Cryptography in a quantum world

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Wehner, S.D.C.

Publication date 2008

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Citation for published version (APA):

Wehner, S. D. C. (2008). Cryptography in a quantum world.

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Linear algebra and semideﬁnite

programming

Semidefinite programming is a useful tool to solve optimization problems. Since we employed semidefinite programming in Chapters 3, 7, and 11, we briefly state the most important notions. We refer to [BV04] for an in-depth introduction.

A.1 Linear algebra prerequisites

Before turning to semidefinite programming in the next section, we first briefly recall some elementary definitions from linear algebra. We thereby assume the reader is familiar with basic concepts, such as matrix multiplication and addition. Unless explicitly indicated, all vector spaces V considered here are over the field of complex numbers. We use V = Cd _{to denote a d-dimensional complex vector}

space, and Cd×d _{to denote the space of complex d}_{× d matrices. A set of vectors}

|v1, . . . , |vd ∈ V is linearly independent if

_d

i=1ai|vi = 0 implies that a1 =

. . . = a_d = 0. A basis of a d-dimensional vector space V is a set of linearly independent vectors |v₁, . . . , |v_d ∈ V , the basis vectors, such that any vector

|u ∈ V can be written as a linear combination of basis vectors. We use v| to

The inner product of two vectors |u, |v ∈ V with |u = (u₁, . . . , u_d) and

|v = (v1, . . . , vd) is given by u|v =

iu∗ivi. The 2-norm of a vector is given by

|||v|| =v|v. Unless otherwise indicated, all norms of a vector are 2-norms in

this text. We also use |||v||_V to denote emphasize that the norm is deﬁned on a vector space V . Two vectors |u, |v ∈ V such that u|v = 0 are orthogonal. If, in addition,|||u|| = |||v|| = 1 then they are also called orthonormal.

A Hilbert space is deﬁned as a vector space V with an inner product, where the vector space is complete. We refer to [Con90] for a formal deﬁnition of the

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188 Appendix A. Linear algebra and semideﬁnite programming

notion of completeness and merely note that informally a vector space is complete if for any sequence of vectors in said space approaching a limit, the limit is also an element of the vector space. A bounded operator is an operator A : V → V such that there exists a c∈ R satisfying ||A|v||_V ≤ c|||v||_V for all v ∈ V . The

smallest such c is also called the operator norm of A.

The transpose of a matrix A is written as AT _{and given by A}T

ij = Aji, where

A_ij denotes the entry of the matrix A at column i and row j. Similarly, the conjugate transpose A† of A is of the form A†_ij = A∗_ji. We use I to denote the identity matrix deﬁned as I = [I_ij] with I_ij = δ_ij. A matrix U is called unitary if U U†= U†U =I. Furthermore, M is called Hermitian if and only if M = M†. Any Hermitian matrix can be decomposed in terms of its eigenvalues λ_j and eigenvectors |u_j as M = _jλ_j|u_ju_j|, where |u_ju_j| is a projector onto the

vector |u_j. We also call this the eigendecomposition of M. The support of M is the space spanned by all its eigenvectors with non-zero eigenvalue.

The tensor product of an m× n-matrix A and an m × n matrix B is given by the mm× nn-matrix A⊗ B = ⎛ ⎜ ⎜ ⎜ ⎝ A₁₁B . . . A_1nB A₂₁B . . . A_2nB . .. A_n1B . . . A_nnB ⎞ ⎟ ⎟ ⎟ ⎠.

The tensor product is also deﬁned for two vector spaces V and V. In particular, if the basis of the d-dimensional vector space V is given by {|v₁, . . . , |v_d} and the basis of the d-dimensional vector space V is given by {|v₁, . . . , |v_d}, then

W = V ⊗V denotes the d·d-dimensional vector space W with basis{|v_i⊗|v_j |

i∈ [d], j ∈ [d]}.

The direct sum of an m× n-matrix A and an m × n matrix B is given by the m + m × n + n matrix A 0 0 B .

Two vector spaces V and V deﬁned as above can also be composed in an analo-gous fashion yielding a d + d dimensional vector space W = V ⊕ V, where any

|w ∈ W can be written as |w = |v ⊕ |v_{for some |v ∈ V and |v}_{∈ V} _with

|v ⊕ |v_{= (v}

1, . . . , vd, v1, . . . , vd) for|v = (v1, . . . , vd) and |v = (v1, . . . , vd).

The trace of a matrix A is given by the sum of its diagonal entries Tr(A) =

iAii. Note that Tr(A + B) = Tr(A) + Tr(B), and Tr(AB) = Tr(BA). If A is

an Hermitian matrix, then Tr(A) is the sum of its eigenvalues.

Finally, the rank of a matrix A is denoted as rank(A) and given by the maximal number of linearly independent columns (or rows) of A.

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A.2 Deﬁnitions

We now turn to the definitions relevant for our discussion of semidefinite pro-gramming. A Hermitian matrix M is positive semidefinite if and only if all of its eigenvalues are non-negative [HJ85, Theorem 7.2.1]. Throughout this text, we use M ≥ 0 to indicate that M is positive semidefinite. We know from [HJ85, Theorem 7.2.11]:

A.2.1. Proposition. For a Hermitian matrix M ∈ Cd×d _{the following three}

statements are equivalent: 1. M ≥ 0,

2. x†M x≥ 0 for all vectors x ∈ Cd_,

3. M = G†G for some matrix G∈ Cd×d_.

M is called positive deﬁnite if and only if all of its eigenvalues are positive: we

have x†M x > 0 for all vectors x ∈ Cd_{. We use M > 0 to indicate that M is}

positive deﬁnite. We also encounter projectors, where a Hermitian matrix M is a

projector if and only if M2 = M . Note that this implies that M ≥ 0. We say that two projectors M₁ and M₂ are orthogonal projectors if and only if M₁M₂ = 0.

Furthermore, we use Sd _{to denote the set of all Hermitian matrices,} _Sd ₌

{X ∈ Cd×d _{| X = X}†_{}, and S}d

+ = {X ∈ Sd | X ≥ 0} for the set of all positive

semideﬁnite matrices. A set T is a cone, if for any α ≥ 0 and T ∈ T we have

αT ∈ T . A set T is convex, if for any α ∈ [0, 1] and T₁, T₂ ∈ T we have αT₁+ (1− α)T₂ ∈ T . A set T is called a convex cone, if T is convex and a cone: for any α₁, α₂ ≥ 0 and T₁, T₂ ∈ T we must have that α₁T₁+ α₂T₂ ∈ T . Note that Sd

+ is a convex cone: Let α1, α2 ≥ 0, and A, B ∈ S+d. Then for any x ∈ Cd we

have

x†(α₁A + α₂B)x = α₁x†Ax + α₂x†Bx≥ 0.

Hence, α₁A + α₂B ∈ Sd

+. The following will be of use in Chapter 3.

A.2.2. Proposition. Let A, B ∈ Sd. Then A ≥ 0 if and only if for all B ≥ 0

Tr(AB)≥ 0.

Proof. Suppose that A ≥ 0. Note that we can decompose B =_jλ_j|u_ju_j|

where for all j λ_j ≥ 0 since B ≥ 0. Hence, Tr(AB) = _jλ_jTr(A|u_ju_j|) =

jλjuj|A|uj ≥ 0, since A ≥ 0.

To prove the converse, suppose on the contrary that for all B ≥ 0 we have Tr(AB) ≥ 0, but A ≥ 0. If A ≥ 0, then there exists some vector |v such that

v|A|v < 0. Let B = |vv|. Clearly, B ≥ 0 and Tr(AB) = v|A|v < 0 which is

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190 Appendix A. Linear algebra and semideﬁnite programming

A.3 Semideﬁnite programming

Semideﬁnite programming is a special case of convex optimization. Its goal is to solve the following semideﬁnite program (SDP) in terms of the variable M ∈ Sd

maximize Tr(CM )

subject to Tr(A_iM ) = b_i, i = 1, . . . , p, and M ≥ 0

for given matrices C, A₁, . . . , A_p ∈ Sd_{. The above form is called the standard form}

of an SDP, and any SDP can be cast in this fashion (possibly at the expense of additional variables) [BV04]. To gain some geometric intuition about this task, note that M ≥ 0 means that M must lie in the cone Sd

+. The constraints

T r(A_iM ) = b_i determine a set of hyperplanes which further limit our possible solutions. A matrix M is called feasible, if it satisﬁes all constraints.

An important aspect of semideﬁnite programming is duality. Intuitively, the idea behind Lagrangian duality is to extend the objective function (here Tr(CM )) with a weighted sum of the constraints in such a way, that we will be penalized if the constraints are not fulﬁlled. The weights then correspond to the dual variables. Optimizing over these weights then gives rise to the dual problem. The original problem is called the primal problem. For the above SDP in standard form, we can write down the Lagrangian as

L(M, λ₁, . . . , λ_p, K) = Tr(CM ) + p i=1 λ_i(b_i− Tr(A_iM )) + Tr(KM ) = Tr((C − i λ_iA_i+ K)M ) + i λ_ib_i,

where K ≥ 0. The dual function is then

g(λ₁, . . . , λ_p, K) = sup M Tr((C − i λ_iA_i+ K)M ) + i λ_ib_i = iλibi if C− iλiAi+ K = 0 ∞ otherwise

From C−_iλ_iA_i+ K = 0 and K ≥ 0, we obtain that K = −C +_iλ_iA_i ≥ 0.

This gives us the dual problem as

minimize _iλ_ib_i

subject to _iλ_iA_i ≥ C,

where the optimization is now over the dual variables λ_i.

We generally use d∗ to denote the optimal value of the dual problem, and p∗ for the optimal value of the primal problem. Weak duality says that d∗ ≥ p∗.

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Let’s see why this is true in the above construction of the dual problem. Let M(∗) and {λ(∗)_i } be the optimal solutions to the primal and dual problem respectively. In particular, this means that M(∗) and{λ(∗)_i } must satisfy the constraints. Then

d∗− p∗ = i λ(∗)_i b_i− Tr(CM(∗)) = i λ(∗)_i Tr(A_iM(∗))− Tr(CM(∗)) = Tr −C + i λ(∗)_i A_i M(∗) ≥ 0,

by Proposition A.2.2 since M(∗) ≥ 0 and _iλ(∗)_i A_i ≥ C. An important

conse-quence of weak duality, is that if we have d∗ = p∗ for a feasible dual and primal solution respectively, we can conclude that both solutions are optimal. If solutions exist such that d∗ = p∗, we also speak of strong duality. We know from Slater’s conditions [BV04], that strong duality holds if there exists a feasible solution to the primal problem which also satisﬁes M > 0.

A.4 Applications

In many quantum problems, we want to optimize over states, or measurement operators. Evidently, semideﬁnite programming is very well suited to this case: When optimizing over a state ρ, we ask that ρ ≥ 0 and Tr(ρ) = 1. When optimizing over measurement operators M₁, . . . , M_k belonging to one POVM, we ask that M_j ≥ 0 for all j ∈ [k] and _jM_j =I. Concrete examples can be found in Chapters 3, 7, and 11.

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