State and Process Tomography : In Spekkens' Toy Model

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Linköpings universitet

Linköping University | Department of Electrical Engineering

Master’s thesis, 30 ECTS | Applied Physics

2020 | LiTH-ISY-EX--20/5273--SE

State and Process Tomography

–

In Spekkens’ Toy Model

Andreas Andersson

Supervisor : Niklas Johansson Examiner : Jan-Åke Larsson

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Abstract

In 2004 Robert W. Spekkens introduced a toy theory designed to make a case for the epis-temic view of quantum mechanics. But how does Spekkens’ toy model differ from quan-tum theory? While some differences are well-established, we attempt to approach this question from a tomographic point of view. More specifically, we provide experimentally viable procedures which enables us to completely characterize the states and gates that are available in the toy model. We show that, in contrast to quantum theory, decompositions of transformations in the toy model must be done in a non-linear fashion.

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Acknowledgments

I would like to thank my supervisor Niklas Johansson, not only for enabling this project to happen, but also for the patient and fruitful discussions throughout the process. I would also like to express special gratitude to my beloved girlfriend and family for the invaluable support through every high and low point of the project.

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List of Figures

2.1 The Bloch sphere picture of an arbitrary state ρ. . . . 9

2.2 Unknown quantum black box that takes input states and produces output states. . 12

2.3 Geometrical picture of the six pure STM and QSL states. . . 18

2.4 Visual representation ofI,X,Y andZ. . . 21

2.5 Visual representation ofHandS. . . 21

3.1 The ontic states of a single-qubit simulation depicted as boxes. . . 23

3.2 Permutations corresponding to theI,X,Y andZgates. . . 24

3.3 The ontic states of a two-qubit simulation depicted as boxes. . . 25

3.4 The permutations corresponding to tI I, I X, . . . , Z Y, Z Zu. . . 26

3.5 The permutations corresponding to theHgate. . . 28

3.6 The permutations in terms of the Pauli operators. . . 28

3.7 The permutation corresponding to U = HSin terms of the Pauli operators. . . 32

3.8 Schematic view of a transformation composed of a probabilistic mixture of re-versible transformations. . . 34

3.9 ~r= (aˆx, b ˆy, c ˆz)points to a mixed state. . . 37

3.10 Unknown STM black box that takes input states and produces output states. . . 47

3.11 Visual representation ofHS. . . 54

A.1 Two dimensional visualization of an arbitrary χ matrix. . . . 75

A.2 The χ matrix corresponding to the H transformation. . . . 76

A.3 The real part of the χ matrix corresponding to the S transformation . . . . 77

A.4 The imaginary part of the χ matrix corresponding to the S transformation. . . . 77

A.5 Visualization of the χ matrix corresponding to the CNOT gate. . . . 78

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List of Tables

3.1 Decomposition of theHgate. . . 28

3.2 Decomposition of theS gate. . . 29

3.3 The Pauli operator basis, t rEmu16_m=1, for two-qubit simulations . . . 29

3.4 All ontic states mapped by theCN OT gate. . . 30

3.5 Decomposition of theCN OT gate. . . 30

3.6 Maps of all ontic states when U= p1H +p2S. . . 35

3.7 Determination of(x, p)based on the measurements. . . 38

3.8 The possible Pauli operators corresponding to transformations of eigenstates. . . . 69

3.9 The Pauli operators corresponding to inversion of states. . . 70

3.10 All Pauli operators are viable descriptions of this map. . . 70

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1 Introduction

1.1 Motivation

Since the formulation of Moore’s law in 1965, computer science has been able to hold true to the idea that computational power roughly doubles every second year for a constant cost. This trend is generally expected to end in the near-future due to unwanted quantum effects as a consequence of the ever smaller hardware components. Quantum computation is specu-lated to bring a new paradigm to computation, possibly enabling continuous development of computational power by doing calculations based on quantum physics rather than classical physics.

Extensive resources are worldwide being invested in the technically challenging field of quantum computation. Meanwhile, researchers at Linköping University have devised a framework that has proven to efficiently simulate quantum algorithms such as the famous Deutsch-Jozsa algorithm [7]. This framework is known as Quantum Simulation Logic (QSL) and seems to come close to quantum computation. QSL is an extension of Spekkens’ toy model (STM) [14], which is an older simulation framework. Up to this point it is unclear exactly how QSL and STM differs from quantum theory, motivating the importance of further studies of the subject.

1.2 Aim

The main purpose of this thesis is to further develop an understanding of STM and how it differs from quantum theory. Although some differences are well-established, we expect state and process tomography to close some missing gaps in our understanding of the theory. Quan-tum state and process tomography are tools that fully describes quanQuan-tum states and processes respectively. By adapting these tools to STM we hope to gain insight to the transformations

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1.3. Research Questions

available in the model and how they differ from the analogues in quantum theory. By build-ing a better understandbuild-ing of STM we also hope to lay a foundation for further studies of the extended model; QSL.

1.3 Research Questions

With the above-mentioned purpose of this thesis in mind, the main research question is nat-ural:

1. How does STM differ from quantum theory?

Before we can expect to have an answer to this question, further research of state and process tomography and their eventual applications in STM is needed. To be more specific we need to answer the following two sub questions:

2. Is it possible to perform tomography in STM? 3. How can tomography be performed in STM?

The outcomes of question (2) and (3) will be crucial with regards to what kind of answer that can be formulated for (1).

1.4 Delimitations

The main focus of this report will be on STM, but since QSL is a similar model some of the results will be applicable also there. Beyond that, the work is mainly limited to state and process tomography; the answer that will be provided to the main research question (1) will therefore have to be within the scope of those tools.

Additionally, when we study processes in quantum theory and STM we will mainly limit ourselves to transformations that correspond to gates rather than measurements.

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2 Preliminaries

"If you are not completely confused by quantum mechanics, you do not understand it"

– JOHN WHEELER

2.1 Quantum Theory

In what follows, a brief introduction of quantum theory will be presented. The content of this introduction is specifically chosen with the aim to facilitate necessary knowledge related to quantum information, rather than approaching the topic from a more conventional "quan-tum physics" point of view. For this purpose ref. [9] generally serves as an excellent source, spanning a wide range of topics.

Mathematical Formalism

Before explicitly approaching quantum theory, some basic understanding of the mathemati-cal framework is needed. As we will see, linear algebra is the foundation of this framework.

Notation

The conventional way of representing vectors in quantum theory is by using bra-ket notation, also known as Dirac notation. A ket vector is denoted|yand it usually represents a column vector of complex numbers. Every ket vector has a corresponding dual vector called bra, which is denoted x|. A bra vector xΨ| is the transposed complex conjugate, or hermitian conjugate of|Ψy, meaning that

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2.1. Quantum Theory

Here (:) denotes the hermitian conjugate [4].

Inner Product

Equa-tion 2.2, where φ˚

i denotes the complex conjugate of the number φi[4].

xφ|ψy ” n ÿ i=1

φ˚_iψi. (2.2)

Note that the result is equivalent to the scalar product:

h φ˚₁ φ˚₂ . . . φ˚_n i        ψ1 ψ2 .. . ψn        . (2.3)

The two vectors are said to be orthogonal if xΦ|Ψy = 0. Furthermore, a set of vectors are said to be orthonormal if they are mutually orthogonal and of norm one. The norm of a vector is defined as

k|Ψyk”axΨ|Ψy. (2.4)

A natural choice of orthonormal basis in two dimensions is referred to as the computational basis. These basis vectors are defined as

|0y ” " 1 0 # and |1y ” " 0 1 # . (2.5)

A Hilbert space is a complete, possibly infinite dimensional vector space wherein an inner product can be formulated. Throughout this report we will restrict ourselves to finite dimen-sional vector spaces. As a consequence of this restriction, the existence of an inner product turns out to be sufficient for the space to qualify as a Hilbert space. These spaces are of par-ticular interest in quantum information since they, as we will see, constitute the framework in which finite quantum bit systems can be described.

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2.1. Quantum Theory

Outer Product

The outer product between|Ψy and|Φy is denoted|Ψy xΦ|and is defined as        ψ1 ψ2 .. . ψm        h φ˚₁ φ˚₂ . . . φ˚_n i ”        ψ1φ˚₁ ψ1φ₂˚ . . . ψ1φn˚ ψ2φ˚₁ ψ2φ₂˚ . . . ψ2φ_n˚ .. . ... . .. ... ψmφ₁˚ ψmφ˚₂ . . . ψmφ˚_n        . (2.6)

The outer product is therefore a useful way of constructing linear operators [4].

Linear Operators

Suppose that A is a linear operator. By definition this means that

A ÿ i ai|Φiy =ÿ i aiA|Φiy. (2.7)

Every linear operator can be represented in the form of a matrix which motivates that the terms linear operator and matrix can be used interchangeably [4]. Within the context of quan-tum theory the famous set of Pauli operators are of outmost importance:

I ” " 1 0 0 1 # , σx”X ” " 0 1 1 0 # , σy”Y ” " 0 ´i i 0 # , σz”Z ” " 1 0 0 ´1 # . (2.8)

The Pauli matrices are known as observables since their eigenvalues are associated with mea-surable, physical quantities. As we will see in section 2.1, these matrices are convenient tools for several applications.

The hermitian conjugate, or adjoint, of an operator A is commonly denoted A:_{and follows} the definition

(|Φy , A|Ψy) = (A:_|_{Φy ,} _|_Ψy₎_, _(2.9) where(¨, ¨)denotes the inner product.

If A = A:_{, A is said to be hermitian or self-adjoint. Hermitian operators have real-valued} eigenvalues; it is not surprising that the Pauli matrices (Equation 2.8) are hermitian, keeping in mind that nature provides us with real numbers upon measurement.

Furthermore, A is unitary if AA: ₌ _A:_A ₌ _{I, implying that A}: ₌ _A´1_{. Unitary operators} successfully describe the evolution of quantum systems, such as state changes due to quantum gates, as we will see in section 2.1.

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2.1. Quantum Theory

Any unitary matrix is normal, since they commute with their adjoint: AA:₌ _A:_{A. This} con-dition turns out to be sufficient for the operator to have a spectral decomposition. The spectral decomposition of an operator A can be defined in terms of a sum of outer products

A ”ÿ i

λi|iy xi|, (2.10)

where λiare the eigenvalues of A and|iy is an orthonormal basis in the vector space on which A acts.

Operator Functions

By using the spectral decomposition of an operator in accordance with Equation 2.10, an operator function f :CnˆnÞÑCnˆncan be defined as

f(A) =ÿ

i

f(λi)|iy xi|, (2.11)

where A is an n dimensional matrix. This way we can construct counterparts to some of the most well-known functions fromC to C, such as the square root, logarithm and exponential function. The most important operator function for our purposes is however the trace func-tion. The trace of a matrix is simply an operator function that takes the sum of the diagonal elements of the matrix

f(A) =tr(A)”ÿ i

Aii. (2.12)

Some important properties of the trace is

1. tr(A+B) =tr(A) +tr(B)

2. tr(AB) =tr(BA)

3. tr(U AU:_{) =}_tr₍_A₎ 4. tr(A|Ψy xΨ|) =xΨ|A|Ψy

The third property can be thought of as the fact that the trace function is invariant under unitary transformation. The fourth property is important since it gives the trace function a practical interpretation; the average value of the measurement outcomes of the observable A (with the understanding that|Ψy is the quantum state of interest). These concepts will be

further expanded upon in the upcoming sections.

Tensor Product

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2.1. Quantum Theory

The tensor product of matrices is also well-defined. Suppose that M and N are operators (matrices) onΩm andΩn respectively. Then M b N is an operator onΩmbΩn. The tensor product has a matrix formulation known as the Kronecker product; Suppose that M is an i ˆ j matrix and N is a k ˆ l matrix. Then

M b N ”        M11N M12N . . . M1jN M21N M22N . . . M2jN .. . ... . .. ... Mi1N Mi2N . . . MijN        , (2.13) which is an ij ˆ kl matrix. Operator Space

Let LVdenote the group of linear operators which maps elements of the Hilbert space V to V. Keeping in mind that every linear operator has a matrix representation, it can be verified that the sum of two linear operators on V is a linear operator on V. Furthermore, every operator zA is linear if A is a linear operator and z PC. Lastly, we note that there is a zero element 0 in this group of operators, which completes the list of criteria in order for LV to qualify as a vector space. More specifically, LV is in fact a Hilbert space since a natural inner product can be defined between operators A and B:

(A, B)HS”tr(A:B). (2.14)

Equation 2.14 is commonly referred to as the Hilbert-Schmidt inner product, or trace inner prod-uct. The operators A and B are thus said to be orthogonal if(A, B)HS =0. A popular choice of basis for the space of 2 ˆ 2 matrices is the Pauli operator basis. Let t rEmudenote the set of 2 ˆ 2 Pauli operators, including the identity: tI, X, Y, Zu. It can then be verified that

(Er_m, rE_n)_HS =dδ_mn, (2.15) where d= 2 is the dimension of the matrices and δmn =1 if m = n and δmn =0 otherwise. For higher dimensions the basis elements can be generated using the tensor product between all combinations of the two dimensional Pauli matrices. The sixteen operators in the set tI b I, I b X, . . . , Z b Y, Z b Zu will for example form a basis for all 4 ˆ 4 matrices, in perfect analogue with Equation 2.15.

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2.1. Quantum Theory

Quantum States

The concept of quantum states is fundamental to quantum theory and the applications within quantum information. In what follows, some of the most useful formulations of quantum states will be presented.

State Vectors

All quantum systems are associated with a Hilbert space. The associated Hilbert space is known as the state space of the system, in which a state vector fully describes the quantum state. The most fundamental quantum system is the quantum bit, or qubit. An arbitrary single-qubit state|Ψy can be represented as a unit state vector (a ket). In the computational basis,

we write

|Ψy=a|0y+b|1y . (2.16)

|a|2+|b|2=1, (2.17)

which may be interpreted as the fact that the probability of finding|Ψy in the state|0y or|1y is 1. A pure state is simply a state that can be expressed by a state vector.

Higher dimensional state vectors describes multiple-qubit systems. The two-qubit analogue version of the computational basis can be generated from the tensor products between all combinations of|0y and|1y, yielding

|00y=       1 0 0 0       , |01y=       0 1 0 0       , |10y=       0 0 1 0       , |11y=       0 0 0 1       . (2.18)

In this sense, quantum states are said to compose under the tensor product. This means that the state space of a combined system is nothing but the Hilbert space generated by the tensor product between the Hilbert spaces of the respective subsystem (see section 2.1).

Density Matrices

It is not always the case that the state of a system is completely known. Suppose that there is a large set of possible state vectors that may describe the system, each associated with a

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2.1. Quantum Theory

the ensemble tpi, |Ψiyu, meaning that the system is in the state with index i with probability pi. The mixed states are not to be confused with states in superposition, as the probabilities pi are classical probabilities. A convenient way of handling mixed states is by using the density operator formalism:

ρ ” ÿ

i

pi|Ψiy xΨi|. (2.19)

The density operator can equivalently be referred to as the density matrix which is an Hermi-tian matrix. Note that the density operator formalism enables description of pure states just as well as mixed states, while the state vector formalism is restricted to pure states.

For single-qubit states an equivalent way of expressing the density operator is

ρ= I+~r ¨~σ

2 . (2.20)

Here,~σis a vector containing the Pauli matrices X, Y and Z, while~r is a three-dimensional positional vector. The positional vector facilitates a geometrical interpretation of single-qubit states. This interpretation is often depicted as the Bloch sphere (Figure 2.1), which is a unit sphere in which all single-qubit states may be associated with a unique point. The point that corresponds to a specific ρ is simply located where the associated positional vector~r points. Note that an orthogonal basis such as the computational basis is depicted to be antipodar in the Bloch sphere;|0y and|1y are eigenstates to the Z observable and are thus depicted at+ˆz and ´ ˆz, respectively. The corresponding basis vectors for ˘ ˆx and ˘ ˆy are|˘yand|˘iy, where

|˘y ”?1 2(|0y ˘|1y) and |˘iy ” 1 ? 2(|0y ˘ i|1y). (2.21) ˆz „ |0y ´ˆz „ |1y ~r „ ρ ˆx ˆy

Figure 2.1: The Bloch sphere picture of an arbitrary state ρ.

All pure states are found on the surface of the sphere (|~r| = 1) while the mixed states are found inside the sphere (|~r|ă1). In theory the Bloch sphere provides an infinite number of

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2.1. Quantum Theory

states, however, upon measurement in the computational basis we will always find the state to be|0y or|1y.

Quantum State Tomography

Quantum state tomography is a method for experimental determination of the density operator ρof an unknown quantum state. For now, let ρ denote the state of a single-qubit. It is possible to expand ρ using the normalized set of Pauli operators: I/?2, X/?2, Y/?2 and Z/?2, which forms an orthonormal set with respect to the Hilbert-Schmidt inner product. This means that any single-qubit state ρ can be expressed as a linear combination of the set of Pauli matrices. The coefficient of each matrix can be determined by performing a large number of measurements of the corresponding observable and calculating the average value of these outcomes. In practice this requires a separate copy of ρ for each measurement since each measurement causes the quantum state to collapse.

Let a1, a2, . . . , am be the measurement outcomes of an arbitrary observable A. The average value of these measurement outcomes estimates the true average, tr(Aρ):

m ÿ i=1

ai

m « xΨ|A|Ψy=tr(Aρ). (2.22) Therefore, an arbitrary single-qubit state ρ can be expanded as follows:

ρ= tr(ρ)I+tr(Xρ)X+tr(Yρ)Y+tr(Zρ)Z

2 . (2.23)

Note that comparison between Equation 2.23 and Equation 2.20 facilitates a geometrical con-nection between the average measurement outcomes in the respective basis and the corre-sponding elements of the positional vector~r.

The concept of quantum state tomography can be generalized to the n-qubit case by the ex-pansion ρ=

ÿ

v1, ..., vn tr(σv1b ¨ ¨ ¨ b σvnρ)σv1b ¨ ¨ ¨ b σvn 2n , where viP t1, 2, 3, 4u. (2.24)

Intuitively, we might think of this expansion as the sum of all basis elements in the operator space of interest, where each term is weighted using the average measurement outcome of ρ in the corresponding basis.

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2.1. Quantum Theory

Quantum Operations

The mathematical formalism of quantum operations is of great convenience when considering discrete state changes of quantum systems:

ρ ÝÑ E (ρ)

tr(E (ρ)). (2.25)

E is a linear map that fully describes the quantum operation and is commonly formulated in terms of its operator-sum representation:

E (ρ) = ÿ k EkρE_k:, where ÿ k E:_kEkďI. (2.26)

If tr(E (ρ)) = 1, E is said to be trace preserving. All unitary quantum operations are trace preserving since their operation elements tEkufulfills řkE

:

kEk = I. The most general case, ř

kE :

kEk ďI, occurs when operations corresponding to measurements of the system are con-sidered [2]. For the purposes of this report trace preserving operations will suffice, as we are mainly concerned with quantum gates which indeed are unitary:

E (ρ) =ρ1= ÿ k EkρE_k:, where ÿ k E:_kEk= I. (2.27) Intuitively it is often helpful to think of unitary transformations in terms of rotations on the Bloch sphere. In fact, any arbitrary unitary single-qubit transformation can be factorized in terms of rotations about the three axes: U=eiαRz(β)Ry(γ)Rz(δ), where eiαis a phase factor. As an alternative — yet equivalent — definition of quantum operations, there exists an ax-iomatic approach. DefiningEto be a map from the set of density operators of the input space to the set of density operators of the output space while fulfilling the following axioms will suffice [9, 12].

‚A1: tr(E (ρ)) is the probability that the operation associated withE occurs when the initial state is ρ.

‚A2: E is a convex-linear map on the set of density operators. ‚A3: E is a completely positive map.

It turns out that A1, A2 and A3 holds true if and only ifEhas an operator-sum representation in accordance with Equation 2.26 [9].

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2.1. Quantum Theory

Quantum Process Tomography

Now consider the problem of determining an unknown quantum operation that takes input states and produces output states. The process associated to the unknown operation may be thought of as a black box of which we have no prior knowledge.

ρ

_{Quantum Black Box}

E (

_ρ

) =

ρ

1

Figure 2.2: Unknown quantum black box that takes input states and produces output states.

The goal of fully determining the dynamics of the black box can be summarized as the task to find the set of operation elements tEiusuch thatE (ρ) =ř_iEiρE:_i. In 2008 an experimental procedure known as process tomography was outlined by Isaac L. Chuang and M. A. Nielsen with the purpose of accomplishing this task [2]. We will now outline this procedure for a d dimensional state space.

First, prepare d2pure quantum statesΨ_jD in such a way that their density matrices are lin-early independent and thereby constitutes a basis for all d ˆ d matrices.

! Ψ_j D @ Ψj )d2 j=1” ! ρj )d2 j=1. (2.28)

These states will now serve as input states to the unknown processE, which we wish to characterize. Each outputE (ρj)may be recovered using state tomography. At this point we can predict the output of any given input staterρsince the input may be expressed as a linear combination of our basis states;rρ=

ř

jcjρj. By linearity this means thatE (ρr) = E ( ř

jcjρj) = ř

jcjE (ρj), where the set of coefficients cjcan be calculated and allE (ρj)are known from state tomography.

Up to this point however, we have no explicit information about the composition ofE. To proceed, we will choose an operator basis t rEiud

2

i=1for the space of operators which may act on our state space. These will be the building blocks in terms of which we will understandE. Since we have chosen t rEiud

2

i=1to span the space containing any Ei, we can express Eiin terms of our operator basis:

Ei= ÿ m eimEr_m, and E:_i = ÿ m e˚_inEr:_n where e_im, e˚_inPC. (2.29)

Recall thatE (ρ) =ř_iEiρE_i:. Using Equation 2.29, we may equivalently write

E (ρ) = ÿ mn eime˚inEr_mρ rEn:” ÿ mn χmnEr_mρ rE:n where χmn =eime˚in, (2.30)

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2.1. Quantum Theory

which is known as the chi matrix representation of E. Note that χ can be regarded to be a complex-valued matrix indexed by m and n. This matrix completely describesE once the set of operators t rEiuhas been chosen. The task of determining the dynamics of the black box may therefore be reformulated as the task of finding the χ matrix. To proceed, we make the observation that each output stateE (ρj)can be formulated in terms of the basis states (Equation 2.28):

E (ρj) = ÿ

k

λjkρk. (2.31)

One way of solving for λjk is to perform the Hilbert-Schmidt inner product with ρkon both sides of the equation. Furthermore, we can introduce a complex valued matrix β such that each entry βmn_jk fulfills

r

EmρjEr:_n = ÿ

k

βmn_jk ρ_k. (2.32)

Inserting Equation 2.31 and Equation 2.32 into Equation 2.30, it is clear that

ÿ k ÿ mn χmnβmn_jk ρ_k= ÿ k λ_jkρ_k, (2.33) meaning that ÿ mn βmn_jk χmn=λ_jk, for each k. (2.34) At this point, we can consider β to be a jk ˆ mn matrix, while χ and λ are mn- and jk dimen-sional vectors respectively. It turns out that Equation 2.34 is both necessary and sufficient for χto correctly characterizeE. To solve for χmnwe may first find the generalized inverse of β. The generalized inverse κ is a matrix that fulfills

βjkmn= ÿ st, xy

βst_jkκxy_st βmnxy, (2.35)

which together with Equation 2.34 yields the sought after χ matrix

χmn= ÿ

jk

κmn_jk λjk. (2.36)

This concludes the experimental procedure known as quantum process tomography. We will now explore this procedure further by considering an explicit example. To do this theoreti-cally we will need to assign a unitary transformation to our unknown black box in order to

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2.1. Quantum Theory

generate proper outputs from our inputs. Our unitary transformation of choice will be the Hadamard gate (H): H ” ?1 2 " 1 1 1 ´1 # . (2.37)

The quantum process thus become

E (ρ) =HρH:. (2.38)

Since this is a single-qubit example we need to prepare 22=4 input states. These states must be chosen so that their density matrices collectively forms a basis for the space of all 2 ˆ 2 matrices. We choose

tρju4j=1 = !

|0y x0|, |1y x1|, |+y x+|, |+iy x+i|), (2.39) which forms a non-orthogonal basis with respect to the Hilbert-Schmidt inner product. Now apply the quantum processE (ρj) =HρjH:for j=1, . . . , 4. In an experimental setting, these outputs would be recovered by state tomography. We now make use of Equation 2.31:

E (ρj) =řkλjkρk ùñ λjk= E (ρj), ρk

HS =tr E (ρj):ρk. (2.40) Next, we choose the four 2 ˆ 2 Pauli matrices to be the set of operators t rEiud

2

i=1:

t rEiu=tI, X, Y, Zu. (2.41)

By once again utilizing the Hilbert-Schmidt inner product, we can calculate the β matrix from Equation 2.32: r EmρjEr:_n= ÿ k βmn_jk ρk ùñ βmnjk =tr ρ : kEr_mρjEr:_n. (2.42)

From the β matrix, we can calculate its generalized inverse κ. For this example we will not focus on how this is done; most computer algebra packages will provide support for this. We now calculate each element of the χ matrix indexed by m and n using

χmn= ÿ

jk

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2.1. Quantum Theory

which finally yields

χ= 1 2       0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1       . (2.44)

The procedure presented above will work as a prescription for an experimental procedure, but also if we wish to theoretically predict the outcome of process tomography on an al-ready known process. For the latter case the procedure can be significantly simplified, as outlined in 2013 by Alexander N. Korotkov [8]. To explore this scenario, let U be some the-oretically known transformation on which we wish to perform process tomography. From Equation 2.30, it is clear that

U=ÿ m eimEr_m and U:= ÿ n e˚ inEr:_n if E (ρ) =UρU:. (2.45) By performing the Hilbert-Schmidt inner product with rEmon both sides of the first equation and likewise with rE:non the second, it can be shown that

eim = 1 dtr(U rE : m) and e˚in= 1 dtr(U : r En), (2.46)

where d is the dimension of the corresponding operator space. Recalling that χmn ”eime˚in we thus write

χmn = 1 d2tr(U rE

:

m)tr(U:Er_n). (2.47) Using either of the presented methods we are able to theoretically predict the outcome of process tomography on any quantum process. For our purposes the generators of the Clifford group are of special interest. These are the H gate (Equation 2.37), the phase gate (S) gate and the controlled-not gate (CNOT):

H ” ?1 2 " 1 1 1 ´1 # , S ” " 1 0 0 i # , and CNOT ”       1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0       . (2.48)

By adding the Toffoli gate to the group we achieve a universal set of quantum gates, mean-ing that we can access the complete continuum of quantum states in the state space of our

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2.1. Quantum Theory

system [5]. For single-qubit systems this may be thought of as being able to produce states corresponding to every point on the Bloch sphere.

Toffoli=                 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0                 . (2.49)

Before discussing the results yielded from theoretical process tomography on these gates we point out that there is a handy interpretation of χ matrices. This interpretation is visualized in figure in Appendix A.1. We see that each term in the χ matrix, χmn =eime˚in, is a measure of how the corresponding pair of operators contributes to the process of interest (in the operator-sum formalism). To further articulate the point, the χ matrix is commonly visualized as a three-dimensional histogram where the height of each bar represents the absolute value of the product eime˚in. When needed, the imaginary part of the matrix is often depicted in a separate histogram.

The χ matrices yielded from theoretical process tomography on the H, S and CNOT gates are visualized in Appendix A.2. As indicated by the labels of the axes, the operator basis t rEiu were chosen to be the Pauli basis for the respective operator space. As an example we see that figure A.2, stemming from theoretical process tomography on the H gate, tells us that

E (ρ) =1 2XρX+ 1 2XρZ+ 1 2ZρX+ 1 2ZρZ= X+Z ? 2 ρ X+Z ? 2 , (2.50)

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2.2. Spekkens’ Toy Model and Quantum Simulation Logic

2.2 Spekkens’ Toy Model and Quantum Simulation Logic

QSL is an entirely classical framework that simulates quantum states and gates. A thorough compilation of QSL has been provided by Niklas Johansson [6], which constitutes the main source of information for this section. Ref. [6] also serve as a great introduction to STM, but for some of the details ref. [14] has been used.

States

QSL is an extension of STM, which is an earlier simulation framework presented in 2004 by Robert W. Spekkens [14]. Spekkens’ framework models quantum states as the state of the knowledge that the observer has about the system. The key principle of this model is the so called knowledge balance principle, stating that the amount of knowledge one can have about an ontic state must be less or equal to the amount of knowledge one lacks about it. The available knowledge of the ontic state is referred to as an epistemic state and may be regarded as a probability distribution over the ontic states. This epistemic point of view has proven to successfully mimic many empirical phenomena that generally are characterized as being quantum. Because of STM:s ability to reproduce phenomena and, beyond that, demystify them by offering a more natural viewpoint compared to the ontic one, Spekkens pursues the thesis that all quantum states are epistemic. That being said, STM (and by extension QSL) are not taken to be actual descriptions of nature; it just happens to simulate many aspects of quantum information efficiently. Even though we mainly target STM in this thesis, we will borrow much of the notation from QSL.

STM models an ontic state in a discrete sample space of four points. QSL represents these four points using a tuple of classical bits:

(x, p), where x, p P t0, 1u. (2.51)

Here x and p are commonly referred to as the computational and phase bit respectively and together they constitute an elementary system. From the knowledge balance principle we know that the maximum knowledge available in this tuple is either the value of x, the value of p, or the relation x ‘ p (where ‘ is the "exclusive or" operation). Preparing the state to have a definitive x value of 0 or 1 (leaving p unknown) corresponds to preparing a quantum state to be|0y or|1y. Likewise; preparing p to take on the values 0 or 1 corresponds to preparing a quantum state to be|+yor|´y. The unknown bit of information is modeled by a random variable R which is equally weighted between the discrete values 0 and 1. The random vari-able envari-ables STM and QSL to store information in the correlation between x and p. More specifically, if x and p are prepared to be correlated random variables, the analogous quan-tum state is|+iy. If they instead are prepared to be anti-correlated the associated quantum state is|´iy. The six above-mentioned states are all of the pure states that STM and QSL can simulate and they correspond to the six epistemic states of maximal knowledge presented by Spekkens in ref. [14]. The states are of maximal knowledge due to the fact that we know one of

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the two bits in each state with certainty, which is exactly as much information that the knowl-edge balance principle allows for. A geometrical interpretation of these states analogous to the Bloch sphere picture in quantum theory is provided in Figure 2.3:

|0y „(0, R) |1y „(1, R) |+iy „(R, R) |´iy „(R, R ‘ 1) |+y „(R, 0) |´y „(R, 1)

Figure 2.3: Geometrical picture of the six pure STM and QSL states.

QSL also models mixed states successfully. These are found in the convex hull of the pure states, symbolized by the octahedron of dotted lines in the figure. The maximally mixed state is naturally modeled by two independent random variables R and R1_{, corresponding to the} intersection point of the ˆx, ˆy and ˆz axes. The set of mixed states in STM is more restricted; the only mixed elementary system is for example the maximally mixed one. [14].

Composition of Elementary States

As reviewed in section 2.1, quantum states compose under the tensor product. STM and QSL states instead compose under the Cartesian product:

(x1, p1)ˆ(x0, p0), or shorter: (x1, p1)(x0, p0). (2.52) Equation 2.52 represents the composition of two elementary states; (x1, p1) and(x0, p0). The ontic states of the composed system can now be thought of as the set of sixteen points generated by all combinations of xi P t0, 1u and pi P t0, 1u for i = 0, 1. Analogously we denote the composition of n elementary systems as

(xn´1, pn´1). . .(x0, p0). (2.53) For compositions of elementary STM systems, that is n ą 1, there are additional states of non-maximal knowledge than the maximally mixed one. We will consider these in greater detail in section 3.4, where we discuss state tomography in STM.

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Transformations in STM

The STM framework provides transformations that are analogous to the quantum gates dis-cussed in section 2.1. First off, the single-qubit Pauli operators can be constructed by making sure that the corresponding eigenstates stays true while all other states gets inverted:

I (x, p)”(x, p), X (x, p)”(x ‘ 1, p),

Y (x, p)”(x ‘ 1, p ‘ 1), Z (x, p)”(x, p ‘ 1). (2.54) (See visual representation in Figure 2.4)

As an example we can consider theZgate and compare the results to the quantum analogues.

Quantum Theory: $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % Z|0y=|0y Z|1y=|1y Z|+y=|ý Z|ý=|+y Z|+iy=|íy Z|íy=|+iy , STM: $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % Z (0, R) = (0, R) Z (1, R) = (1, R) Z (R, 0) = (R, 1) Z (R, 1) = (R, 0) Z (R, R) = (R, R ‘ 1) Z (R, R ‘ 1) = (R, R) . (2.55)

Given Figure 2.3, this is what we should expect in order for STM to accurately simulate the Pauli Z gate. To articulate some more similarities between STM and quantum theory it is easily verified that

X2=Y2=Z2= I, where the STM counterpart holds true: X2= Y2= Z2= I. (2.56) There are however well-established examples of where STM somewhat differs to quantum theory, such as the lack of global phase, ´i, in composition ofX andZ:

XZ=´iY, while X Z = Y. (2.57)

Beyond the Pauli gates STM is equipped with theH,S andCN OT gates, corresponding to the quantum gates which generates the Clifford group (see Equation 2.48). TheHandSgates are defined to act on elementary systems according to

H(x, p)”(p, x) and S (x, p)”(x ‘ 1, p ‘ x). (2.58) (See Figure 2.5 for visual representation)

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Meanwhile theCN OT gate acts on a composite system in the following manner:

CN OT (x1, p1)(x0, p0)”(x1, p1‘p0)(x0‘x1, p0). (2.59) These gates often reproduce identities consistent with quantum theory. Some counterexam-ples are however:

HYH=´Y, while HY H = Y and SYS:=´X while S Y S´1= X, (2.60) with the understanding thatH´1 _{= H} _and _S´1₍_{x, p}_{) = (}_{x ‘ 1, p ‘ x ‘ 1}₎_{. In quantum} theory H inverts states along the ˆy axis, while according to Figure 2.5, the states on ˘ ˆy are left untouched byH. Meanwhile, the same figure provides us with the understanding thatS

inverts states along the ˆz axis, which is not true in the quantum case. SinceHandS cannot be seen as rotations on the Bloch sphere they do not correspond to unitary transformations. In fact,HandScorrespond to antiunitary transformations since they require states to be both rotated and mirrored. Spekkens points out that this category of transformations exists in the toy model as a consequence of the discrete nature of the transformations . In contrast, the evolution of a quantum system is assumed to be continuous in time [14]. One of the funda-mental postulates of quantum theory states that the evolution of a closed quantum system is described by a unitary transformation[9], which constitute a major difference between the two theories.

All above-mentioned STM gates are available in QSL and they will be used within both the-ories throughout this report. The main extension that characterizes QSL is the Toffoli gate, defined to perform the map

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I

ˆz

ˆx

ˆy

X

ˆz

ˆx

ˆy

Y

ˆz

ˆx

ˆy

Z

ˆz

ˆx

ˆy

Figure 2.4: Visual representation ofI,X,YandZ.

H

ˆz

ˆx

ˆy

S

ˆz

ˆx

ˆy

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3 Tomography in Spekkens’ Toy

Model

"I do not like it, and I am sorry I ever had anything to do with it"

– ERWIN SCHRÖDINGER about the probability interpretation of quantum mechanics

This chapter aims to resolve the research questions by expanding upon the concepts provided in chapter 2.

3.1 The Possibility of Process Tomography

In quantum theory we have seen that a process is completely characterized by its chi matrix (χ). We concluded that each matrix element (χmn ” eime˚in) defines how the corresponding pair of operators, rEm and rE:n, contributes to a process in its operator-sum representation;

E (ρ) =ř_mn χmnEr_mρ rE:n. The chi matrix is simply a way of representing the decomposition of a process in terms of some complete set of basis operators (such as the Pauli operator basis). This choice of representation is natural in quantum theory, recalling that a quantum operation — definition wise — must have an operator-sum representation (see section 2.1). In STM however, there exist no such thing as an operator-sum representation. We will also see that the transformations available in STM generally cannot be decomposed independently of the input state which further complicates the task of characterizing the differences between the two theories. In what follows we will investigate the possibility of state and process tomography in STM using the available formalism.

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3.1. The Possibility of Process Tomography

The Pauli Operator Basis

We will start our investigation by returning to the second research question (2):

"Is it possible to perform tomography in STM?"

Our initial focus will be on the possibility of process tomography. In quantum theory we have seen that any quantum process can be decomposed in terms of the Pauli operators. The initial motivation for using the Pauli operators was the fact that they collectively constitute a basis for all operators that may act on the quantum system. We now look for an analogous basis in STM, in terms of which we can decompose any STM transformation. A natural set of candidates are of course the STM Pauli operators (Equation 2.54). In what follows we will see that all valid transformations in STM can be described using only these operators, but only if we allow different ontic states to be mapped by different Pauli operators.

To visually aid our reasoning we will use boxes to depict ontic states, similarly to Spekkens [14].

(0, 0) (0, 1) (1, 0) (1, 1)

Figure 3.1: The ontic states of a single-qubit simulation depicted as boxes.

The reversible transformations available for single-qubit simulations in STM corresponds to permutations of these boxes. Since there are four boxes there exist 4!= 24 possible permuta-tions (i.e. 24 reversible transformapermuta-tions). It can be verified that the permutapermuta-tions presented in Figure 3.2 correspond to the set of STM Pauli transformations. Using this figure we now make a key observation; for every transposition of two ontic states in the system there exist a corresponding Pauli operator. To see this, consider the(0, 0)box of each transformation in the figure. Collectively the Pauli operators clearly yield all possible transpositions involving

(0, 0). This can also be seen algebraically:

r Em(0, 0) = $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % (0, 0) if m=1 (1, 0) if m=2 (1, 1) if m=3 (0, 1) if m=4 , (3.1)

where we define t rEmu4_m=1to be the set of four STM Pauli operators labelled in the following order: tI, X, Y, Zu. By analogous observations of(0, 1),(1, 0)and(1, 1)it becomes clear that every separate ontic state can be mapped to any ontic state in the sample space using only these operators, similarly to Equation 3.1.

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Note thatY = X Z = Z X. In this sense we can map all ontic states to all ontic states using onlyI,X andZ. Note also that compositions of elements in t rEmusimply generates elements in t rEmu:

X X = Y Y = Z Z = I, X Y = Y X = Z, Y Z = Z Y = X, X Z = Z X = Y, (3.2) and so forth. Consequently, the permutations that can be achieved by applying compositions of Pauli operators to the entire sample space of a system are the ones presented in Figure 3.2.

(0, 0) (0, 1) (1, 0) (1, 1)

I

:

X

:

(0, 0) (0, 1) (1, 0) (1, 1)

Y

:

Z

:

(0, 0) (0, 1) (1, 0) (1, 1)

Figure 3.2: Permutations corresponding to theI,X,Y andZ gates.

These 4 transformations make up only a subset of all available transformations in the model. In order to express all 24 transformations in terms of the Pauli operators we must therefore allow different points in the sample space to be mapped by different operators. In other words, we must condition the Pauli operators on the ontic states. The 24 transformations available in the model — the ones corresponding to permutations of the ontic states — can then be constructed by conditioning the Pauli operators in such a way that all separate points in the sample space is mapped to a unique point in the same space. We will sometimes refer to these conditions as state-dependency in the context of process tomography.

This result can be generalized to any number of composite systems in STM. To see how, we will consider the transformations available in two-qubit simulations. The composite system will be denoted(x1, p1)ˆ(x0, p0), which yields 16 ontic states in total. A convenient way of visualizing this system is to draw a two-dimensional grid where the four different rows correspond to the set of ontic states(x1, p1)and the four different columns correspond to the different ontic states(x0, p0)(see Figure 3.3).

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3.1. The Possibility of Process Tomography (0, 0) (0, 1) (1, 0) (1, 1) (0, 0) (0, 1) (1, 0) (1, 1)

!

(

x

0

, p

0

)

!

(

x

1

, p

1

)

Figure 3.3: The ontic states of a two-qubit simulation depicted as boxes.

The available transformations now correspond to a subset the possible permutations of rows and columns in Figure 3.3 (there are some permutations that takes valid epistemic states to invalid ones). The Pauli operator basis for this system is generated by all 16 combinations ofI,X, Y and Z, yielding: I I,I X, . . . , Z Y and Z Z. These are not to be confused with sequential application of the operators, but are instead defined to act on the composite system in the following manner:

r

EmEr_n(x₁, p₁)ˆ(x₀, p₀) =Er_m(x₁, p₁)ˆ rE_n(x₀, p₀), where rE_m, rE_nP tI, . . . , Zu. (3.3) Geometrically this means that rEmpermutes the rows while rEn permutes the columns. In the previous example we saw that all possible permutations of columns can be achieved by state-dependent combinations of the Pauli operators. By the same argument we can can conclude that all possible permutations of rows can be described by the same set of transformations. This means that all reversible transformations on the composite system can be explained in terms of the elements in our Pauli basis conditioned on the ontic states, completely analogous to the elementary system case.

To further convince ourselves we can visualize the possible permutations on this system (Fig-ure 3.4). For any two choices of ontic states in this system, there exist a basis element com-posed of Pauli operators such that the chosen states exchange position in the grid. By once again allowing different points in the sample space to be mapped by different basis elements, we can perform any permutation since any ontic state can be mapped to any ontic state. For aesthetical purposes we leave outI in the figure, whileYis left out sinceY = X Z = Z X.

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X

Z

Figure 3.4: The permutations corresponding to tI I, I X, . . . , Z Y, Z Zu.

From symmetry it follows that the possible permutations on a system composed of n ele-mentary systems can be described analogously. Generally, we can associate a Pauli operator basis composed of 4n elements to a such system. The operator basis elements then corre-spond to the 4ndifferent ways of combining n of theI,X,Y andZ operators. To conclude; all transformations that may act on a system composed of n elementary systems can be de-scribed in terms of a Pauli basis composed of 4n elements conditioned on the ontic states. In what follows, we will see some explicit examples of the need for state-dependency when decomposing some common STM transformations.

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3.2. Decomposition ofH,S andCN OT

3.2 Decomposition of

H,

S

and

CN OT

We will now see how some of the most common STM gates can be decomposed. The goal is to find a set of coefficients, temu, such that a transformation U can be expressed on the form

U=ÿ

m

emEr_m, (3.4)

where t rEmuis the Pauli basis in STM. Specifically, if we consider theHgate (Equation 2.58), we must solve H = 4 ÿ m=1 emEr_m where t rE_mu=tI, X, Y, Zu. (3.5) We will first show that this cannot be done state-independently. Combining the fact that

H(x, p)”(p, x)with Equation 2.54, Equation 3.5 must fulfill

H(x, p) =e1(x, p) +e2(x ‘ 1, p) +e3(x ‘ 1, p ‘ 1) +e4(x, p ‘ 1) = (p, x). (3.6) We now assume that the solution is independent of the input state(x, p). Definition wise, the ontic states(0, 0)and(0, 1)must be mapped in accordance with Equation 3.7.

$ & %

H(0, 0) = (0, 0)

H(0, 1) = (1, 0) . (3.7)

For the expansion in Equation 3.6 to accurately reconstructH, the two ontic states must then follow the map

$ & % e1(0, 0) +e2(1, 0) +e3(1, 1) +e4(0, 1) = (0, 0) e1(0, 1) +e2(1, 1) +e3(1, 0) +e4(0, 0) = (1, 0) . (3.8)

According to the first row in Equation 3.8, e1=1 while em‰1=0. On the contrary, the second row requires that e3=1 while e3‰1=0. We have thus reached a contradiction, stemming from the assumption thatHcan be decomposed independently. If we do not enforce state-independency the equations will not have to be simultaneously fulfilled and no contradiction is found. In order for Equation 3.5 to have a solution we must thus allow the em:s to depend

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on the ontic state on which the transformation acts. ApplyingHto all ontic states it is clear that $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % H(0, 0) = (0, 0) H(0, 1) = (1, 0) H(1, 0) = (0, 1) H(1, 1) = (1, 1) . (3.9)

The corresponding permutations can thus be visualized as

(0, 0) (0, 1) (1, 0) (1, 1)

Figure 3.5: The permutations corresponding to theHgate.

Clearly,(0, 1)and(1, 0)are transformed byYwhile(0, 0)and(1, 1)are transformed byI:

(0, 0) (0, 1) (1, 0) (1, 1) Y

I I

Figure 3.6: The permutations in terms of the Pauli operators.

This shows that the permutation corresponding toHcan be formulated in terms of the Pauli basis if we allow different Pauli operators to act on different states. The situation is summa-rized in Table 3.1.

Table 3.1: Decomposition of theHgate. Coefficients Condition Meaning temu=t1, 0, 0, 0u x=p H = I temu=t0, 0, 1, 0u x ‰ p H = Y

Note that(x ‘ p ‘ 1) = 1 for all cases of x = p and(x ‘ p ‘ 1) = 0 when x ‰ p. Likewise

(x ‘ p) = 1 when x ‰ p while (x ‘ p) = 0 when x = p. We thus conclude that H can be completely characterized by the following set of state-dependent coefficients: temu4_m=1 = tx ‘ p ‘ 1, 0, x ‘ p, 0u, which in the form of Equation 3.4 yields

H =

4 ÿ m=1

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Also theSgate can be decomposed in a similar fashion. By definition,Smaps the ontic states in accordance with Equation 3.11.

$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % S (0, 0) = (1, 0) S (0, 1) = (1, 1) S (1, 0) = (0, 1) S (1, 1) = (0, 0) . (3.11)

The behaviour ofScan therefore be summed up as shown in Table 3.2. Table 3.2: Decomposition of theSgate. Coefficients Condition Meaning temu=t0, 1, 0, 0u x =0 S = X temu=t0, 0, 1, 0u x =1 S = Y

In order for the state-dependent coefficients to equate accordingly, the expansion must take the form

S = (x ‘ 1) X + (x) Y, (3.12) meaning that theS is completely characterized by temu =t0, x ‘ 1, x, 0u. Finally we will consider the decomposition of theCN OT gate (Equation 2.59). Since theCN OT gate acts on states composed of two elementary systems, the corresponding Pauli basis consists of 42=16 elements. These will be labelled in accordance with Table 3.3 throughout this report.

Table 3.3: The Pauli operator basis, t rEmu16_m=1, for two-qubit simulations

m 1 2 3 4 5 6 7 8 r Em I I I X I Y I Z X I X X X Y X Z m 9 10 11 12 13 14 15 16 r Em Y I Y X Y Y Y Z Z I Z X Z Y Z Z

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By applying theCN OT gate to all 16 ontic states of the system, we may associate each map with a basis element, as per usual:

Table 3.4: All ontic states mapped by theCN OT gate.

x1 p1 x0 p0 ÞÑ x11 p11 x10 p10 Corresponding basis element

0 0 0 0 0 0 0 0 I I 0 0 0 1 0 1 0 1 Z I 0 0 1 0 0 0 1 0 I I 0 0 1 1 0 1 1 1 Z I 0 1 0 0 0 1 0 0 I I 0 1 0 1 0 0 0 1 Z I 0 1 1 0 0 1 1 0 I I 0 1 1 1 0 0 1 1 Z I 1 0 0 0 1 0 1 0 I X 1 0 0 1 1 1 1 1 Z X 1 0 1 0 1 0 0 0 I X 1 0 1 1 1 1 0 1 Z X 1 1 0 0 1 1 1 0 I X 1 1 0 1 1 0 1 1 Z X 1 1 1 0 1 1 0 0 I X 1 1 1 1 1 0 0 1 Z X

As can be seen in Table 3.4, the basis elements must be conditioned on x1 and p0. More specifically, it can be verified that Table 3.5 fully characterizes the gate.

Table 3.5: Decomposition of theCN OT gate. Coefficients Condition Meaning e1 =1, em‰1 =0 x1=p0‰1 CN OT = I I e2 =1, em‰2 =0 p0‰x1=1 CN OT = I X e13=1, em‰13=0 x1‰p0=1 CN OT = Z I e14=1, em‰14=0 x1=p0=1 CN OT = Z X

Similarly to the previous two examples, we can finally describe the transformation as a sum of conditioned elements in the Pauli basis by letting the set of coefficients equate in accordance with Table 3.5.

CN OT =ř16

m=1emEr_m =

(x1‘p0‘1 ‘ x1p0)I I + (x1‘x1p0)I X + (p0‘x1p0)Z I + (x1p0)Z X.

(3.13)

To summarize; STM does not facilitate a representation of the decomposition of a gate that is comparable to the chi matrix in quantum theory. In this sense we cannot explicitly compare

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ideal quantum gates. However, we just gained some fundamental insight to theH, S and

CN OT gate by fully characterizing these transformations using the available formalism. The set of state-dependent coefficients, temu, is in fact as close as we will get to a chi matrix in STM. To see this, recall that a chi matrix in quantum theory is fully defined by two sets of state-independent coefficients, temuand tenu(Equation 2.30). In order to show that process tomography is possible for all transformations in STM, we must show that all transformations have an analogous decomposition.

Compositions of Transformations

To bring us closer to an answer to research question (2), we will now consider elementary system transformations that are composed of more than one gate. Specifically we will con-sider compositions of theHandS gate; the STM analogues to the generators of the Clifford group for single-qubit systems. To be clear, in this context the term composition simply refers to a sequence of gates, each acting on the system in a specific order. To explore this scenario, we start by defining U to be a transformation composed of two gates which can act on an elementary system(x, p):

U(x, p)”C1C0(x, p), (3.14) where C0, C1P tH, Su. We now apply the gates sequentially:

U(x, p)”C1C0(x, p)”C1(x1, p1)”(x2, p2). (3.15) We have seen that bothHandS are viable transformations in STM and QSL since they cor-respond to permutations of the ontic states of the system (see e.g. Figure 3.5). Clearly, we can therefore think of the effective map(x, p) ÞÑ (x2_{, p}2₎_{as a permutation as well, since} the sequence of two gates must correspond to a sequence of two permutations. By the same argument we can in fact add an arbitrary number ofHandS gates to the composition of U; the effective transformation(x, p)ÞÑ(x1...1_{, p}1...1₎_{will always correspond to a permutation:}

Cn´1. . . C0(x, p) =Cn´1. . . C1(x1, p1) =¨ ¨ ¨= (x1...1, p1...1). (3.16) This result may seem trivial, but the fact that U(x, p) = (x1...1, p1...1)simply corresponds to a permutation on(x, p)is of outmost importance, since it implies that every ontic state in the sample space of the system is mapped to another point in the same sample space. Earlier in this chapter it was argued that these kind of maps can be expressed in terms of a the Pauli basis, where the elements are allowed to be conditioned on the ontic states. This shows that any transformation composed of onlyHandS gates can be expressed as a state-dependent sum of the STM Pauli operators. In the upcoming section we will expand upon this result to show that any reversible transformation in STM can be decomposed analogously. First we

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will however consider an explicit example; U= HS. By definition, this composition of gates yields the map

HS (x, p) = H(x ‘ 1, p ‘ x) = (p ‘ x, x ‘ 1). (3.17) This means that

$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % HS (0, 0) = H(1, 0) = (0, 1) HS (0, 1) = H(1, 1) = (1, 1) HS (1, 0) = H(0, 1) = (1, 0) HS (1, 1) = H(0, 0) = (0, 0) . (3.18)

Visually, Equation 3.18 corresponds to the permutation

(0, 0) (0, 1) (1, 0) (1, 1) Z X

Y

Figure 3.7: The permutation corresponding to U= HSin terms of the Pauli operators. Algebraically we thus get the following decomposition:

U= (x ‘ xp)I + (p ‘ xp)X + (xp)Y + (x ‘ p ‘ 1 ‘ xp)Z. (3.19)

The Generators of All Reversible Gates in STM

As mentioned earlier in this chapter, there exist 4!=24 different transformations that corre-spond to single-qubit transformations in STM and QSL. Spekkens refers to this set of trans-formations as the set of permutations of 4 elements; S4[14]. As pointed out in ref. [6], this set of transformations can be generated using only theHandS gate. More specifically, for every available transformation U in S4, there exist a composition of HandS which generates that transformation:

U=ź

i

Ci, (3.20)

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3.3. Probabilistic Mixtures of Transformations

of the system. This proves that all reversible transformations corresponding to single-qubit transformations can be decomposed accordingly.

This result can be expanded to two-qubit simulations if we add theCN OT gate to the set of gates which generates U. By this addition we complete the list of generators of the Clifford group in STM and QSL, meaning that any simulation corresponding to a reversible two-qubit transformation can be generated using only these three gates [6]. Of course we now wonder; can these transformations be decomposed analogously to the reversible elementary system transformations? The answer to this question is yes, and it is surprisingly easy to con-vince ourselves of this fact. We have already seen that every separate point in the sample space of a composite system can be mapped to any point in the same space using the Pauli basis conditioned on the points. Moreover, theCN OT gate is reversible which means that it can be seen as a permutation of the ontic states. This means that any composition of only theH,S andCN OT gates can be seen a permutation of the ontic states. Since all reversible transformations on two elementary systems can be generated by this type of composition, we conclude that these transformations can be decomposed in terms of the corresponding Pauli basis where the elements are conditioned on the ontic states. Essentially, we arrived at this conclusion by following the same reasoning as in the elementary system case (see Equation 3.14 - Equation 3.16). This result is especially powerful within STM since all of the reversible transformations on n elementary systems can be generated using only these three gates. Recall that QSL extends beyond STM through the Toffoli gate which acts on three ele-mentary systems. Since the Toffoli gate cannot be generated using theH,SandCN OT gates, we can currently only guarantee that process tomography on reversible transformations on one and two systems is possible in QSL.

To conclude, we have seen that the reversible transformations on any number of elementary systems in STM can be decomposed in terms of a corresponding Pauli basis where the el-ements are conditioned on the ontic states of the system. We have also seen that the same result is true in QSL if we only consider transformations of one or two elementary systems.

3.3 Probabilistic Mixtures of Transformations

Finally we will consider probabilistic mixtures of reversible gates. As we will see, this type of transformation generally maps pure states to mixed ones, meaning that this section mainly is useful within the QSL framework. In QSL we can generate probabilistic mixtures of trans-formations as convex sums of valid transtrans-formations where each coefficient is associated with the probability of the corresponding transformation:

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We have already shown that all reversible transformations can be decomposed in terms of the Pauli basis so each Ciin Equation 3.21 can be rewritten by using that Ci=řmeimEr_m:

U= p1ÿ m e1mEr_m+¨ ¨ ¨+p_n ÿ m enmEr_m= ÿ i, m pieimEr_m. (3.22)

This shows that U can be expressed as a convex combination of state-dependent Pauli basis elements. The transformation is depicted in Figure 3.8.

(x, p)(

p1(x1, p1)

(

pn(x1...1, p1...1)(

Figure 3.8: Schematic view of a transformation composed of a probabilistic mixture of re-versible transformations.

We can clearly see that U maps pure states to mixed states. As a consequence, this type of trans-formation is irreversible. STM does not allow for convex combinations of epistemic states with unequal weight [14], so only a small subset of transformations on the form of Equation 3.21 is available in the model. In section 3.4 we will further discuss the different requirements on mixed states in STM which ultimately limits the number of valid mixed transformations in the model.

In what follows we will consider an explicit example of how an irreversible transformation may be decomposed. The example turns out to be valid in QSL but not in STM due to the nature of the allowed mixed states in the two models:

U=p1H +p2S, where ÿ

i

pi =1 and 0 ď p1, p2ď1. (3.23) As per usual, we expose all ontic states in the sample space of the system to U:

$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ U(0, 0) =p1(0, 0) +p2(1, 0) U(0, 1) =p1(1, 0) +p2(1, 1) U(1, 0) =p1(0, 1) +p2(0, 1) . (3.24)

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From Equation 3.24 it is quite straight forward to associate the correct mixture of Pauli oper-ator with the respective ontic state. As a result, we get Table 3.6.

Table 3.6: Maps of all ontic states when U=p1H +p2S. Condition Meaning

(x, p) = (0, 0) U= p1I +p2X

(x, p) = (0, 1) U= p1Y +p2X

(x, p) = (1, 0) U= p1Y +p2Y

(x, p) = (1, 1) U= p1I +p2Y

Algebraically we can thus express U by making sure that the state-dependent coefficients equates accordingly: U= (x ‘ p ‘ 1 ‘ xp)hp1I +p2X i + (p ‘ xp)hp1Y +p2X i + (x ‘ xp)hp1Y +p2Y i + (xp)hp1I +p2Y i , (3.25)

which completes the decomposition. However, we may choose to rewrite Equation 3.25 to get a more familiar form. We start by factoring out the probabilities and Pauli operators. Equation 3.25 then takes the form

U= p1 h (x ‘ p ‘ 1 ‘ xp+xp)I + (p ‘ xp+x ‘ xp)Yi+ p2 h (x ‘ p ‘ 1 ‘ xp+p ‘ xp)X + (x ‘ xp+xp)Yi. (3.26)

Note that the state-dependent coefficient can be rewritten since

$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % (x ‘ p ‘ 1 ‘ xp+xp) = (x ‘ p ‘ 1) (p ‘ xp+x ‘ xp) = (x ‘ p) (x ‘ p ‘ 1 ‘ xp+p ‘ xp) = (x ‘ 1) (x ‘ xp+xp) = (x) , (3.27)

for all x, p P t0, 1u. Inserting these coefficients to Equation 3.26 we recover the decomposed form ofHandSas separate terms in the expansions (review Equation 3.10 and Equation 3.12 to see this): U= p1 h (x ‘ p ‘ 1)I + (x ‘ p)Yi+p2 h (x ‘ 1)X + (x)Yi=p1H +p2S (3.28) To summarize this chapter so far, we may conclude that process tomography is possible in STM, which partially answers research question (2). To fully answer the question, we also

State and Process Tomography : In Spekkens&apos; Toy Model

Linköping University | Department of Electrical Engineering

Master’s thesis, 30 ECTS | Applied Physics

2020 | LiTH-ISY-EX--20/5273--SE

State and Process Tomography

In Spekkens’ Toy Model

Andreas Andersson

Upphovsrätt

Copyright

Acknowledgments

Contents

List of Figures

List of Tables

1

Introduction

1.1

Motivation

1.2

Aim

1.3

Research Questions

1.4

Delimitations

2

Preliminaries

2.1

Quantum Theory

Mathematical Formalism

Quantum States

Quantum State Tomography

ÿ

Quantum Operations

Quantum Process Tomography

ρ

Quantum Black Box

E (

ρ

) =

ρ

2.2

Spekkens’ Toy Model and Quantum Simulation Logic

States

Composition of Elementary States

Transformations in STM

I

ˆz

ˆx

ˆy

X

ˆz

ˆx

ˆy

Y

ˆz

ˆx

ˆy

Z

ˆz

ˆx

ˆy

H

ˆz

ˆx

ˆy

S

ˆz

ˆx

ˆy

3

Tomography in Spekkens’ Toy

Model

3.1

The Possibility of Process Tomography

The Pauli Operator Basis

I

:

X

:

Y

:

State and Process Tomography : In Spekkens' Toy Model

_{Quantum Black Box}

_ρ