “ tr |A|. (5.1.9) Now, applying Equation (5.1.9) onto the last term of Equation (5.1.7) for the decomposition ˜w₁“?

ρ₁U₁ and ˜w₂“?

ρ₂U₂, yields

max

˜ w1, ˜w2

Re” tr´

˜ w^:₁w˜₂¯ı

“ max

U₁,U₂Re” tr´

U^:₁? ρ₁?

ρ₂U₂¯ı

“ max

U Re rtr p? ρ1

?ρ2U qs

“ trb?

ρ1ρ2

?ρ1. (5.1.10)

Thus, inserting Equation (5.1.10) back into Equation (5.1.7), one obtains the Hilbert space distance between two parallel amplitudes w1 and w2 which is (equal to the Bures distance [43]) given by

kw1´ w2k²“ 2 ´ 2 trb?

ρ1ρ2

?ρ1. (5.1.11)

The equality in Eqution (5.1.10) is obtained for U “ U2U^:₁ “ U^:^?_ρ

1,?ρ₂

which is the adjoint of the unitary operator of the decomposition

?ρ₁?

ρ₂“ |? ρ_q?

ρ₂|U^?ρ1,?

ρ2. (5.1.12)

This can be flipped around and be expressed as

U₂U^:₁“ b

ρ^´1₂ b

ρ^´1₁ b?

ρ₁ρ₂?

ρ₁, (5.1.13)

Equation (5.1.13) is the relation that the unitary operators U1and U2must satisfy to minimize the distance between associated amplitudes w1 and w2 in order to achieve w1k w2.

Geometric Phases In Quantum Systems Of Pure And Mixed State