New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels

by Jeremy Levick

A Thesis presented to The University of Guelph

In partial fulfilment of requirements for the degree of Doctor of Philosophy in Mathematics

Guelph, Ontario, Canada c Jeremy Levick, June, 2015 ABSTRACT

New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels

Jeremy Levick Advisor: University of Guelph, 2015 Dr. Rajesh Pereira

The Perfect-Mirsky conjecture states that the set of all possible eigenvalues of an n by n doubly stochastic matrix is the union of certain specific n − 1 regular polygons in the complex plane. It is known to be true for n = 1, 2, 3 and recently a counterexample was found for n = 5. We show that the conjecture is true for n = 4. We also modify the original conjecture for n > 4 in order to accommodate the n=5 counterexample, as well as provide evidence for our new conjecture. An important class of quantum channels are the Pauli channels; channels whose Kraus operators are all tensor products of Pauli matrices. A private quantum channel is a quantum channel that is non-injective for some set of inputs. Private quantum channels generalize the classical notion of the one-time pad to the quantum setting. PQCs are most useful when the set of inputs they privatize is isomorphic as an algebra to an algebra of qubits. We use the theory of conditional expectations to show that in certain cases, the set of algebras privatized by a channel is the set of algebras quasiorthogonal to the range of the channel. Quasiorthogonality is an idea from operator algebras that has begun to find applications elsewhere in quantum information. Using these ideas, we prove that conditional expectation

log(n) Pauli channels with commuting Kraus operators can always privatize 2 qubits, where n is the number of Kraus operators. We also use ideas from Fourier analysis on finite groups to give a method for finding privatized algebras general Pauli channels. Using character theory, we extend this analysis beyond Pauli channels, and show how to find private algebras for qutrit and other channels. Finally, using ideas from Lie theory, we show a connection between quasiorthogonality and the Cartan decomposition of a Lie algebra, giving new methods for finding algebras privatized by a channel. We also use the KAK decomposition of a Lie algebra to begin looking at ways to encode algebras so they are privatized. Dedication

This thesis is dedicated to my parents and grandparents, particularly the memory of my grandfather Hillel Katzman. Without their love and guidance and support throughout my life, this thesis would not have been possible.

iv Acknowledgements

I would like to thank my advisor Rajesh Pereira, for his invaluable help and support over the years. It has been wonderful having him as an advisor, a mentor, and a role model. I would also like to thank my co-advisor David Kribs for his support, as well as Pal Fischer, and John Holbrook, and the members of my committee: Bei Zheng, Alan Willms, and especially Chi-Kwong Li, who have also contributed to thesis, and whose suggestions have helped to improve it. Susan McCormick and Carrie Tanti in the Mathematics and Statistics office have always been there to help me navigate administrative matters, and have only ever been extremely helpful. My fellow graduate students made working on this thesis not just tolerable, but fun. My officemates Sarah Plosker and Tyler Jackson, as well as Andrew Skelton, Keith Poore, and especially Bryce Morsky have been wonderful to work with, and great friends.

v Contents

List of Figures viii

1 Introduction 1 1.1 Outline ...... 1 1.2 Group Theory ...... 2 1.2.1 Representations ...... 4 1.3 Lie Groups and Lie Algebras ...... 8

2 The Four Dimensional Perfect-Mirsky Problem 10 2.1 Positive Matrices ...... 10 2.2 Stochastic and Doubly Stochastic Matrices ...... 11 2.2.1 Doubly Stochastic Matrices and Majorization ...... 12 2.2.2 Multivariate and Directional Majorization ...... 13 2.3 The Inverse Eigenvalue Problem for Non-Negative Matrices ...... 15

2.3.1 Karpelevich and the Region Kn ...... 16 2.4 The Perfect-Mirsky Conjecture ...... 18 2.4.1 The Rivard-Mashreghi Counterexample ...... 20 2.4.2 The Four Dimensional Perfect-Mirsky Conjecture ...... 21

3 Private Quantum Channels and Matrix Conditional Expectations 27 3.1 Quantum Information ...... 27 3.1.1 Quantum States, Qubits, and Density Matrices ...... 27 3.2 Quantum Channels ...... 29 3.2.1 Completely Positive Trace-Preserving Maps ...... 29 3.3 Private Quantum Channels ...... 31 3.4 Algebras and Conditional Expectations ...... 34 3.5 Quasiorthogonality ...... 36

vi 3.5.1 Group Theory of Pauli Subalgebras ...... 38 3.6 Private Pauli Channels ...... 44 3.7 Conditional Expectations onto Operators Spanned by Group Elements . . . 48

4 Group Theory, Fourier Analysis, and Lie Theory and Private Quantum Channels 52 4.1 Fourier Analysis and Private Quantum Channels ...... 52 4.2 Lie Theory and Private Pauli Channels ...... 57 4.3 Global Cartan and KAK Decompositions ...... 60 4.4 Cartan Decompositions of su(2n) and Quasiorthogonal Algebras of Paulis . . 61 4.4.1 Entanglers and Encoding Private Subalgebras ...... 63

5 Future Work and Conclusions 68 5.1 Perfect-Mirsky ...... 68 5.2 Private Quantum Channels ...... 69

Bibliography 69

vii List of Figures

2.1 The region K4 of all eigenvalues of 4 × 4 stochastic matrices ...... 19 S5 S6 2.2 The Rivard-Mashreghi point is not in k=1 Πk, but is in k=1 Πk ...... 21

2.3 Close-up of the Rivard-Mashreghi point in K4 ...... 25 2.4 The associated eigenvector is convexly dependent ...... 26

viii Chapter 1

Introduction

1.1 Outline

This thesis is concerned with two different problems, the first from matrix analysis, the sec- ond from the intersection of matrix analysis with quantum information theory. The first problem is that of characterizing the region of all possible eigenvalues of all possible n × n doubly stochastic matrices. A famous conjecture on the matter, the Perfect-Mirsky conjec- ture, is known to be true for n ≤ 3, and false for n = 5. The first part of the thesis proves that the Perfect-Mirsky conjecture is true for n = 4, and formulates a new conjecture to account for the falsity of the Perfect-Mirsky conjecture in the case n = 5. We also discuss some lines of reasoning that support our new conjecture. The second part of the thesis concerns the theory of private quantum channels in quantum information theory. As implied by the name, a private quantum channel is a quantum chan- nel which privatizes some inputs. That is, the channel fails to be injective for certain inputs, and therefore an observer cannot distinguish all possible inputs based only on their out- put. In particular, we discuss the connection between private quantum channels and matrix subalgebras, showing conditions under which subalgebras isomorphic to qubit algebras can be privatized. This section relies on a connection between private channels and conditional expectations, and we also make use of the theory of Fourier analysis on certain finite Abelian groups arising from these conditional expectations. We also use Lie theory to investigate the problem of encoding and decoding qubit algebras to be private for certain channels. This introduction contains some of the necessary mathematical background, particularly the group theory and Lie theory necessary for Chapters 3 and 4. In Chapter 2, we discuss the Perfect-Mirsky conjecture. First we introduce some important

1 ideas from the theory of positive matrices, and discuss the related problem of finding the region of all eigenvalues of all n × n stochastic matrices. We discuss important similari- ties between the stochastic and doubly stochastic cases; we also discuss the counterexample disproving the Perfect-Mirsky conjecture in n = 5, and the relationship between doubly stochastic matrices and majorization. Finally, we combine these ideas to obtain a proof of the Perfect-Mirsky conjecture for n = 4, and we end with a new conjecture to replace the Perfect-Mirsky conjecture, and provide some evidence and heuristic reasoning explaining why this new conjecture might be true. Chapter 3 discusses private quantum channels. After a brief review of the important ideas from quantum information, we explain what a private quantum channel is, and give some examples. We explain the connection between private quantum channels and conditional expectations onto matrix subalgebras, and explain how privacy is related to the operator- algebraic idea of quasiorthogonality. We use these ideas to prove limits on the number of qubits privatized by certain Pauli channels, and show explicit encodings achieving these bounds. Lastly, we look at how group-theoretic properties of the Pauli group play a role in the preceding analysis. We proceed to look further into the relationship between group theory and conditional ex- pectations and privacy. We show how representations of groups with certain properties give rise to conditional expectations, and their associated private subalgebras. We also use tools from the theory of bicharacters, and Fourier analysis on finite Abelian groups to show how to find private subalgebras for channels arising from these groups. Finally in Chapter 4 we turn to the connection between Lie theory and private quantum channels, showing how Cartan decompositions of a Lie algebra relate to quasiorthogonality. We show how the KAK decomposition from Lie theory can be used to encode and decode qubit algebras so as to be private for certain quantum channels.

1.2 Group Theory

We develop some of the group theory necessary for the later sections of this thesis. We will usually write the group operation multiplicatively, so g h = gh. Recall that two elements a, b of a group commute if ab = ba. If all the members of a group commute, we say the group is commutative or Abelian.

Definition 1.2.1. Let G be a group. The center of G, Z(G), is the set of elements of G

2 that commute with everything in G:

Z(G) = {g ∈ G : gh = hg, for all h ∈ G}.

Definition 1.2.2. Let G be a group with subset H. The centralizer of H in G, written

CG(H), is the set of elements of G that commute with everything in H:

CG(H) = {g ∈ G : gh = hg for all h ∈ H}.

Remark 1.2.3. If H is a normal subgroup, then so is CG(H). The center of G is always normal. The quotient G/Z(G) called the central quotient of G.

If g, h ∈ G, we can test how much g and h fail to commute by means of the commutator:

[g, h] = ghg−1h−1.

If h ∈ CG(< g >), then [g, h] = e. If g, h do not commute, then [g, h] is some non-identity element of G. Some simple identities for the commutator that will be useful later are the following:

1. [g, h] = [h, g]−1

2. gh = [g, h] hg

Definition 1.2.4. For a group G, the commutator of G, denoted [G, G], is the set of com- mutators between all elements of G:

[G, G] = {[g, h]: for all g, h ∈ G}.

Remark 1.2.5. The commutator of G is not always a subgroup of G. The group generated by [G, G] is called the commutator subgroup or the derived subgroup. We will usually denote it < [G, G] >. Sometimes if the set of commutators is identical to the commutator subgroup, we will just write [G, G] The commutator subgroup is always normal, and the quotient G(0) = G/ < [G, G] > is always Abelian, which can be seen easily by noting that the quotient of the normal homo- morphism from G to G(0) has in its kernel all elements of the form ghg−1h−1. If ν is the

3 natural homomorphism, then ν(ghg−1h−1) = e is equivalent to the condition ν(gh) = ν(hg). The quotient of G by its commutator subgroup is for this reason called the Abelianization of G. Any subgroup of G that contains the commutator subgroup must also have an Abelian quo- tient with G.

1.2.1 Representations

A group G is often abstract and hard to get a handle on; it is generally useful to get a handle on a group by finding some concrete way to represent the elements of a group. The most useful way to do this is to find groups of matrices whose multiplication table preserves features of the multiplication of G. This motivates the idea of group representations:

Definition 1.2.6. Let G be a group. A representation of G is a group homomorphism ρ : G → GL(n) into the group of n × n invertible matrices. The number n is called the dimension of the representation.

Definition 1.2.7. Let ρ : G → GL(n) be a representation of some group G. If there exists a subspace of V ⊆ Cn such that ρ(g)v ∈ V for all g ∈ G, v ∈ V , then the representation is said to reducible.

The idea behind reducibility is that the action of ρ(G) on Cn is entirely contained inside some subspace, and thus by restricting to the vector space V , we lose none of the features that make ρ a representation. For this reason, we are most interested in the irreducible representations of G; those representations that are not reducible.

Definition 1.2.8. Let G be a group with representation ρ : G → GL(n). For all g in G the linear function Tr(g) is called an irreducible character

Proposition 1.2.9. (Schur’s Lemma) Let G be a group and ρ an n-dimensional irreducible representation of G. Let Z(G) be the center of G. Then ρ(Z(G)) ⊆ {cIn : c ∈ C}.

As is usual in mathematics, it is useful to consider the structure-preserving maps from a group to C. In this case, we seek to understand the group homomorphisms χ : G → C. If G is an Abelian group, then Z(G) = G. Thus, Schur’s lemma asserts that any irreducible representation of an Abelian group G must be represented by constant multiples of the identity. Of course, such a representation is irreducible if and only if it is one-dimensional.

4 Hence, all irreducible representations of Abelian groups are simply homomorphisms from G → C.

Definition 1.2.10. Let G be an Abelian group. The set of homomorphisms χ : G → C is called the set of characters of G. We denote this set Gˆ.

The terminology is consistent with that of irreducible characters, since a one-dimensional representation takes values in C; thus for such a representation Tr(ρ(g)) = ρ(g). Character theory is an old and well-developed area of group theory. We will sketch some of the important results that we will make use of later.

Proposition 1.2.11. The characters from G to C form a group under pointwise multiplica- tion. We call the group Gˆ the dual group of G.

This is easy to see by considering two characters χ1, χ2. Their pointwise product must

take G 3 e → 1 ∈ C. Since each χi does this, their product must as well.

Multiplicativity follows also from the multiplicativity of each of χ1, χ2.

Proposition 1.2.12. There is one distinct character of G for each conjugacy class in G.

Corollary 1.2.13. If G is Abelian, then |Gˆ| = |G|.

This follows from the fact that each element of an Abelian group is self-conjugate. Even more is true in certain cases:

Proposition 1.2.14. If G is a locally compact Abelian group, then Gˆ is isomorphic as a group to G.

Remark 1.2.15. Although the proposition above asserts an isomorphism between G and its dual, this isomorphism is not canonical; there is usually more than one natural isomorphism between the two groups. ˆ There is, however, a natural isomorphism between G and Gˆ, the double-dual of G, when ˆ G is locally compact and Abelian. The natural isomorphism comes from considering g in Gˆ to act on χ ∈ Gˆ by evaluating χ at some element g ∈ G:

g(χ) = χ(g).

5 The relationship between G and its dual is the start of Fourier analysis. Let f : G → C be a function from G to C. The set of such functions CG naturally forms a vector space under pointwise addition. The Fourier transform of such a function is a function fˆ : Gˆ → C. The Fourier transform of f is defined by the equation X fˆ(χ) = χ(g)f(g). (1.2.1) g∈G

Definition 1.2.16. Let G be a locally compact Abelian group. The group ring of G, C[G], is the set of linear combinations of elements of G with complex coefficients, where addition and multiplication are defined by

X X X cgg + dgg = (cg + dg)g (1.2.2) g∈G g,h∈G g∈G X X X X ( cgg)( dgg) = ( cgdh−1g)h (1.2.3) g∈G g∈G h∈G g∈G

Representations are also a convenient setting to understand the concept of the group ring. P Let x ∈ C[G], x = g∈G xgg. Multiplication by x on the left yields a map from the C[G] into itself:

Lx(a) = xa. The elements of G form a basis for C[G], so we will find a matrix representation for x in this basis.

X Lx(h) = xggh g∈G X = xgh−1 g. g∈G

Thus, we may form a matrix X whose rows and columns are indexed by elements of G, such

that Xg,h = Xgh−1 .

The vector space generated by the matrices Xg for g ∈ G can be viewed as the group ring of G, under the natural group multiplication extended by linearity. If G is locally compact and Abelian, the group ring is simultaneously diagonalizable. The matrix achieving the diagonalization is none other than the character table of G, with rows

6 and columns ordered to match the ordering of Xg. P To see this, pick as a different basis for C[G] the elements g∈G χ(g)g for each character ˆ χ ∈ G; this basis is an eigenbasis for the action of Lx. Note that this essentially amounts to the claim that, in the Abelian case, the group ring is the direct sum of the (one-dimensional) irreducible representations of G. This is in fact true in general.

Proposition 1.2.17. Let G be a group, not necessarily Abelian. We may define the group ring C[G] as above, and represent each g ∈ G by means of its action by left-multiplication on C[G]. Then C[G] is a direct sum of the irreducible representations of G. P Corollary 1.2.18. An element x = g∈G cgg ∈ C[G] is invertible if and only if ρ(x) = P g∈G cgρ(g) is invertible for each irreducible representation ρ of G.

Example 1.2.19. Let G = Z/nZ under addition, the cyclic group of order n. The characters 2kπi on G are easily seen to be the functions χi such that χk(1) = e n . The value of χ on other members of G follows from the fact that χ is a homomorphism. The group ring of the cyclic group of order n is the set of circulant matrices:

  c1 c2 ··· cn−1 cn   cn c1 ··· cn−2 cn−1 A =  . .  . (1.2.4)  . .. .   . ··· . .  c2 c3 ··· cn c1

The character table is the matrix

(i−1)(j−1) (F )ij = ω (1.2.5)

with ω a primitive nth root of unity. The matrix F is called a Fourier matrix. For any circulant matrix C, FCF ∗ is diagonal.

Definition 1.2.20. Let H be a subgroup of G. The annihilator of H, Hann is the subgroup of Gb defined by ˆ Hann = {χ ∈ G : χ(h) = 1 ∀ h ∈ H}.

For any finite Abelian group G, and any subgroup H, it is well known that |Hann||H| = |G| [26].

7 1.3 Lie Groups and Lie Algebras

Lie groups are another well-studied structure in mathematics. We will be less concerned with Lie groups than with their associated Lie algebra, so we will sketch some of the important ideas in the theory of Lie groups in order to motivate the Lie algebras we will use later. A Lie group is a group that also has the structure of a differentiable manifold. To avoid the details required to make this definition precise, we will say that intuitively, a Lie group is a “smooth, continuous group”, such that the multiplication function from G × G → G is a continuous function. An easy and motivating example is the group GL(n) of n × n (real or complex) invertible matrices. For complete definitions, see the introduction of [23], which is a good source for more information on the relationship described below between Lie groups and their corresponding Lie algebras. Other important examples include the n × n orthogonal matrices, denoted O(n); the restric- tion of O(n) to its determinant-1 subgroup SO(n), called the special orthogonal group; and their complex counterparts U(n) of unitary matrices and SU(n) (the special unitary group) of unitary matrices with determinant 1. Because a Lie group G has a continuous structure, we can try and understand it locally. In particular, we can look at the tangent space around the identity I. The tangent space around the identity of a Lie group has the structure of a Lie algebra:

Definition 1.3.1. A set g with a bilinear operation [·, ·]: g × g → g is called a Lie algebra if the bilinear map, called the Lie bracket, obeys the conditions:

1. The Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

2. Alternativity: [x, x] = 0

Any Lie algebra is related to a Lie group by means of the exponential map: e : g → G, given by exponentiation the elements of g. In the case of a Lie algebra of matrices, this is simply X P∞ Xi the exponential e = i=0 i! .

Example 1.3.2. The Lie algebra corresponding to the Lie group GL(n) is gl(n), which as a set is simply the n × n (real or complex) matrices.

Example 1.3.3. The Lie algebra su(n) for the Lie group SU(n) is the algebra of skew- Hermitian matrices with zero trace. We will see that this is an important case in quantum information, especially when n = 2k.

8 Any matrix algebra under the matrix commutator [A, B] = AB − BA is a Lie algebra.

Given a basis {Xi}i∈I for a Lie algebra g, we can take the Lie bracket of basis elements and express the result in the basis: X k [Xi,Xj] = cijXk. (1.3.1) k∈I

k The cij are called the structure constants of the Lie algebra. Using the structure constants we may form a bilinear form for the Lie algebra:

k Definition 1.3.4. Let g be a Lie algebra with basis {Xi}i∈I and structure constants {cij}i,j,k∈I . The Killing form is a bilinear form defined by

X l k B(Xi,Xj) = cikclj. (1.3.2) k,l∈I

The Killing form is important in the classification of Lie algebras.

Definition 1.3.5. Let g be a Lie algebra. Let I ⊆ g be a subspace such that [I, g] ⊆ I. Then I is said to be an ideal of the Lie algebra.

Let I and J be two ideals of the Lie algebra g with intersection 0. Then B(I,J) = 0. The vanishing of the Killing form will turn out to be important to us later. We will also mention another interpretation of the Killing form that we will use later, based on the idea of representations of Lie algebras. As with groups, we often wish to find a set of matrices that is isomorphic under the matrix commutator as a Lie algebra to a given Lie algebra g. In the case that we can find a Lie algebra homomorphism ρ : g → Mn(C), we say that ρ is a representation of g. One important representation of a Lie algebra is the adjoint representation, Ad : g → Mn(C), given by AdX (Y ) = [X,Y ]. The Killing form can be re-expressed as

B(X,Y ) = Tr(AdX AdY ). (1.3.3)

9 Chapter 2

The Four Dimensional Perfect-Mirsky Problem

2.1 Positive Matrices

The Perron-Frobenius theorem is an important result in matrix analysis with many inter- esting applications. In this section, we only sketch the basics, introducing deeper ideas as needed later. Perhaps the most naive way of defining matrix positivity is to say that a matrix is positive

if all of its entries are positive real numbers. If A ∈ Mn(R), with aij > (≥)0 for all i, j ∈ [n] we write A > (≥)0. The Perron-Frobenius theorem is an important theorem about such matrices:

Theorem 2.1.1. Let A ∈ Mn such that A > 0. Then the following all hold:

1. A has a positive real eigenvalue λ1 such that |λ1| > |λk| for all other eigenvalues λk of A

2. The eigenvector corresponding to A can be chosen to be all positive

If A ≥ 0 then instead the following hold:

1. A has a positive real eigenvalue λ1 such that |λ1| ≥ |λk| for all other eigenvalues λk of A.

2. The eigenvector corresponding to λ1 can be chosen to have only non-negative entries.

10 In fact, the theorem can be sharpened considerably, by noting that the stricter conditions are true for a class of entry-wise non-negative matrices called irreducible.

Definition 2.1.2. A matrix A is said to be reducible if there exists a permutation P such that P T AP is block upper triangular.

If a matrix cannot be put in such a form, it is said to be irreducible.

2.2 Stochastic and Doubly Stochastic Matrices

The Perfect-Mirsky conjecture is a conjecture about a certain class of matrices: the doubly stochastic matrices. The stochastic matrices are two simple types of positive matrix, with many interesting properties.

Definition 2.2.1. A matrix A ∈ Mn(R) is row (column) stochastic if the sum across each row (column) of these entries is 1:

n n X X A ≥ 0, Aij = 1 ∀i;( Aij = 1 ∀i). j=1 i=1

row col We denote the set of n × n row (column) stochastic matrices by Ωn (Ωn ).

Definition 2.2.2. A matrix A ∈ Mn(R) is doubly stochastic if it is both row and column stochastic: n n X X A ≥ 0, Aij = Aij = 1 ∀i, j i=1 j=1

We denote the set of all n × n doubly stochastic matrices Ωn. A few simple facts about these matrices will be useful to us. First of all, note that the vector e = (1, 1, ··· , 1)T is a right eigenvector for all row and doubly stochastic matrices, and eT is a left eigenvector for all column and doubly stochastic matrices. All stochastic matrices (row, column, and doubly) satisfy the conditions of the Perron-Frobenius theorem; it is clear that the eigenspace associated with e is the eigenspace of the largest modulus eigenvalue, hence all stochastic matrices satisfy |λ| ≤ 1 for all eigen- values λ.

11 Lemma 2.2.3. All eigenvectors v associated to an eigenvalue λv 6= 1 for a column stochastic P matrix must satisfy i vi = 0. Proof. For each row, the eigenvalue equation yields

n X Aijvj = λvvi. j=1

Summing these equations for each i and using the fact that A is column stochastic we find

n n n X X X Aijvj = λv vi i=1 j=1 i=1 n n X X vj = λv vi. j=1 i=1

Pn Hence if λ 6= 1, i=1 vi = 0. In particular, all doubly stochastic matrices have this feature as well.

2.2.1 Doubly Stochastic Matrices and Majorization

One reason for the importance of doubly stochastic matrices in particular is their relationship with the notion of majorization.

Definition 2.2.4. Let α, β ∈ Rn, α, β ≥ 0. We say that α is majorized by β (equivalently, β majorizes α), denoted α ≺ β if the following holds:

k k X ↓ X ↓ αi ≤ βi ∀ k : 1 ≤ k ≤ n (2.2.1) i=1 i=1 n n X X αi = βi. (2.2.2) i=1 i=1

The notation α↓ means the entries of α have been rearranged to be in descending order. Thus, the biggest entry of β is larger than the biggest element of α, the sum of the first two largest entries in β is greater than the sum of the two largest entries in α, and so forth.

n The majorization relation on pairs of vectors in R≥0 with the same element sum is a partial order, and this partial order has many uses. Intuitively, it corresponds to measuring

12 which of α or β is more “spread out”, or “evenly distributed”. For example, in economics, the majorization order is closely linked to the Gini coefficient, a measure of how equal the income distribution is in a country. Usually, we consider the majorization order on probability vectors, whose sum is 1; we are always free to normalize our vectors if this is not the case. These remarks should make it intuitively plausible that the next three statements are true:

1 1 n Remark 2.2.5. The vector n e = n (1, 1, ··· , 1) ∈ R≥0 is the least element in the majorization order.

n Remark 2.2.6. Any vector ei = (0, ··· , 0, 1, 0, ··· 0) ∈ R≥0 is a greatest element in the majorization order.

n Remark 2.2.7. Let α ∈ R≥0, and let σ ∈ Sn be a permutation from the symmetric group on n symbols. Then α and ασ = (ασ(1), ασ(2), ··· , ασ(n)) are equal in the majorization order. Note that this means that the greatest element in the ≺ ordering are the extreme points of the n-probability simplex, and the least element is the centroid of the n-probability simplex, and that the ordering respects permutations of the vertices of the simplex.

From this we see that α ≺ β if and only if α ∈ Conv{βσ : σ ∈ Sn}. The relationship between doubly stochastic matrices and majorization follows from this fact and the next theorem.

Theorem 2.2.8. (Birkhoff’s Theorem) [5] The set Dn of doubly stochastic matrices of size n × n is the convex hull of the n × n permutation matrices.

Combining Birkhoff’s theorem with the other results mentioned so far, we obtain

n Pn Pn Theorem 2.2.9. (Hardy, Littlewood, Polya) [16] Let α, β ∈ R≥0, i=1 αi = i=1 βi = 1. Then the following are equivalent:

1. α ≺ β

2. α ∈ Conv{βσ : σ ∈ Sn}

3. α = Dβ for some doubly stochastic D ∈ Dn

2.2.2 Multivariate and Directional Majorization

The notion of majorization can be extended from a partial order on vectors to a partial order on matrices. The equivalent conditions that characterize majorization in the single

13 vector case are no longer equivalent in the matrix case, so we have different ways of defining majorization between matrices. Most of the material in this chapter can be found in the standard reference of Marshall and Olkin [28]. The most obvious way of generalizing majorization, based on the doubly stochastic matrix characterization of majorization, is the following:

Definition 2.2.10. Let A, B ∈ Mn×m(R≥0). We say A is multivariate majorized by B (or

B multivariate majorizes A), denoted A ≺m B if there exists a doubly stochastic matrix

D ∈ Mn(R) such that A = BD.

Another possible definition of majorization in the higher dimensional case is

Definition 2.2.11. Let A, B ∈ Mn×m(R). We say A is directionally majorized by B (or B T T n directionally majorizes A), denoted A ≺d B if α A ≺ α B for all α ∈ R .

An equivalent characterization of directional majorization is from the economics litera- ture, due to Koshevoy [24]. To explain it, we first need to define the Lorenz zonotope of a matrix:

n Definition 2.2.12. Let xi ∈ R . Denote by x¯i the vector xi with a 1 appended as its  ···  T   first component: xi = (1, xi1, ··· , xin) . Let X = x¯1 x¯2 ··· x¯k, where the xi are all ··· members of Rn. Then the Lorenz zonotope of X, denoted LZ(X) is the convex hull of the columns of X: k X LZ(X) = Conv{ tix¯i|ti ∈ [0, 1]} i=1 Using this notion, we define Koshevoy’s equivalent characterization of directional ma- jorization.

Remark 2.2.13. Let X and Y be matrices as above; then X ≺d Y if and only if LZ(X) ⊂ LZ(Y ). Koshevoy proved that Lorenz majorization (as he called it) and directional majorization are equivalent [24].

We also introduce an equivalent characterization of directional majorization due to Mala- mud:

14 k k n Remark 2.2.14. [27] Let X = {xi}i=1,Y = {yj}j=1, xi, yj ∈ R . We say that Y majorizes

X, denoted X ≺d Y if the following conditions hold:

Conv{xi1 : i1 ∈ {1, ··· , k}} ⊂ Conv{yj1 : j1 ∈ {1, ··· , k}}

Conv{xi1 + xi2, : i1, i2 ∈ {1, ··· , k}} ⊂ Conv{yj1 + yj2 : j1, j2 ∈ {1, ··· , k}} . .

Conv{xi1 + xi2 + ··· + xil : 1 ≤ i1 ≤ i2 · · · ≤ il ≤ k} ⊂ Conv{yi1 + ··· + yil : 1 ≤ i1 ≤ · · · ≤ il ≤ k} . . k k X X xi = yij . i=1 j=1

Following [15], we call this majorization Malamud majorization. It is clear that the relation

X ≺d Y is exactly the condition that LZ(X) ⊂ LZ(Y ), and so Malamud majorization is exactly Lorenz majorization. Koshevoy’s result establishing the equivalence of Lorenz majorization and directional majorization also establishes the equivalence of Malamud ma- jorization and directional majorization. T T T Note that A ≺m B implies A ≺d B, since α A = α DB ≺ α B. In general, directional majorization does not imply multivariate majorization, and the two notions are distinct. However, a result of Bhandari gives a condition for when the two definitions are equivalent, that will be useful to us: Theorem 2.2.15. Let X and Y be 2 × n matrices. If none of the columns of Y is in the

convex hull of any of the others, then X ≺d Y implies X ≺ Y . Proof. See [4] for details

This result will turn out to be very useful for our purposes. In particular, it applies to vectors v ∈ Cn of complex numbers, if we regard such a vector as a member of Rn×2, under the identification Re(vi) = a1i, Im(vi) = a2i.

2.3 The Inverse Eigenvalue Problem for Non-Negative Matrices

There are two distinct questions regarding eigenvalues for a set of matrices. The first is the inverse eigenvalue problem: given a set of n × n matrices X, for which v ∈ Cn does there

15 exist an A ∈ X such that the eigenvalues of A are the components of v. The second problem is that of identifying which complex numbers x ∈ C can appear as the eigenvalue for some A ∈ X. In this section we describe a connection between the inverse eigenvalue problem for non-negative matrices, and the inverse eigenvalue problem for stochastic matrices. We follow the approach of Johnson [18]. Let A be an n × n matrix with non-negative entries. If A is not irreducible, then A is

permutationally equivalent to a block lower-triangular matrix, whose diagonal blocks Ai are irreducible, the irreducible components of A:

  A1 ······ 0    ∗ A2 ··· 0    T  ∗ ∗ · · · 0  P AP =    . . .   . ··· .. .    ∗ ∗ · · · An

Clearly, the spectrum of A is the union of the spectra of Ai.

We may choose the ordering so that A1 has the largest Perron-Frobenius eigenvalue, λ1 ≥ λk for all other k, and then normalize by λ1. Let xi be the Perron-Frobenius eigenvector of Ai, −1 and let Di = diag(xi). Then Di AiDi has constant row-sums λi.  −1  D1 A1D1 0 ··· 0  −1   A12 D2 A2D2 ··· 0    ˆ  A 0 ··· 0  Let A =  13 , where A1i are chosen to be positive and  . . .   . ··· .. .    −1 A1n 0 ··· Dn AnDn ˆ have row sums of 1−λi. Then A is a stochastic matrix with the same spectrum as A. Hence, the problem of finding which vectors in Cn are the spectrum of some positive n × n matrix is equivalent after a normalization to solving the inverse-eigenvalue problem for row stochastic matrices.

2.3.1 Karpelevich and the Region Kn Thus, to solve the (normalized) inverse eigenvalue problem for positive matrices, one need only solve the stochastic inverse eigenvalue problem. This problem has only been partially solved, but a simpler problem was solved by Karpelevich in 1949, building on ideas of

Dmitriev and Dynkin [3, 12, 19]. Karpelevich found the region Kn, the subset of C such

16 n that λ ∈ C if and only if there exists a stochastic matrix S ∈ Mn(R) and a vector v ∈ C such that Sv = λv. That is, Karpelevich solved the second type of eigenvalue problem men- tioned above. Karpelevich’s solution is rather complicated, and so we will not explicate it in detail, but we will briefly describe the region, and then move on to mention some of the basic ideas used.

th Proposition 2.3.1. The region Kn has as its boundary certain arcs connecting adjacent k roots of unity for k ≤ n. The arcs are solutions to parametric equations, defined as follows: a1 a2 iπ b iπ b Let p1 = e 1 , p2 = e 2 be two adjacent roots of unity for some k, m ≤ n, taken in coun- terclockwise order. Essentially, a1 and a2 are adjacent Farey fractions with denominator less b1 b2 than or equal to n. n n If b2b c ≥ b1b c, then we proceed as follows; if not, the complex conjugate arc will satisfy b2 b1 this condition, so there is no loss of generality in assuming the condition holds.

Let r1 = b2, r2 = a2, and let ri be the remainders obtained by performing the Euclidean

algorithm on r1, r2. Thus:

r1 = q1r2 + r3

r2 = q2r3 + r4 . .

rl−2 = ql−2rl−1 + 1.

Since Farey fractions are in reduced form, the Euclidean algorithm will return rl = 1 for some l. n If l is even and b2b c = 1, then the equation to be solved to obtain the arc connecting p1 to b2 p2 is

xq(xp − t)r − (1 − t)r = 0 (2.3.1)

where r = rl−1, and q, p are the unique solutions to

a2p ≡ 1 mod b2

a2q ≡ −r mod (b2).

17 The solutions to 2.3.1 for 0 ≤ t ≤ 1 form the arc from p1 to p2. n On the other hand, if either l is not even, or b2b c= 6 1, then the equation to be solved is b2

(xb − t)d − (1 − t)dxq. (2.3.2)

n Here, d = b2b c, b = b2, and q is the unique solution to b2

a2q ≡ −1 mod (b2)

Once again, solving this equation for 0 ≤ t ≤ 1 yields the arcs connecting p1 to p2.

For more on Karpelevich’s result, see [14] or [31]. Importantly, Dmitriev and Dynkin noticed that there was a strong connection between the eigenvalue problem for stochastic matrices and polygonal geometry in the plane.

row n Proposition 2.3.2. Let A ∈ Ωn be an n × n row stochastic matrix, and let v ∈ C be an eigenvector for A with eigenvalue λ. Then there exists a polygon K ⊆ C such that λK ⊆ K.

n Pn Proof. Let K = Conv{vj}j=1. Since Av = λv, we have that j=1 Aijvj = λvi for each i. The n expression on the left is a convex combination, and hence asserts that λvi ∈ Conv{vj}j=1 =

K. Since this is true for each vi, and the extreme points of K are a subset of the points {vi}, this is equivalent to the claim that λK ⊆ K.

Dmitriev and Dynkin, and then Karpelevich, analyzed how the vertices of λK were allowed to lie inside K, and from analysis of this sort, Karpelevich was able to find implicit equations for the boundary arcs of the λ that were allowed. Since any polygon K and λ ∈ C such that λK ⊆ K leads to an eigenvalue/eigenvector pair of some stochastic matrix, the allowable λ are exactly the elements of the set Kn.

2.4 The Perfect-Mirsky Conjecture

The Perfect-Mirsky conjecture concerns the next obvious question: since we know the subset of C which contains all eigenvalues of n × n stochastic matrices, what is the solution of the analogous doubly stochastic eigenvalue problem? That is, can we characterize the region n ωn = {λ ∈ C : ∃D ∈ Ωn, v ∈ C , Dv = λv} containing all eigenvalues of all n × n doubly

18 Figure 2.1: The region K4 of all eigenvalues of 4 × 4 stochastic matrices

stochastic matrices? An easy observation is the following:

Proposition 2.4.1. The region ωn is star-shaped.

n Proof. Let λ ∈ ωn, with D ∈ Ωn and v ∈ C such that Dv = λv. Pn If λ 6= 1, then by Lemma 2.2.3 i=1 vi = 0. Let J be the n × n matrix all of whose entries are 1; then Jv = 0. For t ∈ [0, 1],

1 (tD + (1 − t) J)v = tDv (2.4.1) n = λtv. (2.4.2)

1 tD + (1 − t) n J is clearly doubly stochastic, with eigenvalue tλ, and thus the ray from 0 to λ is contained in ωn.

i2πm Definition 2.4.2. We denote by Πk the region Conv{e k : 0 ≤ m ≤ k − 1}. This is a regular k-gon circumscribed in the unit circle with one vertex located at 1.

Recall that a circulant matrix C is a matrix such that Cij = ci−j where i − j is taken

mod n, and ck k ∈ {0, ··· , n − 1} are in C. Equivalently, let Xn ∈ Mn(C) be given by (Xn)ij = δi,j+1, where the indices are taken Pn i mod n. Then C is a circulant if it can be expressed as C = i=0 ciXn. Circulant matrices are a representation of the group algebra of the cyclic group of order n.

Recall that the Fourier matrix for the cyclic group of order n is the matrix F ∈ Mn(C)

19 (i−1)(j−1) th defined by (F )ij = ω where ω is a primitive n root of unity. As discussed in the introduction, F is the character matrix for the cyclic group of order n, and Pn−1 hence diagonalizes any C, whose eigenvalues are i=0 χ(i)ci where χ(·) is some character Pn−1 of the group Z/nZ. Since there is always a character such that χ(i) = ωi, i=0 ciωi is an eigenvalue of the circulant with entries (c0, ··· , cn−1). Pn−1 Perfect and Mirsky realized that if ci ≥ 0 and i=0 ci = 1, then C is a doubly stochastic matrix, and for any point λ ∈ Πn, we can find a doubly stochastic circulant matrix C such that λ is an eigenvalue of C.

We can also obtain any point in Πk, k ≤ n, by choosing C = Ck ⊕ In−k, with Ck the appropriate k × k circulant. Sn Thus, k=1 Πk ⊆ ωn. The Perfect-Mirsky conjecture, stated in [36] which also contains the observation above, is the conjecture that this containment goes both ways:

Sn Conjecture 2.4.3. (Perfect-Mirsky) ωn = k=1 Πk.

The Perfect-Mirsky conjecture is trivially true for n = 1, 2, and Perfect and Mirsky were able to give a proof for n = 3, but higher dimensions eluded them.

2.4.1 The Rivard-Mashreghi Counterexample

In 2005, Rivard and Mashreghi gave the following counterexample, showing that the Perfect- Mirsky conjecture is false for n = 5 [29]. They numerically solved the eigenvalues of the following matrix, and found that one of them did not lie in the Perfect-Mirsky region:

 0 0 0 1 0   0 0 1 0 1   2 2     0 1 1 0 0  . (2.4.3)  2 2   0 1 0 0 1   2 2  1 0 0 0 0

20 The eigenvalue, eigenvector pair

 1     −0.58 − 0.30i    v ∼  0.29 − 0.09i       −0.28 − 0.76i  −0.42 + 1.16i λ ∼ −0.28 − 0.76i

S5 is a counterexample, as λ can be seen to lie outside the region k=1 Πk. It does however, lie S6 in k=1 Πk, and no counterexamples to the Perfect-Mirsky conjecture are known for n > 5.

S5 S6 Figure 2.2: The Rivard-Mashreghi point is not in k=1 Πk, but is in k=1 Πk

Thus, the Perfect-Mirsky conjecture is true for n ≤ 3, and false for n = 5. The truth of the Perfect-Mirsky conjecture for n ≥ 6 is not known.

2.4.2 The Four Dimensional Perfect-Mirsky Conjecture

In this section we will prove that the Perfect-Mirsky conjecture is true for n = 4. We will also formulate a new conjecture to take into account Rivard and Mashreghi’s counterexam- ple, and provide some heuristic reasons explaining why their counterexample is possible, and why we believe our new conjecture may be true. Since any doubly stochastic matrix is a row stochastic matrix, it is true that for K = n Conv{vi}i=1, λK is a convex combination of the vertices of K, and thus λK ⊆ K. Lemma 2.2.3 asserts that K must have barycentre 0 if v is to be a non-trivial eigenvector of a doubly stochastic matrix.

21 Remark 2.4.4. Notice that unlike in the stochastic case, this implication does not go both ways. If λK ⊆ K, then it is easy to construct a stochastic matrix A such that Av = λv, where

vi are the extreme points of K. But there is no guarantee that, given a K with barycentre 0, and a λ ∈ C such that λK ⊆ K that we can find convex combinations of the vertices of K to obtain λK in such a way that the resulting matrix is doubly stochastic, rather than only stochastic. This is the first difficulty in extending the result of Karpelevich to doubly stochastic matrices: we only have necessary conditions on λ, not sufficient conditions.   Re(v1) Im(v1)   ! Re(v2) Im(v2) Re(λ) −Im(λ) Let V =  . . . Let Aλ = . Note that, using [27], the  . .  Im(λ) Re(λ)  . .  Re(vn) Im(vn) claim that λK ⊆ K is equivalent to the claim that VAλ ≺d V .

n Definition 2.4.5. Let K = Conv{vi}i=1 be an n-gon in the convex plane. We say that K is

(the vi are) convexly independent if none of the vi lie in the convex hull of any of the others.

Otherwise, K is (the vi are) convexly dependent.

n Pn Remark 2.4.6. Let K = Conv{vi}i=1 with i=1 vi = 0 be convexly dependent. In particular, assume m ≤ n of the vertices vi are in the convex hull of the other n − m. Let I = {i : vi ∈ Conv{vj}j6=i}. Then K = Conv{vi}i/∈I = K and thus K has n − m extreme points. Geometrically then, K is not an n-gon with barycentre 0, but an (n−m)-gon with barycentre P i∈I vi..

Definition 2.4.7. Let v be as above. We define K2(v) = Conv{vi + vj}1≤i

Remark 2.4.8. K is a convex polygon in the complex plane, with at most n vertices.

n Proposition 2.4.9. Let v = {vi}i=1 be a set of convexly independent complex numbers, arranged in counterclockwise order. Then K2(v) = Conv{vi +vi+1}0≤i≤n−1 where the addition in the index is taken mod n.

Proof. We will show that z is an extreme point of K2 if and only if z = vj + vj+1.

Let vj and vj+1 be two adjacent vertices of K, then [vj, vj+1] is a face of K and lies on a supporting hyperplane L, given by a real linear functional f(z) = Re(az), such that

f(vj) = f(vj+1) = b > 0. All other vertices of the polygon lie on the same side of L as 0, and

so f(vk) < b for k 6= j, j + 1. Then f(vj + vj+1) = 2b > f(vk + vm) for any (k, m) 6= (j, j + 1),

22 and so vj + vj+1 is an extreme point of K2.

Now, we show the converse. Suppose vj + vk is an extreme point of K, then let f be a real linear functional on C which achieves its maximum on K2(v) at vj + vk and without loss of generality let f(vj) ≥ f(vk). Since vj is the only vertex of K contained in the open half-plane

{z : f(z) > f(vk)}, vj must be adjacent to vk.

Remark 2.4.10. K2 is a balanced polygon when n = 4, meaning K2 = Conv{±z1, ±z2} where

z1 = v1 + v2 = −v3 − v4 and z2 = v2 + v3 = −v1 − v4.

Theorem 2.4.11. Let z1 and z2 be complex numbers with z1z2 6= 0 and let K = Conv{z1, z2, −z1, −z2}.

If λ is a complex number such that λK ⊂ K then λ ∈ Π4.

T Proof. Recall that K2 = Conv{±z1, ±z2}. Let z1 = a + bi, z2 = c + di. Let v1 = (a, b) , T T T v2 = (c, d) , and let T be the linear transformation taking (1, 0) → v1 and (0, 1) → v2. x −y Finally, for λ = x + iy, let Λ = ( y x ).

Conv{±(1, 0), ±(0, 1)} is the unit ball of the `1 norm. Hence K2 is the convex hull of the −1 image of the `1 unit ball under T . Thus if λK2 ⊆ K2, kT ΛT k`1 ≤ 1, where we use the matrix norm induced by the `1 vector norm. The theorem can be proved by means of the following theorem:

Theorem 2.4.12. Let A ∈ R2×2 be a matrix whose eigenvalues are a complex conjugate −1 pair, a ± bi, b 6= 0. Then the `1 norm of the associated transformation kT AT k1 achieves −1 a −b
its minimum when T AT = b a . Proof. Since T AT −1 is similar to A, it has the same trace and determinant, hence there exist x, y, z ∈ R such that −1 a+x y
T AT = z a−x . Equating the determinant of T AT −1 and A we get

a2 − x2 − yz = a2 + b2 =⇒ x2 + b2 = −yz.

23 Hence, looking at the `1 norm, we find:

−1 kT AT k`1 = max{|a + x| + |z|, |a − x| + |y|} 1 ≥ (|a + x| + |a − x| + |y| + |z|). 2

Since |a + x| + |a − x| ≥ 2|a + x + a − x| = 2|a| and (p|y| + p|z|)2 = |y| − 2p|yz| + |z| ≥ 0, we get

1 √ (|a + x| + |a − x| + |y| + |z|) ≥ |a| + p|yz| = |a| + x2 + b2 2

≥ |a| + |b| = kAk`1 .

The above theorem shows that a 2 × 2 invertible real linear transform achieves its `1 minimum when its matrix representation is that of a complex number. Thus,

−1 |x| + |y| = kΛk`1 ≤ kT ΛT k`1 ≤ 1

−1 where the last inequality holds because if λK2 ⊆ K2, then necessarily T ΛT preserves the

`1 unit ball. Hence, if λ = x + iy and λK2 ⊆ K2, then |x| + |y| ≤ 1 and λ ∈ Π4. Moreover, if λ ∈ R, then |x| ≤ 1 and so λ ∈ [−1, 1].

Theorem 2.4.13. [25] The set of all eigenvalues of 4 by 4 doubly stochastic matrices is S4 k=1 Πk.

Proof. Let λ be the eigenvalue of a 4 by 4 doubly stochastic matrix and let v be the corre- sponding eigenvector. If the entries of v are convexly dependent, then v is the eigenvalue of S3 a 3 by 3 stochastic matrix by Remark 2.4.6 and λ ∈ k=1 Πk. If the entries of v are convexly independent, then K2(v) is a balanced quadrilateral by Remark 2.4.10 and hence λ ∈ Π4 by Corollary 2.4.12 and Theorem 2.4.11.

We wish to remark that the use of the polygon K2 is motivated by Malamud’s charac- terization of directional majorization. Recall that λK ⊆ K if and only if the 2 × n matrix of vertices of λK is directionally majorized by the 2 × n matrix of vertices of K.

24 Malamud’s characterization states that this is true if and only if

Conv{λvi} ⊆ Conv{vi}

Conv{λ(vi + vj)} ⊆ Conv{vi + vj}

Conv{λ(vi + vj + vk)} ⊆ Conv{vi + vj + vk}.

P4 Because i=1 vi = 0, only the first and second condition are independent. The first condi- tion says that λK ⊆ K, and the second that λK2 ⊆ K2.

The considerations above lead us to formulate the following conjecture, which we made in [25]:

Conjecture 2.4.14. The region ωn = Kn−1 ∪ Πn. Remark 2.4.15. This conjecture is compatible with Rivard and Mashreghi’s counterexample; the eigenvalue they present is well within the relevant region of the 4 × 4 Karpelevich region, as our conjecture would suggest. The eigenvector associated with the eigenvalue is not

Figure 2.3: Close-up of the Rivard-Mashreghi point in K4 convexly independent, which is also compatible with our conjecture. Note also that because, in the case where the eigenvector is not convexly independent, with k of the {vi} being in the convex hull of the others, that K = Conv{vi} is geometrically an (n − k)-gon with non-zero barycentre, and so if λK ⊆ K, λ ∈ Kn−k. Also note that Kk ⊂ Kn−1 for all k ≤ n − 1, and also that Πk ⊆ Kn−1 for k ≤ n − 1. Therefore, to prove ωn ⊆ Kn−1 ∪ Πn, we must only prove that if v is convexly independent, λK ⊆ K implies λ ∈ Πn. Remark 2.4.16. We also note that the Rivard Mashreghi counterexample is a convex combi- nation of just two permutation matrices. We expect that if there are to be counterexamples

25 to the Perfect-Mirsky conjecture in dimensions higher than 5, they should occur at matrices that are convex combinations of small numbers of permutation matrices. The polygon geometry interpretation explains why: n In the limit, ∪i=1Πk approaches the full circle, and so if the boundary of ωn is not contained n n in ∪i=1Πk, the region ωn \ ∪i=1Πk becomes smaller and smaller. Hence a counterexample λ must be found close to the true boundary ωn. Let v be the eigenvector associated to the counterexample λ. Then Av is a convex combi- nation of the vertices of K(v), and since λ lies close to the boundary of ωn, the vertices of K(A(v)) must lie close to the edges of K(v). If A is a permutation matrix, then it takes each vertex of K(v) to a vertex of K(v). If it is a convex combination of two permutations, then some vertices of K(v) are taken to edges of K(v). For more terms in the convex com- bination, the vertices of K(v) are taken to interior points of K(v), and thus λ is not close to the boundary of ωn.

Figure 2.4: The associated eigenvector is convexly dependent

26 Chapter 3

Private Quantum Channels and Matrix Conditional Expectations

3.1 Quantum Information

3.1.1 Quantum States, Qubits, and Density Matrices

A classical bit is a system that can take one of two values, traditionally denoted 0 and 1. Thus, for example, a coin that can land either heads or tails is a bit. Quantum informa- tion extends this to quantum systems. It is well-known that one of the most interesting features of quantum mechanics is that distinct states can be put in superposition with one another to achieve a unique state. Mathematically, the superposition of states corresponds to a linear combination of the states with complex coefficients, whose absolute values sum to 1. Famously, the chance of observing the ith outcome corresponds to the square of the ith component in the superposition. Thus, a quantum state may be represented by an n-dimensional vector with complex com- ponents of norm 1, under the equivalence relation identifying two vectors if they are scalar multiples of one another. Here, n refers to the number of distinct possible outcomes for a measurement of the system. Thus, the natural generalization of the bit to the quantum world is the qubit, which rather than taking only the classical values 0 and 1, can also take any superposition of those values. A qubit then, is a system taking values in the unit ball of C2/ ∼ where ∼ is the equivalence relation v ∼ w if v = λw for some λ ∈ C. Such a quantum state is called a pure state. Evolution from one pure state to another is

27 accomplished by means of unitary matrices:

v = Uw.

Given two quantum systems, taking values in Cn and Cm respectively, their composite system takes values in Cn ⊗ Cm ' Cnm, the tensor product of the two systems. The tensor product of two vector spaces Cn and Cm, is defined as follows: n m n,m If {ei}i=1, {fj}j=1 are bases for the two systems respectively, then {ei ⊗ fj}i=1,j=1 is a Pn basis for the tensor product space. The tensor product of two elements i=1 ciei = v ∈ n Pm m C , j=1 djfj = w ∈ C is given by

n m X X v ⊗ w = cidjei ⊗ fj. i=1 j=1

Similarly, we may define the tensor product of two operators, A ∈ L(Cn), B ∈ L(Cm) by

(A ⊗ B)ei ⊗ fj = (Aei) ⊗ (Bfj) and extending by linearity. Given a basis, the tensor product of two operators corresponds to their Kronecker product:

Let A = (A)ij, B = (B)kl. Then the Kronecker product of A and B is the block matrix   A11B A12B ··· A1nB    A12B A22B ··· A2nB    A ⊗ B =  . . .  .  . ··· .. .    An1B An2B ··· AnnB

Thus, two qubit space is the space C2 ⊗C2 ' C4, and n-qubit space is the space (C2)⊗n ' n C2 . Importantly, because we may take superpositions of states in this new system, we can achieve quantum states that cannot be written as products of states v ∈ Cn, w ∈ Cn. Such states are said to be entangled. There is no way to write down a pure state to represent what an observer with access only to Cn would see when looking at an entangled state. For this reason, we need to extend our idea of a quantum state to encompass mixed states. A mixed state is simply a probabilistic mixture of pure states. To represent mixed states

28 mathematically, we need the idea of a density matrix:

Definition 3.1.1. A density matrix is an n×n complex, positive semidefinite matrix ρ that satisfies Tr(ρ) = 1.

The density matrix corresponding to a pure state v is simply the outer product of v with itself: vv∗. It is easy to see that this is indeed a density matrix. A general density matrix is a convex combination of pure state density matrices:

n X ∗ ρ = pivivi . i=1

Definition 3.1.2. Let A be an operator on Cn ⊗ Cm, with A = R ⊗ S. Then the partial trace of A with respect to system 1 (system 2) is the linear operator

Tr1(A) = Tr(R)S (Tr2(A) = Tr(S)R).

n m Since Tri is linear, we can extend it to non-product A ∈ L(C ⊗ C ) by linearity.

Definition 3.1.3. Let ρ be a density matrix acting on the space Cn ⊗Cm. Then the reduced n m density matrix of ρ on C (respectively C ) is the density matrix given by Tr1(ρ) (Tr2(ρ)).

3.2 Quantum Channels

3.2.1 Completely Positive Trace-Preserving Maps

From the previous section we see that an allowable quantum operation must be a linear map that takes density matrices to density matrices. Moreover, if the map acts on only a subsystem of a joint system, that as well must be an allowable quantum operation; in other

words, Φ ⊗ idk must be a linear map preserving density matrices for all k ∈ N, where idk is

the identity map from Mk → Mk. Inspired by this, we define:

Definition 3.2.1. A positive map is a linear map Φ from Mn → Mm such that if H ∈ Mn

is positive semi-definite, Φ(H) is positive semi-definite in Mm.

Definition 3.2.2. A linear map Φ: Mn → Mm is said to be k-positive if Φ⊗idk : Mn⊗Mk →

Mm ⊗ Mk is positive.

Since M1 ' C, Mn ⊗ M1 ' Mn, and thus a 1-positive map is the same as a positive map.

29 Definition 3.2.3. If a linear map is k-positive for all k ∈ N it is said to be completely positive.

We will give some important and useful equivalent conditions for complete positivity that are much easier to work with than the above definition. For now though, we note a completely positive map satisfies all the criteria necessary to be a valid operation except that it may take matrices of trace 1 to matrices of arbitrary trace. Thus, to complete our characterization of quantum operations, we require:

Definition 3.2.4. A completely positive map Φ: Mn → Mm is said to be trace-preserving if Tr(Φ(A)) = Tr(A) for all A ∈ Mn.

A completely positive trace-preserving map (or CPTP map) is also called a quantum channel.

Definition 3.2.5. A CP map Φ: Mn → Mm with the property that Φ(In) = Im is said to be unital.

Notice that in some senses, a unital CPTP map is a natural analogue of a doubly stochas- tic matrix in the positive map setting. The condition Φ(I) = I is analogous to the row- stochastic condition Ae = e, and the trace preserving condition Tr(Φ(ρ)) = ρ is analogous Pn Pn to the column-stochastic condition i=1(Av)i = i=1 vi. An important characterization of completely positive maps is due to Choi :

Proposition 3.2.6. (Choi’s theorem) [9] Let Φ: Mn(C) → Mn(C). Then Φ is completely positive if and only if the matrix   Φ(E11) Φ(E12) ··· Φ(E1n)    Φ(E21) Φ(E22) ··· Φ(E2n)    CΦ =  . . .  (3.2.1)  . ··· .. .    Φ(En1) Φ(En2) ··· Φ(Enn)

th th is positive semidefinite, where Eij is the n × n matrix with a 1 in the i row and j column,

and 0s elsewhere. The matrix CΦ is called the Choi matrix of Φ.

If Φ is CP then CΦ is positive, and so must have a set of mutually orthogonal eigenvectors n {vi}i=1 each with positive real eigenvalue. Let Ki be matrices such that √ vec(Ki) = λvi

30 n2 where vec : Mn(C) → C is the operation acting on a matrix A by stacking the columns of A on top of each other to create a vector. Then it can be shown that n X ∗ Φ(ρ) = KiρKi . (3.2.2) i=1 The form 3.2.2 is called the operator-sum or Kraus decomposition of Φ, and the operators

{Ki} are called the Kraus operators of Φ. Kraus operators are not necessarily unique, but

may be chosen with the same freedom with which one may choose eigenvectors of CΦ. Since any linear map of the form 3.2.2 is CP, Choi’s theorem shows that a linear map Φ having a Kraus decomposition is equivalent to Φ being CP.

Proposition 3.2.7. A CP map Φ with Kraus operators {Ki} is trace-preserving if and only if n X ∗ Ki Ki = I. (3.2.3) i=1 Φ is unital if and only if n X ∗ KiKi = I. (3.2.4) i=1

3.3 Private Quantum Channels

Definition 3.3.1. Let Φ: Mn(C) → Mm(C) be a quantum channel. Φ is said to be a private quantum channel (or private) if there exists a subset S ⊆ Mn(C) and a ρ0 ∈ Mm(C) such that

Φ(ρ) = ρ0 ∀ρ ∈ S

Private quantum channels were introduced by Ambainis et al. in [1]. Their definition was slightly different than ours, among other things requiring quantum channel to be a Pn ∗ random unitary channel, of the form Φ(ρ) = i=1 piUiρUi , with Ui unitary and pi ≥ 0 ∀i Pn and i=1 pi = 1. They were motivated in their definition by an attempt to find a quantum information version of the so-called one-time pad in classical communication: Given an n-bit binary string s that we wish to encode, the one-time pad is a random n-bit string x. Two parties Alice and Bob who wish to share information privately must each be in possession of x. Then, before sending s to Bob, Alice takes the bit-wise addition x ⊕ s, thus randomizing the bits of s by x. To an observer, with no knowledge of x, the string x⊕s

31 looks random. However, Bob, who also has x, and take s ⊕ x ⊕ x = s to recover the original message. Ambainis and his co-authors envisaged a private quantum channel working similarly: Alice wishes to send a density matrix ρ to Bob. As with the one-time pad, she and Bob must share a private key, in this case, the digit k. She encodes the density matrix using the ∗ unitary Uk, so that she sends Bob UkρUk . To ensure no one else can decode the message, she requires that to an outsider, who doesn’t know k, the channel is completely uninformative.

To someone who doesn’t know k, but only knows the set of {Ui}i∈I from which Uk was drawn, the channel has the form: X ∗ Φ(ρ) = piUiρUi i∈I where pi is the probability that the eavesdropper assigns to the private key being i. If Φ(ρ) gives some fixed output, then the eavesdropper can learn nothing about what went into the channel, since each input gives the same output. But Bob, who knows the key, can simply ∗ ∗ ∗ apply Uk to the state he receives, to obtain Uk UkρUk Uk = ρ. The restriction to random unitary channels in [1] is to ensure that Bob can in fact recover the original state. However, though all our examples in this work are in fact random unitary channels, we do not require the channel to be a random unitary, as we are more interested in the mathematical aspects of channels with fixed outputs, regardless of their actual practical applications to quantum communication. As a simple example, consider the completely depolarizing channel:

Example 3.3.2. The map Φ: Mn(C) → Mm(C) defined by

Tr(ρ) Φ(ρ) = I (3.3.1) m m

1 is clearly private for all ρ ∈ Mn(C), with ρ0 = m I.

If the subset S is the set of all density matrices supported on some subspace of C ⊆ Cn, then we say that C is a private subspace for Φ. If C is isomorphic as a vector space to C2n then Φ can privatize n qubits. Explicitly, a private subspace is a subspace C ⊆ Cn such that, for any ρ of the form ! ∗ 0 0 0

with Φ(ρ) = ρ0, where the block-partitioning of ρ corresponds to the orthogonal decomposi- tion Cn = C ⊕ C⊥.

32 Pm ∗ Proposition 3.3.3. [17] Let Φ be a random unitary channel Φ(ρ) = i=1 piUiρUi with commuting Kraus operators. That is, UiUj = UjUi for all i, j. Then Φ cannot have a non-trivial private subspace.

The proof of the theorem follows from trying to express two basis vectors |0Li, |1Li in terms of the basis that diagonalizes the Ui. It is impossible for any such subspace to have dimension greater than 1. It may seem that if Φ has no private subspaces, then there is no way to privatize quantum information, but this is not true. The following example due to Jochym-O’Connor et al. [17] shows that even for a channel with commuting Kraus operators, it is still possible to privatize a qubit of information.

Example 3.3.4. Let Φ: M4(C) → M4(C) be given by

1  Φ(ρ) = ρ + (I ⊗ Z)ρ(I ⊗ Z) + (Z ⊗ I)ρ(Z ⊗ I) + (Z ⊗ Z)ρ(Z ⊗ Z) . (3.3.2) 4

This map is the project-to-diagonal map, sending ρ to the diagonal matrix with the same diagonal entries as ρ. Any set of matrices with constant diagonal is privatized by Φ, with

ρ0 = cI. The matrices I ⊗ X, Y ⊗ Y , Y ⊗ Z satisfy this, and moreover, are closed under multiplication up to a scalar, and hence along with I ⊗ I, are a basis for the algebra they generate. Further, this algebra is isomorphic as a matrix algebra to the 2×2 matrix algebra. Hence, Φ can privatize a qubit of information.

Note that this does not contradict Proposition 3.3.3–the algebra privatized by Φ is not a private subspace. If so, there must be some proper subspace of V ⊆ C4 such that

(I ⊗ X)V ⊆ V (I ⊗ X)V ⊥ = 0 (Y ⊗ Y )V ⊆ V (Y ⊗ Y )V ⊥ = 0 (Y ⊗ Z)V ⊆ V (Y ⊗ Z)V ⊥ = 0.

This is clearly not so, since all of I ⊗ X, Y ⊗ Y , and Y ⊗ Z are invertible. Hence this privatized set is a private subalgebra, but not a private subspace.

33 3.4 Algebras and Conditional Expectations

Definition 3.4.1. A subset A ⊆ Mn(C) is said to be an algebra if it satisfies the following properties:

1. x + y ∈ A ∀x, y ∈ A

2. xy ∈ A ∀x, y ∈ A

3. λx ∈ A ∀λ ∈ C, x ∈ A An algebra is said to be unital if I ∈ A.

Definition 3.4.2. Let A ⊆ Mn(C) be a unital algebra. A conditional expectation onto A is a linear map Φ: Mn(C) → Mn(C) that satisfies the following properties: 1. Φ(a) = a ∀a ∈ A

2. Φ(a1xa2) = a1Φ(x)a2 ∀a1, a2 ∈ A, x ∈ Mn(C)

3. Φ(x) ≥ 0 ∀x ≥ 0

A conditional expectation onto an algebra is always a completely positive map; if the algebra is a unital algebra, then Φ is a unital CP map. If we demand that the conditional expectation is trace-preserving, then there is a unique map that satisfies these properties. This unique map is in fact the orthogonal projection on the vector subspace spanned by the algebra, where the Hilbert space is defined by the trace inner product. Moreover, this map is always a quantum channel.

Proposition 3.4.3. Let A ⊆ Mn(C) be a matrix algebra. Then A is unitarily equivalent to an algebra of the form m M Iik ⊗ Mjk k=1

where Iik is the ik × ik identity matrix, and Mjk is a jk × jk matrix algebra. In other words, A is unitarily equivalent to a direct sum so that every element a ∈ A has the form:   M1 0 0 ··· 0    0 M2 0 ··· 0    a =  .   0 0 .. ··· 0    0 0 0 ··· Mm

34 where   Mjk 0 ··· 0    0 Mjk ··· 0    Mk =  .   0 0 .. 0   

0 0 ··· Mjk

with Mjk appearing ik times.

0 Definition 3.4.4. Let A be a subalgebra of Mn(C). The commutant of A, A , is the subalgebra consisting of all matrices that commute with every element of A:

0 A = {x ∈ Mn(C): ax = xa ∀ a ∈ A}

00 Proposition 3.4.5. For any unital subalgebra A of Mn(C), A = A. Moreover, if A Lm 0 unitarily decomposes as A = k=1 Iik ⊗ Mjk then A is unitarily equivalent to

m 0 M A = Mik ⊗ Ijk k=1

Pm Simply by observing the block matrix structure, it is clear that k=1 ikjk = n. Also, it Pm 2 0 Pm 2 is easy to observe that dim(A) = k=1 jk and dim(A ) = k=1 ik.

Let i = (i1, i2, ··· , im) and j = (j1, j2, ··· , jm). Then, by the Cauchy-Schwarz inequality we have that

|i · j|2 = n2 ≤ |i|2|j|2 m m X 2 X 2 = ( ik)( jk) k=1 k=1 = dim(A) dim(A0).

Definition 3.4.6. Let A be a subalgebra of Mn(C), and let a ∈ A. Moreover, suppose A Lm decomposes as k=1 Iik ⊗ Mjk . Let LMa : Mn(C) → Mn(C) be the linear map given by

LMa(x) = ax, the left-regular representation of a. The left-regular trace of a, tra(a) is the

trace of LMa as an operator.

n More precisely, if {xi}i=1 is a basis for A, then represent a ∈ A as the matrix A = aij Pn where axj = i=1 aijxi. The left regular trace of a is the trace of the matrix A. Alternatively, Lm Lm if A' i=1 Iki ⊗ Mqi as in Proposition 3.4.3, then a ' i=1 Iki ⊗ Mi where Mi is qi × qi.

35 Pm The left regular trace of a is then i=1 kiTr(Mi).

Proposition 3.4.7. (Pereira) [35] Let A be a unital subalgebra of Mn(C), and let Φ be the trace-preserving conditional expectation onto A. Then the Kraus operators for Φ must be an orthonormal basis for A0, where orthonormality is defined relative to the inner product given by the left-regular trace for A.

3.5 Quasiorthogonality

Definition 3.5.1. Let A, B be two subalgebras of Mn(C). A and B are said to quasiorthog- onal to one another, denoted A ⊥B if any of the following equivalent conditions hold:

1 1 1 1. n Tr(ab) = n Tr(a) n Tr(b) ∀a ∈ A ∀b ∈ B

1 1 1 2. n Tr((a − n Tr(a)I)(b − n Tr(b)I)) = 0 ∀a ∈ A ∀b ∈ B Notice that this means that the traceless parts of the two algebras are orthogonal to one 1 another. Equivalently, translating the two algebras by n I makes them orthogonal in the Hilbert-Schmidt inner product. Equivalently, the two algebras, apart from their intersection 1 in the subspace spanned by n I are orthogonal.

If A, B are unital subalgebras then they are also subspaces of Mn(C). The trace-

preserving conditional expectations ΦA and ΦB are the orthogonal projections onto the

subspaces A and B. span{In} ⊆ A ∩ B, and A ⊥B if and only if the inclusion is actu- ally an equality.

Proposition 3.5.2. Let A and B be unital subalgebras of Mn(C) quasiorthogonal to one another, and let Φ be the trace-preserving conditional expectation onto A. Then A ⊥B, if Tr(b) Φ(b) = n I for all b ∈ B Tr(b) Proof. Assume that Φ(b) = n I. As Φ is trace-preserving, we have

1 1 Tr(ab) = Tr(Φ(ab)) n n 1 = Tr(aΦ(b)) n 1 Tr(b) = Tr(a I) n n Tr(b) Tr(a) = . n n

36 Proposition 3.5.3. Let A and B be quasiorthogonal unital subalgebras of Mn(C). Then dim(A) dim(B) ≤ n2.

dim(A) dim(B) Proof. Let {ai}i=1 and {bj}j=1 be bases for A and B respectively. Then

∗ ∗ ∗ ∗ ∗ Tr((aibj ) (akbl )) = Tr((bj ai)(akbl)) (3.5.1) ∗ ∗ = Tr(aiakbjbl ) (3.5.2) ∗ ∗ = Tr(aiak)tr(bjbl ). (3.5.3)

dim(A),dim(B) Thus the set of pairwise products {aibj}i,j=1 are orthonormal. So altogether there 2 must be no greater than n = dim(Mn(C)) such products. Hence dim(A) dim(B) ≤ n2.

Thus, given the conditional expectation Φ onto A, a subalgebra of Mn(C) with qua- siorthogonal subalgebra B, we have the following relationships: The Kraus operators for Φ form a basis for A0, while B is privatized by Φ. Moreover, dim(A) dim(B) ≤ n2 ≤ dim(A) dim(A0) and thus the dimension of any privatized algebra is bounded above by the number of Kraus operators for Φ.

Proposition 3.5.4. Let A, B be two unital subalgebras of Mn(C), and let ΦA, ΦB be the trace-preserving conditional expectations onto A, B respectively. Then A ⊥B if and only if 1 ΦA ◦ ΦB(ρ) = ΦB ◦ ΦA(ρ) = n In for all density matrices ρ.

Tr(ρ) Tr(b) Proof. Assume that Φ = ΦA(ΦB(ρ)) = n I for all ρ. Then Φ(b) = ΦA(b) = n I for all b ∈ B, and by Proposition 3.5 we must have that A ⊥B. Tr(b) Tr(a) Now assume that ΦA(b) = n I for all b ∈ B, and ΦB(a) = n I for all a ∈ A. Tr(ΦB(ρ)) Tr(ρ) Then, since ΦB(ρ) ∈ B for all ρ, Φ(ρ) = n I = n I. Switching as and bs yields the symmetric result.

Hence, if ΦA is the trace-preserving conditional expectation onto a unital subalgebra A,

the algebras B that are privatized by ΦA are exactly those algebras that are quasiorthogonal to A. Quasiorthogonality already appears in the theory of quantum information. For example,

given a set of mutually quasiorthogonal algebras Ai, it is well-known how to construct a collection of mutually unbiased bases, a fact apparently first noticed by Schwinger in [37];

37 see also Section 2.2 in [38]. The theory of mutually unbiased bases is important in quantum state tomography, for example [39]. Quasiorthogonality also underlies the notion of comple- mentarity in quantum mechanics; this suggests a relationship between quasiorthogonality and the uncertainty principle. For more information on quasiorthogonality in quantum in- formation, see [32,33,38]. It is interesting then, that the phenomenon of privacy in quantum channels is also a mani- festation of quasiorthogonality.

3.5.1 Group Theory of Pauli Subalgebras

The Pauli Group

The Pauli matrices are well-known in quantum mechanics and quantum information; they are self-inverse Hermitian matrices whose sets of eigenvectors are mutually unbiased–a property that allows for the distinctive quantum phenomena associated with quantum spin. Explicitly, the Pauli matrices are ! ! ! ! 1 0 0 1 0 −i 1 0 σ0 = , σ1 = X = , σ2 = Y = , σ3 = Z = 0 1 1 0 i 0 0 −1

Notice that each Pauli matrix commutes with itself and the identity, and anti-commutes with the other two Paulis. The Pauli group is the set P = {±I, ±iI, ±X, ±iX, ±Y, ±iY, ±Z, ±iZ}

The n-qubit Pauli group is the set of all n-fold tensor products of elements of P: Pn = Nn { i=1 σi : σi ∈ P}. x Nn y Nn Two elements in Pn either commute or anti-commute. Let ρ = i i=1 σi, τ = i i=1 τi, where x, y ∈ {0, 1, 2, 3} and let J = {1 ≤ i ≤ n : σi 6= I, τi 6= 1, σi 6= τi}. Then σ and τ commute if and only if |J| is even.

In most of what follows, we want to ignore scalar factors, so we consider the group Pn = n Pn/{±I, ±iI}. This is an Abelian group of size |Pn| = 4 . Note that the criterion for σ and τ to commute is independent of constant multiples, and only depends on the images of σ and

τ in the natural map to Pn. Note that Pn is the central quotient of Pn, as the group of scalar multiples of the identity is exactly the center of Pn; the claim that commutation relations do not depend on scalar multiples can be rephrased as: commutation or anti-commutation in Pn is well-defined on equivalence classes in the central quotient Pn.

38 We define the following matrix:   1 1 1 1   1 1 −1 −1 H1 =   1 −1 1 −1   1 −1 −1 1

H1 encodes the commutation relations between elements of P1: σiσj = H1i,j σjσi.

n N N Proposition 3.5.5. Let I,J ⊆ {0, 1, 2, 3} , and let ρ = i∈I σi, τ = j∈J σj ∈ Pn, and let H = H⊗n. Then ρτ = H τρ. n 1 ni1,··· ,in,j1,··· ,jn

Proof. This is easily proved by induction. We have already shown this is true for n = 1.

Assume the proposition holds for n − 1. Let I|n−1,J|n−1 denote the restrictions of I and J to their first n − 1 entries, and denote by ρ| , τ| the tensor products N σ , n−1 n−1 i∈I|n−1 i N σ . j∈J|n−1 j Then ρσ = H H σρ = H σρ. 1injn n−1i1,··· ,in−1 ,j1,··· ,jn−1 ni1,··· ,in,j1,··· ,jn

Notice that in the course of the proof, we label rows and columns of Hn by labels that

are in direct isomorphism with elements of Pn; we will refer to the row or column associated

to or indexed by a specific element of Pn. We may of course equally refer to submatrices

indexed or associated to subsets of Pn.

2 Definition 3.5.6. The Klein four group V is the group {e, g1, g2, g3} where gi = e for all i

and gigj = gk

The group P is isomorphic to V .

The character table of the Klein four group is exactly H1. In other words, each row of H1

is a character of V : χi(gj) = Hij, where the i and j reflect our choice of ordering of the ˆ n n elements of V and V . Thus, the character table of the group V is the matrix Hn. Here V n is the direct product of V with itself n times: an element of V is (v1, ··· , vn) where each

vi ∈ V , and multiplication is pointwise. n Since the group P is isomorphic to V , Pn is isomorphic to V . Hence, Hn is equally well

regarded as the character table of the group Abelian group Pn, where each column is la- ˆ belled by an element of Pn and each row is labelled by an element of Pn, which is of course

39 isomorphic to Vˆn.

Since the group Pn is Abelian, all its characters are one-dimensional, and all subgroups are normal.

Let H be a subgroup of Pn, and let I be the set of column indices associated to H. The column indices J associated to the annihilator Hann is the largest set of indices such that

Hn[I|J] is an all 1 matrix, by definition. Thus, the subset of Pn associated to J is the com- mutator of the subset associated to I. From this we deduce that for any subset of A ⊆ Pn, |A||A0| = 4n.

n Theorem 3.5.7. Any commuting subset A of Pn of size less than 2 can be extended to a commuting set A0 containing A, of size 2n.

n Proof. The proof relies on the fact that |Hann||H| = |4 | when H is a subgroup of Pn, and

the fact that the matrix Hn is symmetric. As A is commuting, the submatrix of Hn indexed

by the elements of A is an all 1 submatrix. Thus, the subset of Pn associated to A, call it

KA, is a subgroup. The set A, if it is not a group, generates a group, < A >, which must also be Abelian. By Lagrange’s theorem, the size of < A > is a power of 2, say | < A > | = 2k, k < n. 2n−k Then |Kann | = 2 . Pick any row indexed by an element h ∈ Kann that is not

already indexed by an element of K; this is possible since |Kann | > |K|. Then

the submatrix of Hn indexed by K ∪ {h} in both its rows and columns is an all 1s

submatrix, as is the submatrix indexed by < K, h >. The associated subset of Pn is the Abelian group generated by all pairwise products of elements of < A > with the element of k+1 Pn associated to h; this group has size 2 . Iterating the procedure yields larger and larger Abelian subgroups until n |Kann | = |K| which is only possible when both groups are of order 2 .

n Corollary 3.5.8. Every maximal Abelian subalgebra of Pn is of order 2 .

0 n Corollary 3.5.9. Any subalgebra A ⊆ span{Pn} has the property that |A||A | = 4 .

Proposition 3.5.10. The group Pn is homomorphic to the group of polynomials of degree n less than 2 over the binary field F2 under addition.

40 2 Proof. The map ψ : P1 → F2 given by

ψ(I) = 00 ψ(X) = 01 ψ(Y ) = 11 ψ(Z) = 10

2n is clearly a homomorphism; we extend ψ to a map from Pn to F2 in the obvious way, th concatenating ψ(σik ) where σik ∈ P is the k tensor factor. 2n 2n Let φ : F2 → F2[x]≤2n be the map that takes the string S = Πi=0si of elements in F2 to the Pn−1 i Pn−1 2n−i−1 polynomial given by i=0 s2ix + i=0 s2n+1x . Obviously, this is a homomorphism as well.

Considering the image of the map φ as a ring, we may also multiply polynomials.

(1) n (2) n Proposition 3.5.11. Let σ = Πk=1σ1ik and σ = Πk=1σ2ik be two elements of Pn. Let 2n−1 p(x) = φ ◦ ψ(σ1), q(x) = φ ◦ ψ(σ2). σ1 and σ2 commute if and only if the coefficient on x on p(x)q(x) is 0.

2 Proof. Let ψ(σi) = Πk=1nsik . We will first show that σ1 and σ2 commute if and only if P2n P2n k=1 σ12k σ22k = k=1 σ12k+1 σ22k+1 . (1) (2) |I| (2) (1) Because σ1 and σ3 anti-commute, σ σ = (−1) σ σ , where I = {ik : σ1ik 6= σ2ik , σ1ik 6= P2n I, σ2ik 6= I}. Each term of the sum k=1 σ12k σ22k + σ12k+1 σ22k+1 is either 0 if either one of

σjik is the identity, or if σ1ik = σ2ik , and 1 otherwise. Thus the sum measures the parity of

|I|; if it is 0, then σ1 and σ2 commute, otherwise they anti-commute.

By virtue of the construction of φ, commutation occurs exactly when the coefficient on x2n−1 in p(x)q(x) = 0: dn Pn dxn (p(x)q(x)) = k=0 σ1i2k σ2i2k + σ1i2k+1 σ2i2k+1 .

Proposition 3.5.12. There exists a homomorphism from Pn to the set of circulant matrices

of dimension 2n + 1 over F2.

41 n 2n Proof. Let σ = Πk=1σik ∈ Pn, and let S = ψ(σ) be the associated string in F2 . Then the map χ such that χ(σ) is the circulant whose first row is:

(0, s0, s2, ··· , s2n−2, s2n−1, s2n−3, ··· , s1).

Proposition 3.5.13. Two elements of Pn, σ1, σ2 commute if and only if Tr(σ1σ2) = 0.

Example 3.5.14. Consider the following two-qubit example. Let σ = X ⊗Y and τ = Y ⊗Z. 4 Then the map Φ from P2 to F2 acts as

p(x) = Φ(σ) = 0x3 + x2 + x + 1 q(x) = Φ(τ) = x3 + x2 + 0x + 1.

The map χ from P2 to the 2n + 1 × 2n + 1 circulants over F2 acts as

0 0 1 1 1   1 0 1 1 0   χ(σ) = 1 1 0 0 1     1 0 1 0 1 1 1 0 1 0 0 1 1 0 1   1 0 1 1 0   χ(τ) = 0 1 0 1 1 .     1 0 1 0 1 1 1 0 1 0

It is clear that στ = τσ, and indeed the x3 coefficient is 0:

p(x)q(x) = x5 + (1 + 1)x4 + (1 + 1)x3 + x + 1 = x + 1 and Tr(χ(σ)χ(τ)) = (1 + 1) = 0.

n Our proof that any Abelian subgroup of Pn can be extended to one of size 2 also suffices n as a proof that if S is a set of polynomials of degree ≤ 2n − 1 over F2, such that the x coefficient of p(x)q(x) is 0, for all p, q ∈ S, then S may be extended to a set of size 2n, where

42 the products of all elements of the extended set have 0 as their xn coefficient. Similarly, it suffices as a proof that any set T of (2n + 1) × (2n + 1) traceless circulants over n F2 with Tr(ab) = 0 for all a, b ∈ T can be extended to a set of size 2 with the same property.

Corollary 3.5.9 is particularly interesting; we know from the inequality 3.5.1 that |A||A0| ≥ n2 for any algebra–but the equality case, which is satisfied in the case of subalgebras gener- ated by subgroups of Pn, tells us something about the decomposition of A as in Proposition 3.4.3:

Proposition 3.5.15. (Pereira) [34] If A is a subalgebra of Mn(C), then the following equiv- alent conditions hold:

1. |A||A0| = n2

∗ Lm P 2 2. ∃r, s, ki ∈ Z and a unitary U such that UAU = i=1 Irki ⊗ Mski , and rs i ki = n

∗ L 3. UAU = M ⊗ N where M is simple and N = i Iki ⊗ Mki , and U is unitary.

.

Proposition 3.5.16. Let A be an Abelian subalgebra of dimension 2k generated by k elements k k of Pn. Then A is unitarily equivalent to I22n−k ⊗ ∆2k , where ∆2k is the algebra of 2 × 2 diagonal matrices.

Proof. By Corollary 3.5.9, A satisfies condition (i) of Corollary 3.5.15. Thus A is unitarily Lm equivalent to i=1 Irki ⊗ Mski . Since A is commutative, ski = 1 for all i, and thus ki = 1. Lm 2n Pm 2 2 Thus A is unitarily equivalent to i=1 Ir, where mr = 2 . Moreover, i=1 s ki = m = dim(A), thus m = 2k and r = 22n−k. L2k So A is unitarily equivalent to i=1 I22n−k = I22n−k ⊗ ∆2k .

Thus, any subalgebra A generated by an Abelian subgroup of Pauli matrices is unitarily k equivalent to the subalgebra I22n−k ⊗ ∆2k , where 2 is the dimension of the subalgebra.

Thus, the commutant of such an algebra is unitarily equivalent to M22n−k ⊗ ∆2k , and the conditional expectation with Kraus operators a basis for A, ΦA0 will privatize any algebra quasiorthogonal to M22n−k ⊗ I2k .

43 3.6 Private Pauli Channels

In this section we seek to generalize and better understand the example of Jochym-O’Connor et. al., using the theory of quasiorthogonality and Fourier analysis on the Abelianization of the Pauli group on n qubits. As a warm-up, we will analyze Jochym-O’Connor’s example from the perspective of condi- tional expectations onto Pauli subalgebras. 1 1 1 1 Recall that the Kraus operators for this map are 4 I ⊗ I, 4 I ⊗ Z, 4 Z ⊗ I, and 4 Z ⊗ Z. Up to scalar factors, these are elements of a commuting set of Paulis, and hence generate an

Abelian subalgebra of M4(C). Φ projects ρ onto its diagonal. The diagonal matrices form an Abelian subalgebra of M4(C). 0 In fact, the diagonal matrices ∆4 is a maximal Abelian subalgebra, since ∆4 = ∆4. Thus

Φ is the conditional expectation onto ∆4, and so the Kraus operators for Φ must be an 0 orthonormal basis for ∆4 = ∆4, which is in fact true. Thus, any subalgebra of Mn(C) quasiorthogonal to ∆4 will be privatized by Φ. In particu- lar, we are looking for a set of representatives from P2/{I ⊗ I,I ⊗ Z,Z ⊗ I,Z ⊗ Z}. The equivalence classes are

[I ⊗ I] = {I ⊗ I,I ⊗ Z,Z ⊗ I,Z ⊗ Z} [I ⊗ X] = {I ⊗ X,I ⊗ Y,Z ⊗ X,Z ⊗ Y } [X ⊗ I] = {X ⊗ I,X ⊗ Z,Y ⊗ I,Y ⊗ Z} [X ⊗ X] = {X ⊗ X,X ⊗ Y,Y ⊗ X,Y ⊗ Y }

Picking I ⊗ I and then one element from each other equivalence class will let us generate an algebra privatized by Φ. Since we want to privatize a qubit, we want to pick elements that anti-commute. Hence, all possible ways of privatizing a qubit using the channel Φ are:

{I ⊗ I,I ⊗ X,X ⊗ Z,X ⊗ Y }, {I ⊗ I,I ⊗ X,Y ⊗ Z,Y ⊗ Y }

{I ⊗ I,I ⊗ Y,X ⊗ Z,X ⊗ X}, {I ⊗ I,I ⊗ Y,Y ⊗ Z,Y ⊗ X}

{I ⊗ I,Z ⊗ X,X ⊗ I,Y ⊗ X}, {I ⊗ I,Z ⊗ X,X ⊗ Z,Y ⊗ Y }

{I ⊗ I,Z ⊗ X,Y ⊗ I,X ⊗ I}, {I ⊗ I,Z ⊗ X,Y ⊗ Z,X ⊗ Y }

{I ⊗ I,Z ⊗ Y,X ⊗ I,Y ⊗ Y }, {I ⊗ I,Z ⊗ Y,X ⊗ Z,Y ⊗ X}

{I ⊗ I,Z ⊗ Y,Y ⊗ I,X ⊗ Y }, {I ⊗ I,Z ⊗ Y,Y ⊗ Z,X ⊗ X}

44 It is clear that each set is quasiorthogonal to the diagonal subalgebra: let B be an algebra

generated by any of the above sets, and let b1, b2 ∈ B. Because the basis elements for the

algebra other than I ⊗ I come from different equivalence classes of Pn/∆4, b1b2 ∈ ∆4 if and −1 only if [b2] = [b1 ]. But all members of Pn are self-inverse, so b2, b1 must come from the same equivalence class, contradicting our choice.

Notice also that these are all maximal choices for private qubit algebras. We know that dim(B) ≤ dim(A) where A is the algebra generated by the Kraus operators. In this case, dim(B) ≤ 4. Since a qubit algebra has dimension 4, this is maximal.

Theorem 3.6.1. Let G ⊆ Pn be an Abelian subgroup of the n-qubit Pauli group, such that

log2(|G|) the image of G in Pn is injective. Then we may always privatize b 2 c qubits.

We will prove this theorem in two steps, first by proving it for the maximal Abelian subalgebras, and then for general Abelian subalgebras. First we prove it is true for the

diagonal subalgebra ∆2n .

n Theorem 3.6.2. Let ∆2n be the diagonal subalgebra of M2n (C), generated by {Zi}i=1, where Nn δij n each Zi = j=1 σj, σi = Z . Then we can always privatize up to b 2 c qubits using the conditional expectation onto ∆2n .

Proof. Essentially we just follow the steps in our motivating example. We need to pick

representatives from the equivalence classes of Pn/∆2n such that we generate an algebra

isomorphic to Mn(C). To do this, let X¯ = X ⊗ I Y¯ = Y ⊗ Y Z¯ = Z ⊗ Y

¯ ¯ Then the algebra generated by < X, Z > is isomorphic to M2(C), and quasiorthogonal b n c b n c N 2 ¯ δij N 2 ¯δij to ∆4. Let Xi = j=1(X ), Zi = j=1(Z ). Then the algebra generated by Xi,Zj for n n n 1 ≤ i, j ≤ b 2 c is quasiorthogonal to ∆2 and clearly encodes b 2 c qubits. Note that this encoding is certainly not unique; in general we can follow the procedure

outlined in the example of ∆4, and pick distinct representatives from the cosets of Pn/∆2n

in such a way to ensure that the algebra they generate is isomorphic to M2n (C).

Also notice that this construction is clearly maximal, since if B is quasiorthogonal to ∆2n ,

45 n n n 2 dim(B) ≤ 2 = 4 , and so at most 2 qubits can be privatized. From here, we can easily extend to the maximal Abelian case:

0 Proof. Suppose G generates a maximal Abelian subalgebra. This means that AG = AG. We know from Corollary 3.5.9 that every such algebra generated by independent Paulis satisfies 0 n Lm dim(AG) dim(AG) = 4 . Hence, AG is unitarily equivalent to i=1 I1 ⊗ M1 ' ∆2n . Thus by Theorem 3.6.2, the result follows.

In order to finish the proof, we want to understand the form of non-maximal Abelian subalgebras, and their quasiorthogonal subalgebras. Recall that an Abelian subalgebra with
∗ Pauli generators has the form A = U I2n−k ⊗ ∆2k U , and thus has commutant unitarily

equivalent to M2n−k ⊗ ∆2k .

Theorem 3.6.3. If B ∈ L(Cs ⊗ Cl) is a unital ∗-subalgebra quasiorthogonal to the algebra ˆ ˆ Ms ⊗ ∆l, then any b ∈ B must have the form b = cI + b, where b is a block matrix of the form   0 b12 ··· b1l    b21 0 ··· b2l  ˆ   b =  . . .   . ··· .. .    bl1 bl2 ··· 0

where each bij is an s × s matrix.

Proof. Let b ∈ B, and let a = M ⊗ ∆ where M ∈ Ms(C), ∆ ∈ ∆l. Partition b conformally l th to a, so that {bij}i,j=1, the (i, j) block of b is an s × s matrix. Then:

Tr(b)Tr(a) Tr(ba) = . sl

Hence, l Pl Pl X ( i=1 Tr(bii))( j=1 diTr(M)) Tr(b M) = . ii sl i=1

This holds for all di ∈ C, M ∈ Ms(C), so set di = δij for some j ≤ l. Then

Pl Tr(b )Tr(M) Tr(b M) = i=1 ii jj sl Tr(Tr(Pl b )IM) = i=1 ii . sl

46 Using linearity of the trace again we obtain

l X Tr((slbjj − Tr( bii)I)M) = 0. i=1

Pl Thus slbjj = Tr( i=1 bii)I, and so each diagonal block of b is the same multiple of the identity.

Any obvious question is, which subalgebras are unitarily equivalent to algebras of the above form? We can formulate a necessary condition.

Lm Theorem 3.6.4. If A' i=1 Ik1 ⊗Mqi is unitarily equivalent to an algebra of block matrices, whose diagonal s×s blocks are all the same constant multiple of the identity, then necessarily sqi ≤ ki.

Lm Proof. There exists a unitary U taking a ∈ A to an element of the form i=1 Ik1 ⊗Mi, where

Mi ∈ Mqi (C), and a V taking a to a matrix with constant diagonal s × s blocks. Combine Lm these unitaries to form W which takes i=1 Ik1 ⊗ M to a matrix with constant diagonal s × s blocks:

  cI ∗ ··· ∗ m   M  ∗ cI ··· ∗  W ( I ⊗ M )W ∗ =   k1 i  . . .   . ··· .. .  i=1   ∗ ··· ∗ cI

l,ki Partition W into matrices {Wij}i,j=1, where each Wij ∈ Mqi×s(C). Then for each i and Pki ∗ each Mi j=1 WijMiWij = cIs. As the Wij are partitioned pieces of a unitary matrix, this map is a completely positive map; in fact, it is the completely depolarizing channel from

Mqi (C) to Ms(C), and hence requires at least sqi Kraus operators. But there are ki Kraus

operators, and hence ki ≥ sqi.

We conjecture that this condition is both necessary and sufficient. However, even without being able to characterize these algebras, we can still prove that every trace preserving conditional whose Kraus operators are an Abelian subgroup G can privatize log(dim(G)) 2 qubits.

Lemma 3.6.5. The algebra < Zi >, i ≤ k is a basis for the algebra I2n−k ⊗ ∆2k .

47 Nn−k Nn IJ (i) Proof. Any product of elements Πk∈K Zk can be written as ( i=1 I) ⊗ ( i=n−k+1 Z ) where J ⊆ {n − k + 1, ··· , n} and IJ is the indicator function on J, IJ (i) = 1 if i ∈ J, and

0 otherwise. Over all such subsets, the right-hand tensor product is a basis for ∆2k .

k n Proof. Let G be a non-maximal Abelian subgroup, dim(G) = 2 ≤ 2 . G ' I2n−k ⊗ ∆2k '

∆2k . Hence this reduces to the maximal Abelian case on M2k (C). In fact, the techniques we have described above work even for non-Abelian algebras. Let Φ be a trace-preserving conditional expectation with Kraus operators equally weighted

elements of a subgroup H ≤ Pn. Then by choosing distinct elements from the equivalence

classes of Pn/H we can find a basis for a quasiorthogonal algebra privatized by Φ of dimension 4n/|H|.

3.7 Conditional Expectations onto Operators Spanned by Group Elements

The example of Pauli subalgebras suggests that the theory of conditional expectations and privacy is especially nice in the case when the algebras involved lie in the linear span of representations of some group with “nice” properties. In this section we seek to formalize this intuition, by developing a theory that manages to generalize the Pauli subalgebra case.

To begin, we develop the Pauli case a little more: Recall the matrix H1, which both encodes the commutation relations for equivalence classes of elements of the Pauli group mod scalar multiplication. This matrix is also a character matrix, and as such has certain nice proper- ties. Let us understand this better. Any two elements of the n-qubit Pauli group either commute or anti-commute: στ = ±τσ. Equivalently, the set of all commutators of elements of P, [P, P] = {±I}. Pm Let Φ be any channel whose Kraus operators are weighted Pauli operators: Φ(ρ) = i=1 piPiρPi.

If P ∈ Pk, then

m X Φ(P ) = piPiPPi i=1 m X 2 = [P,Pi] piPi P i=1 m X = ( ±pi)P. i=1

48 Thus Φ(P ) = λP , where λ is determined only by pi and [P,Pi] for each i. Moreover,

[P,Pi] is independent of scalar multiples of P,Pi; it is the same for any [P ] , [Pi] in the same equivalence class as P,Pi in Pk mod the scalar algebra. This explains why the matrix Hn is so useful: Hn records [[P ] , [Pi]] for all elements of Pn, and thus Φ privatizes P if and only if T the row corresponding to [P ] in the vector Hnp is 0, where p = (p1, ··· , pm) .

Thus, if p = IA is the characteristic function on a unital algebra A, Hnp = dim(A)IA0 . Hence, the characteristic function of any maximal Abelian subalgebra is an eigenvector of

Hn. Let G be a finite group, and let Z(G) be its center. Also, let G(0) be the derived subgroup,

G/ [G, G]. Let H be a subgroup of G, and let CG(H) = {g ∈ G : gh = hg, h ∈ H} be the centralizer of H in G.

Proposition 3.7.1. If H is a subgroup of G such that [H, g] ⊆ CG(H) for all g, then

[h1, g][h2, g] = [h2h1, g] for all h1, h2 ∈ H, g ∈ G.

Note that this holds in particular in the cases that [H,G] ⊆ Z(G), and more generally, if [G, G] ⊆ Z(G). Groups G such that [G, G] ⊆ Z(G) were studied by Baer [2]. Baer was interested in groups with Abelian central quotient, i.e., G such that G/Z(G) is Abelian. Since G/H is Abelian if and only if [G, G] ⊆ H, the two conditions are equivalent. Many of the following results are essentially to be found in Baer’s paper, albeit in slightly different form, and with different focus.

Proof. An arbitrary element in [H, g] has the form hgh−1g−1. If we assume that any such element is in the centralizer of H, then

−1 −1 −1 −1 h1gh1 g h2 = h2h1gh1 g .

−1 −1 −1 −1 Now [h1, g][h2, g] = h1gh1 g h2gh2 g and using Equation 3.7 we obtain −1 −1 −1 −1 −1 −1 −1 h2h1gh1 g gh2 g = h2h1gh1 h2 g = [h2h1, g].

Proposition 3.7.2. Let H be a normal subgroup of G, so that CG(H) is normal as well, and let v : G → G/CG(H) be the normal map from G to cosets of G by the commutator subgroup of H. Let h be an arbitrary element of H. Then there exists a k ∈ CG(H) such −1 that [h, g1] = k [h, g2] k for all g1, g2 with the same image under v.

49 Proof. If v(g1) = v(g2), they are in the same coset of CG (H), and thus there exists an −1 −1 −1 −1 −1 element k ∈ CG (H) such that g2 = kg1. Thus, [h, g2] = hg2h g2 = hkg1h g1 k . Since −1 k ∈ CG(H), we pull it out to the left to obtain k [g1, h] k .

Proposition 3.7.3. Let G be a group where the assumptions of Proposition 3.7.2 hold, and let H be a normal subgroup of G such that [H,G] ⊆ Z(G). We work in the group ring P of G, allowing complex linear combinations of elements of G. Let S(g) = h∈H [h, g]. If

g∈ / CG(H), then S = 0

Proof. If g∈ / CG(H) there exists h1 ∈ H such that [h1, g] 6= e. Pre-multiply S by [h1, g] to obtain

X [h1, g] S = [h1, g][h, g] h∈H X = [h, g] h∈H = S

where we made use of the fact that, since the conditions of Proposition 3.7.2 hold, [h1, g][h, g] =

[hh1, g], where hh1 is just some new element in H.

Thus, ([h1, g] − e)S = 0, but by hypothesis [h1, g] 6= e. Moreover, since [H,G] ⊆ Z(G), the condition that [g, h] − I has no zero divisors always holds, as by Schur’s lemma, any irreducible representation of G must send [g, h] to cI, c 6= I, and thus (c−1)I is invertible for each irreducible representation of G. By Proposition 1.2.18, [g, h] − e must therefore be invertible in the group ring as well.

P −1 Proposition 3.7.4. As above, we work in C[G], to define the function Φ(g) = h∈H hgh .

Then there exists a k ∈ CG(H) such that kΦ(g1) = Φ(g2) whenever g1 and g2 are in the same

coset of G/CG(H).

P −1 P −1 P Proof. Φ(g) = h∈H hgh = h∈H [h, g] ghh = ( h∈H [h, g])g.

If g1 and g2 are in the same coset of G/CG(H) then by Proposition 3.7 there exists k ∈ CG(H) P −1 P −1 such that [h, g1] = k [h, g2] and hence Φ(g2) = k( h∈H [h, g1])k g2 = k( h∈H [h, g1])k kg1 =

kΦ(g1).

Thus the function Φ(g) fixes elements of GG(H), and annihilates elements from outside

CG(H). Hopefully, the connection to conditional expectations and Pauli subalgebras is clear: we

50 merely need a faithful representation π from G to some unitary subgroup of complex matrices for the image of Φ under this representation to fit the role of a conditional expectation. To prove this claim note that

Φ(g) = g for all g ∈ CG(H).

X ∗ Φ(g1kg2) = hg1kg2h h∈H X ∗ = g1( hkh )g2 h∈H

= g1Φ(k)g2 for all g1, g2 ∈ CG(H).

Φ is a random unitary channel, it is completely positive, and thus preserves positive elements.

0 The representation π maps H to a subalgebra AH and GG(H) to the commutant AH . Φ 0⊥ is now a random unitary channel whose kernel is the span of AH , and whose fixed point set 0 is AH . Any subgroup of G all of whose non-identity elements lie outside of CG(H) will have 0 as its image a subalgebra quasiorthogonal to AH . A natural case to examine is when we can

pick one element from each coset of G/CG(H) to form a group; this group will automatically generate such an algebra. The n-qubit Pauli group is a group for which [G, G] ⊆ Z(G), and hence for which all of the above properties hold. Since the commutator of any two Paulis is ±I, the commutator

[H, g] is always in the centre of Pn, and hence certainly in the centralizer of H. Moreover,

by passing to Pn, we find that the image of any subgroup H is normal, as is the image of its

centralizer. Hence elements of Pn/v(CPn (H)) are cosets of the centralizer of H, but stripped of their constant multiples. Hence we may choose elements from here to obtain a group that can form a basis for a quasiorthogonal algebra. Schur’s lemma states that any irreducible representation of a finite group must send the centre of the group to scalar multiples of the identity. Hence, the most direct correspondance between the Pauli matrix case and a group theoretical generalization comes when we have a finite group G such that [G, G] ⊆ Z(G), and a unitary irreducible representation on Cn. Any element in the commutator will be sent to a scalar multiple of the identity, and thus g, h ∈ G/ [G, G] are sent to linearly independent unitaries. Moreover, since the centre contains the commutator subgroup, the central quotient G/Z(G) must be finite Abelian, and hence, by a result of Baer [2], a direct product of cyclic groups (see Remark 6.4 in [2]).

51 Chapter 4

Group Theory, Fourier Analysis, and Lie Theory and Private Quantum Channels

4.1 Fourier Analysis and Private Quantum Channels

Let G be a group such that [G, G] ⊆ Z(G), and let (π, Cn) be a unitary irreducible repre- sentation of G. Then let ΛG = π([G, G]) ⊆ {λI : λ ∈ C} be the set of complex numbers that appear as coefficients in commutators of elements π(g) and π(h). We wish to show that the values of the characters of the Abelian group G/Z(G) must all lie in ΛG, and hence we may understand the group G by doing Fourier analysis on its central quotient.

Definition 4.1.1. Let χ : G × G → ΛG be given by π([g, h]) = χ(g, h)I.

Definition 4.1.2. A bicharacter on a group G is a function B(·, ·): G × G → C satisfying the following:

1. B(e, g) = B(g, e) = 1 for all g, ∈ G

2. B(g, hk) = B(g, h)B(g, k) and B(hk, g) = B(h, g)B(k, g) for all h, k, g ∈ G.

A bicharacter is non-degenerate if for all non-identity g ∈ G there exists some element h ∈ G such that B(g, h) 6= 1

Notice that a bicharacter, when restricted in either argument to a fixed g ∈ G becomes a character on G. See [22] for further details.

52 Theorem 4.1.3. The function χ defined for the group G with the properties above is a bicharacter on G. Moreover, it is non-degenerate and well-defined when restricted to the image of the normal map to the central quotient G/Z(G).

Proof. 1. π([e, g]) = π([g, e]) = π(e) = I hence χ(e, g) = χ(g, e) = 1

2. Since [G, G] ⊆ Z(G) we have that

π([g, hk]) = π(ghkg−1k−1h−1) = π(gh(kg−1k−1g)g−1h−1) = π((kg−1k−1g)(ghg−1h−1) = π(k, g−1)π([g, h]) = π([g, k])π([g, h]).

The proof for the statement in the other argument is essentially identical. For the second claim, note first that χ(k, g) = 1 for all k ∈ Z(G), g ∈ G by definition.

Thus if g1 = kg2, k ∈ Z(G), we see that χ(g1, h) = χ(k, h)χ(g2, h) = χ(g2, h) for all h ∈ G, and χ is constant on cosets of Z(G). Lastly, if h is not in the centre, then there exists a g such that χ(h, g) 6= 1.

From the above, we see that χ([g], [h]) : G/Z(G) × G/Z(G) → C is a character of the Abelian group G/Z(G) when restricted to any [g] ∈ G/Z(G). By the result of Baer cited earlier, χ([g], [h]) for [g] ∈ G/Z(G) are characters of a direct product of Abelian groups. By following the image of this quotient forward along π, we find that π(G)/π(Z(G)) is ex- actly the quotient of π(G) by the scalar multiples of the identity, hence the elements in this group are linearly independent, and provide a basis for the representation of the group ring of G induced by π.

Definition 4.1.4. We define the generalized Hadamard matrix for the group G to be a matrix F whose rows and columns are indexed by elements of G/Z(G) and whose entries are given by (F )[g][h] = χ([g], [h]).

The matrix F is a character matrix of an Abelian group, hence unitary. It is also self- adjoint by conjugate symmetry of χ(·, ·). One may think of F as encoding the commutator

53 of two basis elements in the group ring.

Definition 4.1.5. Let F be the character matrix of a locally compact Abelian group G as above. Columns of F we index by elements of G. Rows are indexed by elements of Gˆ. Also, as G is Abelian, there is an isomorphism ψ : G ↔ Gˆ. Let f : G → (C). The Fourier transform of f is a function fˆ : Gˆ → C given by Z ˆ f(ψ(b)) = Fa,bf(a)dµ a∈G

where µ is the Haar measure of the group. Notice that if G is finite, the integral becomes a sum. An important class of results relating a function on an Abelian group G to its Fourier trans- form are known collectively as “uncertainty principles”. The most famous, of course, is the Heisenberg uncertainty principle, which informally says that given a function f on the locally compact Abelian group R, neither f nor its Fourier transform cannot both be too “concen- trated”.

One uncertainty principle, due to Donoho and Stark on cyclic groups [13], and generalized to arbitrary finite Abelian groups by Matusiak et al. [30], constrains the supports of f and its Fourier transform:

Definition 4.1.6. Let G be a finite Abelian group, and let f : G → C. The support of f is the set supp(f) = {g ∈ G : f(g) 6= 0}.

Proposition 4.1.7. (The Donoho-Stark uncertainty principle on finite Abelian groups) [30] Let G be a finite Abelian group, and let f : G → C, with Fourier transform fˆ. Then:

|supp(f)||supp(fˆ)| ≥ |G|. (4.1.1)

Matusiak et al. also found that for the inequality 4.1.1 to become an equality, then f

must either be the indicator function of a subgroup of H ≤ G, i.e., f = IH , or f must be obtained from a constant multiple of such an indicator function by means of

f(g) = cχ(g)IH (gk)

where c ∈ C and |c| = 1, χ ∈ Gˆ, and k ∈ G.

54 We will now give an example to show how these ideas can be applied to non-Pauli channels.

Example 4.1.8. Consider the matrices X,Z ∈ Mn(C) defined by (X)ij = δi,j+1, (Z)ij = i−1 th δijω where ω is a primitive n root of unity, and all sums are taken mod n. Then n n i j n−1 X = Z = I, and the set {X Z }i,j=1 is an orthonormal set. i n−1 Let G =< X, Z >. XZ = ωZX, and so [G, G] = Z(G) = {ω I}i=1 . Hence G satisfies the conditions specified above. |G/Z(G)| = n2, and so the elements of the central quotient form a basis for Mn(C). As a group G/Z(G) ' Z/nZ × Z/nZ. Let F be the character table of the Abelian group Z/nZ × Z/nZ, which records the commutation relations between basis elements of G. P ∗ Let H ⊆ G/Z(G), and let Φ(ρ) = h∈H phHρH . Then g ∈ G/Z(G) is privatized by Φ if P and only if h∈H phχ(h, g) = 0. If H is a subgroup, then we can find private algebras as before by taking distinct represen- tatives from G/Z(G)
/H.

Example 4.1.9. Let G be as above, with n = 3. Then {X,X2,Z,XZ,X2Z,Z2,XZ2,X2Z2} form a basis for operators on C3. The character table for G/ [G, G] ' Z/3Z × Z/3Z is

IXX2 ZXZX2ZZ2 XZ2 X2Z2 I  1 1 1 1 1 1 1 1 1   2 2 2  X  1 1 1 ω ω ω ω ω ω    X2  1 1 1 ω2 ω2 ω2 ω ω ω     2 2 2  Z  1 ω ω 1 ω ω 1 ω ω    F = XZ  1 ω2 ω ω 1 ω2 ω2 ω 1 . (4.1.2)   2  2 2 2  X Z  1 ω ω ω ω 1 ω 1 ω    Z2  1 ω ω2 1 ω ω2 1 ω ω2    2  2 2 2  XZ  1 ω ω ω ω 1 ω 1 ω  X2Z2 1 ω ω2 ω2 1 ω ω ω2 1

In two qutrit space, the matrix of commutation relations is F ⊗F . Let Φ be the following

55 channel on two-qutrit space:

1 Φ(ρ) = ρ + (I ⊗ Z)ρ(I ⊗ Z)∗ + (Z ⊗ I)ρ(Z ⊗ I)∗ + (Z ⊗ Z)ρ(Z ⊗ Z)∗ + (I ⊗ Z2)ρ(I ⊗ Z2)∗ 9  +(Z2 ⊗ I)ρ(Z2 ⊗ I)∗ + (Z ⊗ Z2)ρ(Z ⊗ Z2)∗ + (Z2 ⊗ Z)ρ(Z2 ⊗ Z)∗ + (Z2 ⊗ Z2)ρ(Z2 ⊗ Z2)∗ .

A choice from the quotient group of G by the group generated by the Kraus operators is the group generated by X2 ⊗ X and XZ2 ⊗ Z, which is isomorphic to the algebra generated by I ⊗ X and I ⊗ Z.

Let G be a group with [G, G] ⊆ Z(G), and let (F )g,h be the generalized Hadamard

matrix of G/Z(G). Let (p)g be a probability vector of size |G/Z(G)|. Given some unitary representation π of G, the vector p represents a random unitary channel with Kraus operators drawn from G/Z(G) via X ∗ Φp(ρ) = pgπ(g)ρπ(g) . g∈G/Z(G)

Φp(g) = (F p)g for any g ∈ G/Z(G); thus Φp has as its eigenvectors the elements of G/Z(G), P with eigenvalue h∈G/Z(G) ph(F )g,h.

Definition 4.1.10. Let G be a group, let f : G → C. f is said to be positive semidefinite −1 if the matrix (F )g,h = f(gh ) is positive semidefinite.

Proposition 4.1.11. (Bochner’s Theorem) [6] Let G be a locally compact Abelian group, and let f : G → C. Then f is positive semidefinite if and only if f is the Fourier transform of a probability measure on G.

Remark 4.1.12. If G is finite, then the claim is that a positive definite function must be Fourier transform of a vector.

Remark 4.1.13. An element g ∈ G/Z(G) is privatized by Φp if and only if its eigenvalue

under Φp is zero. Thus, to investigate the possible privatized elements of Φp corresponds to investigating the possible zero patterns of positive semidefinite functions.

Hence, given a quantum channel Φp as above, the function taking each g ∈ G/Z(G) to

its eigenvalue under Φp is a positive semidefinite function, and every positive semidefinite function on G/Z(G) arises in this way.

56 In the Pauli basis, then, any one-qubit Pauli channel has the form   1 0 0 0   0 p1 + p2 − p3 − p4 0 0    . (4.1.3) 0 0 p − p + p − p 0   1 2 3 4  0 0 0 p1 − p2 − p3 + p4

Equivalently, any Pauli channel has the form Diag(1, a, b, c) where (a, b, c) ∈ Conv{(1, 1, 1), (1, −1, −1), (−1, 1, −1), (−1, −1, 1)}

Proposition 4.1.14. (King, Ruskai) [21] Any unital channel on one-qubit can be written in the Pauli basis as 1 ⊕ T

Moreover, T = O1DO2, where O1, O2 are orthogonal, and D is a diagonal matrix whose diagonal entries (λ1, λ2, λ3) lie the tetrahedron Conv{(1, −1, −1), (−1, 1, −1), (−1, −1, 1)}.

Combined with the observation above, this yields the following corollary:

Corollary 4.1.15. Any unital quantum channel acting on one qubit is a composition of a unitary channel, a Pauli channel, and another unitary channel.

4.2 Lie Theory and Private Pauli Channels

Recall the definition of a Lie algebra:

Definition 4.2.1. A vector space g equipped with an alternating, bilinear operation [·, ·]: g × g → g is a Lie algebra if the bilinear operatation, called the Lie bracket, satisfies the so-called Jacobi identity:

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0.

Any subalgebra of Mn(C) is a Lie algebra with the commutator as its Lie bracket: [X,Y ] = XY − YX. Remark 4.2.2. We warn the reader that this is not the group commutator of the previous section; if there is reason for confusion, we will note which commutator we are referring to, but context should make it clear.

57 Definition 4.2.3. Let g be a Lie algebra of matrices under the commutator. For all x ∈ g the adjoint representation of x, Adx is a matrix representation of x given by the action of x through the commutator:

Adx(y) = [x, y] .

Definition 4.2.4. The Killing form is a bilinear form on a Lie algebra g given by B(x, y) =

Tr(AdxAdy).

The Killing form plays an important role in the classification of Lie algebras. For example, Cartan was able to show that a Lie algebra g is semisimple if and only if the Killing form on g is non-degenerate, that, is, for each x ∈ g there exists a y in g such that

B(x, y) 6= 0.

A simple Lie algebra is one with no non-trivial ideals, and a semisimple Lie algebra is a direct sum of simple Lie algebras. That the Killing form is important can also be seen from its relationship with the structure constants of the Lie algebra: recall Definition 1.3.4, where the Killing form was defined as a quadratic form in the structure constants of the Lie algebra. Since the structure constants contain the information of how two elements of the Lie algebra interact under the Lie bracket, it is reasonable to guess at the Killing form’s important role in classifying Lie algebras. The next proposition shows that in fact, the Killing form also has a connection to qua- siorthogonality, suggesting a close relationship between notions of quasiorthogonality and ideas from the theory of Lie algebras.

Proposition 4.2.5. Let A and B be subalgebras of Mn(C). A is quasiorthogonal to B if and only if B(a, b) = 0 for all a ∈ A, b ∈ B.

Proof. We use the identity vec(XYZ) = X ⊗ZT vec(Y ) to express the adjoint representation of a ∈ A and b ∈ B as matrices:

Ada(x) = ax − xa, hence vec(Ada(x)) = (a ⊗ I − I ⊗ a)vec(x).

58 Thus

B(a, b) = Tr(a ⊗ I − I ⊗ a)(b ⊗ I − I ⊗ b)
= Trab ⊗ I
+ TrI ⊗ ab
− Tra ⊗ b
− Trb ⊗ a
= 2Tr(I)Tr(ab) − Tr(a)Tr(b)
= 2nTr(ab) − Tr(a)Tr(b)
.

The vanishing of the last expression is one of the equivalent conditions for quasiorthogonality of A and B.

In light of Proposition 4.2.5, we examine some relationships between Lie theory and quasiorthogonality. Although the results are well-known in Lie theory, the connection with quasiorthogonality has not been noticed before.

Definition 4.2.6. Let g be a Lie algebra. Let θ : g → g be an involution; that is, θ ◦θ = idg. Assume further that B(x, θ(y)) is a positive semi-definite form on g. The decomposition g = l ⊕ p where l is the subspace {x ∈ g : θ(x) = x} and p = l⊥ = {x ∈ g : θ(x) = −x} is called a Cartan decomposition of g. The involution θ is the associated Cartan involution.

Proposition 4.2.7. Let θ be a Cartan involution on the Lie algebra g with associated de- composition g = l ⊕ p. Then the following hold:

1. [l, l] ⊆ l

2. [p, p] ⊆ l

3. [l, p] ⊆ p

Hence, the action of any x ∈ l under its adjoint representation is to take p to p and l to l, while any y ∈ p takes p to l and l to p. Hence, in a basis given by the Cartan decomposition, Adl is block-diagonal and Adp is block off-diagonal, and hence their trace is 0. Thus B(l, p) = 0. For real, semisimple (see [23] for the definition of a semisimple Lie algebra) Lie algebras there is up to conjugation only one Cartan decomposition, associated with the involution θ(X) = −X∗. Cartan also managed to classify the Cartan involutions for other Lie algebras. Of concern to us is the following fact, which can be found for instance in [10] or [11]:

59 Proposition 4.2.8. Let g = su(n). Up to conjugation of l and p by an invertible matrix T , there are three distinct Cartan decompositions, coming from the following three Cartan involutions:

¯ 1. θ1(X) = −X ¯ 2. θ2(X) = JXJ, where ! 0 I J = k −Ik 0 if n = 2k is even

3. θ3(X) = Ip,qXIp,q where ! Ip 0 Ip,q = 0 −Iq and n = p + q

Remark 4.2.9. Let g = l ⊕ p be a Cartan decomposition arising from the Cartan involution ∗ θ1. Let g = m ⊕ n be a conjugate decomposition, that is, there exists S such that SmS = l. Let θ be the Cartan involution corresponding to the second decomposition. Then

−lT = −ST mT S∗T (4.2.1) = S∗mS (4.2.2) and hence m = −(SST )m(SST )∗; thus θ(X) = −(SST )XT (SST )∗.

Given a Cartan decomposition conjugate to the decomposition arising from θ1, this re- lationship tells us how to find S if we know θ: if θ(X) = −TXT T ∗, we need to find S that satisfies SST = T . See [8,11] for examples of this.

4.3 Global Cartan and KAK Decompositions

Details on the material in this section can be found in chapters VI and VII of [23], with a much more detailed treatment and full proofs. A more streamlined presentation, developed for the purposes of quantum information theory, is in the introduction of [40]. We also note a slight difference in the terminology between the mathematical literature and the quantum

60 information literature. In the quantum information literature, for example [20], [40], the mathematicians’ KAK decomposition is frequently referred to as the Cartan decomposition.

Proposition 4.3.1. Let g be a Lie algebra with Cartan decomposition g = l ⊕ p. Let G be the Lie group associated to g. Then every element in G can be written in the form A = LP where L is in the Lie group of the Lie subalgebra l, and P is an exponential of a member of p.

This is called the global Cartan decomposition.

Example 4.3.2. Let g = gl(n) be the Lie algebra of the general linear group. A possible Cartan decomposition is gl(n) = l ⊕ p where l is the space of skew-Hermitian matrices, and p is the space of symmetric matrices. The Lie group of the Lie subalgebra l is U(n), and the exponential of any Hermitian matrix is positive. Thus any A ∈ GL(n) can be decomposed as A = UP where U ∈ U(n), P ≥ 0. This is the well-known polar decomposition.

Proposition 4.3.3. Let g, l, p be as in Proposition 4.3.1. Let a ⊆ p be an Abelian subalgebra

of p. Then any element in G can be written in the form K1AK2, where Ki is in the Lie group of the Lie algebra l, and A is an exponent of an element of a.

Such a decomposition is a KAK decomposition; although as mentioned above, in quan- tum information papers, this is also referred to as a Cartan decomposition. We will restrict the term “Cartan decomposition” to apply to the Lie algebra decomposition g = l ⊕ p or the associated global decomposition on the level of the Lie group.

4.4 Cartan Decompositions of su(2n) and Quasiorthogo- nal Algebras of Paulis

In this section we consider the Lie algebra su(2n) of traceless Hermitian unitaries. su(2n) has

as basis the n-qubit Pauli matrices Pn. Thus, we may find quasiorthogonal Pauli algebras by means of Cartan decompositions of su(2n).

Recall that up to conjugation, all Cartan decompositions arise from one of θ1, θ2, or θ3. .

n Proposition 4.4.1. Let g = su(2 ), with Cartan involution θ1. Then a basis for l is the

set of all purely imaginary n-qubit Pauli matrices, i.e. σ ∈ Pn with an odd number of Y s appearing in its tensor product. p is then spanned by the real Paulis, and thus has as a basis

61 all elements of Pn with an even number of Y s.

For su(2), this gives the decomposition

su(2) = span{Y } ⊕ span{X,Z}.

Clearly, the only algebras in p are the one-dimensional algebras spanned by X and by Z respectively. It is obvious that each is quasiorthogonal to the algebra Y . 2 For su(2 ), the θ1 Cartan decomposition yields

su(4) =span{I ⊗ Y,X ⊗ Y,Z ⊗ Y,Y ⊗ I,Y ⊗ X,Y ⊗ Z}⊕ span{I ⊗ X,I ⊗ Z,X ⊗ I,X ⊗ Z,Y ⊗ Y,Z ⊗ I,Z ⊗ X,Z ⊗ Z}.

From this we learn that the subalgebra generated by {I ⊗ Y,Y ⊗ X} is quasiorthogonal to that generated by {I ⊗ Z,Z ⊗ I}. Notice that in both of our examples, the subalgebra of p is Abelian. This must in fact always be true: Let A ⊆ p be a subalgebra of p. Then since A is closed under multiplication and linear combinations, [A, A] ∈ p for all a, b ∈ A. But, since p is a subspace in a Cartan decomposition, [A, A] ⊆ l. Since p∩l = {0}, [A, A] = 0 and A is Abelian.

Example 4.4.2. Let g = su(4), and let l = span{I ⊗ X,I ⊗ Y,I ⊗ Z,X ⊗ I,Y ⊗ I,Z ⊗ I}. Then p = span{X ⊗ X,X ⊗ Y,X ⊗ Z,Y ⊗ X,Y ⊗ Y,Y ⊗ Z,Z ⊗ X,Z ⊗ Y,Z ⊗ Z}. This is indeed a Cartan decomposition, as can be verified by taking the commutators of l and p with themselves and with each other. Thus, any subalgebra of p is an Abelian algebra quasiorthogonal to a one-qubit algebra.

We follow the presentation of Example 5.3.10 in [11]. By a dimension count, one can see that the decomposition in Example 4.4.2 must be conjugate to the decomposition arising from θ1. Thus, there exists some T such that

T (span{Y ⊗X,I⊗Y,Y ⊗Z,X⊗Y,Y ⊗I,Z⊗Y })T ∗ = span{I⊗X,I⊗Y,I⊗Z,X⊗I,Y ⊗I,Z⊗I.}

To find this T , we start by identifying the Cartan involution with l = {I ⊗ X,I ⊗ Y,I ⊗ Z,X ⊗ I,Y ⊗ I,Z ⊗ I}.

62 Let θ(ρ) = −(Y ⊗ Y )(ρT )(Y ⊗ Y ). Then

θ(I ⊗ σ) = I ⊗ σ θ(σ ⊗ I) = σ ⊗ I

for all σ ∈ {X,Y,Z}. Let m = Span{Y ⊗ X,I ⊗ Y,Y ⊗ Z,X ⊗ Y,Y ⊗ I,Z ⊗ Y }, and let E be a unitary such that

UlU ∗ = m.

Following the reasoning of Remark 4.2.9, we see that

(ET E)∗lT (ET E) = l and so ET E = Y ⊗ Y. (4.4.1)

Any unitary E satisfying Equation 4.4.1 will encode the algebras I ⊗ X,I ⊗ Y,I ⊗ Z and X ⊗ I,Y ⊗ I,Z ⊗ I as one of the algebras Y ⊗ X,I ⊗ Y,Y ⊗ Z and X ⊗ Y,Y ⊗ I,Z ⊗ Y . Hence, such unitaries E can be used to encode and decode private algebras.

4.4.1 Entanglers and Encoding Private Subalgebras

A matrix E satisfying Equation 4.4.1 is what we call an entangler, following Bullock [8]. We will justify the name by pointing out that such unitaries are the non-local unitaries that can be used to encode algebras isomorphic to a qubit.

n Nn Definition 4.4.3. A unitary matrix U ∈ U(2 ) is said to be a local unitary if U = i=1 Ui,

where each Ui ∈ U(2).

We are most interested in local unitaries that preserve the Pauli group Pn. A local

unitary will have this property if each Ui preserves P.

Example 4.4.4. Let H = √1 ( 1 1 ) be the Hadamard matrix. HXH = Z, HZH = X, and 2 1 −1 HYH = −Y .

1 0 ∗ ∗ ∗ Example 4.4.5. Let U = ( 0 i ). UXU = Y , UYU = X, and UZU = Z. Example 4.4.6. The product HU acts essentially to rotate through the Paulis: (HU)X(HU)∗ = Y , (HU)Y (HU)∗ = Z, and (HU)Z(HU)∗ = X.

63 Clearly, any tensor product of these unitaries will take Pn into Pn, or, after accounting for scalar multiples Pn into Pn. However, not all Paulis are accessible from each other via local unitaries. Trivially, a local unitary cannot change an identity factor in a tensor product, so for example there is no local unitary taking I ⊗ Z to X ⊗ X. In fact, the presence of identities is the only obstruction to the existence of a local unitary transforming σ ∈ Pn to τ ∈ Pn:

For 1 ≤ i ≤ n, let Ii = {σ1 ⊗ σ2 ⊗ · · · ⊗ σi−1 ⊗ I ⊗ σi+1 ⊗ · · · ⊗ σn : σj ∈ {I,X,Y,Z}, j 6= i}. T Let S ⊆ Pn be defined by S = j∈J Ij for some J ⊆ {1, 2, ··· , n}. Then S is closed under local unitaries, and for all σ, τ ∈ S, there is a local unitary transforming σ to τ.

If two elements of Pn are related to each other by some local unitary similarity, we say they are in the same local orbit. Similarly, if A and B are two subalgebras generated by members ∗ of Pn, and there is a local unitary U such that U(A)U = B, then we say the two algebras are in the same local orbit.

We now seek to understand the entanglers, unitaries preserving Pn that are non-local. Recall that we may decompose any matrix in U(n) by means of a KAK decomposition; since we are indifferent to phase factors, we may choose an element of SU(n) rather than U(n) after multiplying by an appropriate eiθ.

Proposition 4.4.7. The decomposition l = {I ⊗ X,I ⊗ Y,I ⊗ Z,X ⊗ I,Y ⊗ I,Z ⊗ I}, 2 {X ⊗ X,Y ⊗ Y,Z ⊗ Z} = a ⊆ p yields a KAK decomposition of SU(2 ) as W = U1 ⊗

U2 U U3 ⊗ U4 where Ui ∈ SU(2). U is a block unitary of the following form:

! I(αβγ − abg) + Z(αβg − abγ) X(aβγ − αbg) − iY (αbγ − aβg) (4.4.2) X(aβγ − αbg) − iY (aβg − αbγ) I(αβγ − abg) + Z(abγ − αβg)

where α = cos(θ), a = isin(θ), β = cos(φ), b = isin(φ), γ = cos(ψ), g = isin(ψ).

Proof. That l ⊕ p is a Cartan decomposition is easily checked; the Lie group corresponding to l is SU(2) ⊗ SU(2).The unitary U comes from the exponential of the Abelian subalgebra a = {X ⊗ X,Y ⊗ Y,Z ⊗ Z} ⊆ p. Each of element of a is self-inverse, so for σ ∈ a

∞ ∞ X (ix)2n X (ix)2n+1 exp(ixσ) = I + σ = cos(x)I + isin(x)σ. (2n)! (2n + 1)! i=0 i=0

64 Thus

U = exp(i(θX ⊗ X + φY ⊗ Y + ψZ ⊗ Z)) = exp(iθX ⊗ X)exp(iφY ⊗ Y )exp(iψZ ⊗ Z) = (αI ⊗ I + aX ⊗ X)(βI ⊗ I + bY ⊗ Y )(γI ⊗ I + gZ ⊗ Z).

Multiplying out the last expression yields the asserted form.

Remark 4.4.8. Any U as in Equation 4.4.2 is a linear combination of I⊗I,X⊗X,Y ⊗Y,Z⊗Z. A large class of the matrices of the same form as in Equation4.4.2 are what we have already called entanglers. In order to find unitaries that move between algebras in different local orbits, we must find entanglers that preserve tensors of Paulis. KAK decompositions were first considered in the context of quantum information by Khaneja and Glaser for the purposes of decomposing unitary gates in quantum circuits [20]. Since then, the importance of these decompositions has grown, with some useful summaries be- ing [7] and [40]. Bullock and Brennen studied entanglers and proved existence theorems in [8].

! I Y Example 4.4.9. Let U be the block matrix √1 . U is a unitary with the form 2 −Y I of Equation 4.4.2. Note also that UU T = iY ⊗ Y , meaning that it is an entangler in the sense of Equation 4.4.1.

We will end by showing how the ideas of the preceding section, which have already been used in quantum information in the context of quantum control (see [10, 11]) can also be used to encode qubits so that they are private for certain channels. We will illustrate the ideas in the case of encoding one-qubit subalgebras of in two-qubit space. The basic idea is to use entanglers and local unitaries to encode algebras isomorphic to qubits. We can classify the algebras in su(2) into local orbits:

Proposition 4.4.10. The local orbits of the four-dimensional Pauli algebras in su(2) have as representatives the following algebras:

1. I ⊗ I,I ⊗ X,I ⊗ Y,I ⊗ Z

2. I ⊗ I,X ⊗ I,Y ⊗ I,Z ⊗ I

3. I ⊗ I, I ⊗ Y , Y ⊗ X, Y ⊗ Z

65 4. I ⊗ I,Y ⊗ I,X ⊗ Y,Z ⊗ Y

5. I ⊗ I,I ⊗ Z,Z ⊗ I,Z ⊗ Z

6. I ⊗ I,X ⊗ X,Y ⊗ Y,Z ⊗ Z

Notice the first four such algebras are all 1-qubit algebras, and the last two are Abelian

algebras, and that all one-qubit algebras other than the standard ones I2 ⊗ M2 and M2 ⊗ I2 are in orbits 3 and 4.

Proposition 4.4.11. Let U be the unitary from Example 4.4.9. Then U takes algebra 1 to algebra 3, and algebra 2 to algebra 4.

This can be seen by a simple computation. Thus, the unitary U performs an encoding of the standard qubit algebras. Recall Example 3.3.4. The private subalgebra for this example was I ⊗ X,Y ⊗ Y,Y ⊗ Z. This algebra is equivalent under a local unitary to the algebra I ⊗ Y,Y ⊗ X,Y ⊗ Z, by performing a unitary on the second qubit that exchanges X and Y . Thus to encode a qubit in the form I ⊗ I, X ⊗ I, Y ⊗ I, Z ⊗ Z so as to be private for the channel in Example 3.3.4, we first apply the matrix U and then the appropriate local unitary. As another example, consider the unitary   1 0 −i 0   1  0 i 0 −1  E = √   . 2  0 i 0 1    1 0 i 0

The matrix E satisfies Equation 4.4.1, and so is an entangler. It too may be used to encode and decode single-qubit algebras; it can be computed that E takes the algebra generated by {X ⊗ Y,Y ⊗ I} to the algebra generated by {I ⊗ X,I ⊗ Y }. In general, to diagonalize an Abelian algebra in two-qubit space, or to encode a qubit, we need only understand which local orbit algebra is in; then by means of an entangler and some local unitaries, the algebra may be brought into the desired form. As a final example, consider the unitary found by Jochym-O’Connor et. al. to privatize the algebra in our motivating example, Example 3.3.4: ! I −iZ U = . X Y

66 The private algebra Jochym-O’Connor et. al. found was span{X ⊗X,Y ⊗I,Z ⊗X}. Notice 1 0 that under the action of the local unitary I ⊗ ( 0 i ) this algebra becomes span{X ⊗ Y,Y ⊗

I,Z ⊗ Y }. This we already know is isomorphic to I ⊗ M2 under the action of E. Hence 1 0 applying I ⊗ ( 0 i ) followed by E must take the algebra generated by {X ⊗ X,Y ⊗ I} to I ⊗ M . And indeed, 2 2 ! I −iZ E(I ⊗ ( 1 0 )) = . 0 i X Y

67 Chapter 5

Future Work and Conclusions

5.1 Perfect-Mirsky

Future work on the Perfect-Mirsky conjecture includes a better analysis of the situation in the cases n ≥ 5. It would useful to find explicit counterexamples to the conjecture for n = 6 or n = 7, and to find more counterexamples for n = 5 to have a better idea of what sorts of behaviour is possible for the eigenvalues of doubly stochastic matrices. To push further with our understanding of the Perfect-Mirsky conjecture, the following are useful goals:

1. To prove that for any polygon K with barycentre 0, such that λK ⊆ K, we can construct a doubly stochastic matrix with eigenvalue λ. This would strengthen a necessary condition for λ ∈ C to be an eigenvalue of a doubly stochastic matrix into a sufficient condition as well. This involves proving that we can always choose the stochastic matrix A describing the way the vertices of λK are convex combinations of the vertices of K to be a doubly stochastic matrix.

2. Adapting our proof to other cases where n is even.

If n is even, then K2, K3, ··· are all balanced polygons and hence define norms in R2. Thus there is some hope that some of the ideas from our proof may be applied in these cases. The difficulty is that the extreme points of the norm are no longer linearly independent; in order to perform a full analysis of these norms, there needs to be some way to keep track among the linear dependencies. Though there are some possibilities for how to deal with this difficulty, it makes the analysis considerably harder.

68 5.2 Private Quantum Channels

There are a number of further directions to push the research on private quantum channels. Some natural questions are:

1. There is also much to be done researching encodings of private codes in n-qubit space for n > 2. Although the tools of KAK decomposition and local unitary orbits still apply, the theory becomes more complicated.

2. There are also many interesting connections between conditional expectations and Lie algebras still to be explored. Such tools as weight-space decompositions of Lie algebras may shed light on the theory of conditional expectations and thus on privacy in quantum systems.

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