theory of the reals [Shi16, Shi17]. For the cp-rank and the cpsd-rank no such results are known, but there is no reason to assume they are any easier. In fact, since no a priori upper bound exists on the cpsd-rank, it is not even clear whether the cpsd-rank is computable in general. It is known that deciding membership in the completely positive cone is NP-hard [DG14].
2.2 Separating cp-rank and cpsd-rank
For the completely positive rank we have the quadratic upper bound (2.2), and completely positive matrices have been constructed whose completely positive rank grows quadratically in the size of the matrix. This is the case, for instance, for the matrices
Mk = Ik 1 kJk 1
kJk Ik
∈ CP2k,
whose cp-rank is known to be equal to k2, see Proposition 2.1. Here Ik ∈ Sk is the identity matrix and Jk ∈ Sk is the all-ones matrix. This means the completely positive rank of these matrices is within a constant factor of the upper bound
2k+1
2 − 4 given in Equation (2.2). The significance of the matrices Mk stems from the Drew-Johnson-Loewy conjecture [DJL94] which was recently disproved [BSU14, BSU15]. This conjecture states that bn2/4c is an upper bound on the completely positive rank of n × n matrices, which means the matrices Mk are sharp for this bound.
It was observed in [PSVW18] that by combining the rank lower bound (2.4) on the completely positive semidefinite rank with (2.3) we obtain the following relation:
Ω(cp-rank(A)1/4) ≤ cpsd-rank(A) ≤ cp-rank(A) for A ∈ CPn.
This leads to the natural question of how fast cpsd-rank(Mk) grows. We show in Proposition 2.2 below that the completely positive semidefinite rank grows lin-early for the matrices Mk, and we exhibit a link to the question of existence of Hadamard matrices. More precisely, we show that cpsd-rankC(Mk) = k for all k, and cpsd-rankR(Mk) = k if and only if there exists a real Hadamard matrix of order k. In particular, this shows that the real and complex completely positive semidefinite ranks can be different.
A real Hadamard matrix of order k is a k × k matrix with pairwise orthogonal columns and whose entries are ±1-valued. Likewise a complex Hadamard matrix of order k is a k × k matrix with pairwise orthogonal columns and whose entries are complex valued with unit modulus. A complex Hadamard matrix exists for any order; take for example
(Hk)i,j= e2πi(i−1)(j−1)/k for i, j ∈ [k], (2.5) the matrix corresponding to the discrete Fourier transform. On the other hand, it is still an open conjecture whether a real Hadamard matrix exists for each order k that is a multiple of 4.
22 Chapter 2. Matrix factorization ranks It is well-known that the completely positive rank of Mk equals k2, for com-pleteness we provide a proof. Here, the support of a vector u ∈ Rd is the set of indices i ∈ [d] for which ui6= 0.
Proposition 2.1 (folklore). The completely positive rank of Mk is equal to k2. Proof. For i ∈ [k] consider the vectors vi = 1/√
k ei⊗ 1 and ui = 1/√
k 1 ⊗ ei, where eiis the ith basis vector in Rk and 1 is the all-ones vector in Rk. The vectors v1, . . . , vk, u1, . . . , ukare nonnegative and form a Gram representation of Mk, which shows cp-rank(Mk) ≤ k2.
To prove the lower bound, suppose Mk = Gram(v1, v2, . . . , vk, u1, u2, . . . , uk) with vi, ui∈ Rd+. In the remainder of the proof we show d ≥ k2. We have (Mk)i,j = δij for 1 ≤ i, j ≤ k. Since the vectors vi are nonnegative, they must have disjoint supports. The same holds for the vectors u1, . . . , uk. Since (Mk)i,j = 1/k > 0 for 1 ≤ i ≤ k and k + 1 ≤ j ≤ 2k, the support of vi overlaps with the support of uj for each i and j. This means that for each i ∈ [k], the size of the support of the vector viis at least k. This is only possible if d ≥ k2.
Proposition 2.2 ([GdLL17]). For each k ∈ N we have cpsd-rankC(Mk) = k.
Moreover, we have cpsd-rankR(Mk) = k if and only if there exists a real Hadamard matrix of order k.
Proof. The lower bound cpsd-rankC(Mk) ≥ k follows because Ik is a principal submatrix of Mk and cpsd-rankC(Ik) = k. To show cpsd-rankC(Mk) ≤ k, we give a factorization by Hermitian positive semidefinite k × k matrices. For this consider the complex Hadamard matrix Hk in (2.5) and define the factors
Xi= eieTi and Yi=uiu∗i
k for i ∈ [k],
where ei is the ith standard basis vector of Rk and ui is the ith column of Hk. By direct computation it follows that Mk= Gram(X1, . . . , Xk, Y1, . . . , Yk).
We now show that cpsd-rankR(Mk) = k if and only if there exists a real Hadamard matrix of order k. One direction follows directly from the above proof:
If a real Hadamard matrix of size k exists, then we can replace Hk by this real matrix and this yields a factorization by real positive semidefinite k × k matrices.
Now assume cpsd-rankR(Mk) = k and let X1, . . . , Xk, Y1, . . . , Yk∈ Sk+be a Gram representation of M . We first show there exist two orthonormal bases u1, . . . , uk
and v1, . . . , vk of Rk such that Xi = uiuTi and Yi = viviT. For this we observe that I = Gram(X1, . . . , Xk), which implies Xi 6= 0 and XiXj = 0 for all i 6= j. Hence, for all i 6= j, the range of Xj is contained in the kernel of Xi. Therefore the range of Xi is orthogonal to the range of Xj. We now have
X
i∈[k]
dim(range(Xi)) = dim X
i∈[k]
range(Xi)
≤ k
and dim(range(Xi)) ≥ 1 for all i. From this it follows that rank(Xi) = 1 for all i ∈ [k]. This means there exist u1, . . . , uk ∈ Rk such that Xi= uiuTi for all i. From
2.2. Separating cp-rank and cpsd-rank 23 I = Gram(X1, . . . , Xk) it follows that the vectors u1, . . . , uk form an orthonormal basis of Rk. The same argument can be made for the matrices Yi, thus Yi = viviT and the vectors v1, . . . , vk form an orthonormal basis of Rk. Up to an orthogonal transformation we may assume that the first basis is the standard basis; that is, ui= ei for i ∈ [k]. We then obtain
1
k = (Mk)i,j+k= hei, vji2= (vj)i
2
for i, j ∈ [k], hence (vj)i= ±1/√
k. Therefore, the k × k matrix whose kth column is√
k vk is a real Hadamard matrix.
The above proposition leaves open the value of cpsd-rankR(Mk) for the cases where a real Hadamard matrix of order k does not exist. Extensive experimentation using a heuristic (see [GdLL17, Section 2.2]) suggests that for k = 3, 5, 6, 7 the real completely positive semidefinite rank of Mk equals 2k, which leads to the following question:
Question 2.3. Is the real completely positive semidefinite rank of Mk equal to 2k if a real Hadamard matrix of size k × k does not exist?
Note that the lower bounds we develop in Chapter 5 are on the complex com-pletely positive semidefinite rank (which is k), therefore they cannot be used to answer the above question.
We also used the heuristic from [GdLL17, Section 2.2] to check numerically that the aforementioned matrices from [BSU14], which have completely positive rank greater than bn2/4c, have small (smaller than n) real completely positive semidefinite rank. In fact, for every completely positive n × n matrix we tried in our numerical experiments, we could always find a cpsd factorization in dimension n, which leads to the following question:
Question 2.4. Is the real (or complex) completely positive semidefinite rank of a completely positive n × n matrix upper bounded by n?
Chapter 3
Quantum information theory
Here we give some basic mathematical background on quantum information theory.
For more details see for example [NC00], or the lecture notes [Wat11, dW11].
Which set of rules governs the physical world around us? Are the laws of classical mechanics the correct model? Or does the world behave according to the laws of quantum mechanics? To answer these questions one can study the predictions that each of these models makes about certain experiments. In this chapter we explore the predictions made about probability distributions arising from measurements to a (quantum) mechanical system. In Part II of this thesis we will study the difference between classical computers (Turing machines) and quantum computers, computers acting according to the laws of quantum mechanics. See Chapter 9 for some background information on the topic of quantum computing.
Below we first explain some basic terminology, leading up to the type of proba-bility distributions that can occur between two parties who simultaneously measure parts of the same physical system. These distributions are called bipartite corre-lations. We then explain the framework of nonlocal games, which can be used to quantify the difference between classical and quantum correlations. Finally we show how bipartite quantum correlations are related to the cone of completely positive semidefinite matrices which we have seen in the previous chapter.
3.1 The basics
A physical system can be described by a state. We can learn information about a state by measuring it, and we can try to alter a state by acting on it. Below we describe the mathematical model, according to the laws of quantum mechanics, of a state and the allowed operations to it. We end the section with an example illustrating the concepts.
Quantum states. The state of a quantum mechanical system with finitely many degrees of freedom is described by a density matrix ρ, that is, a Hermitian positive semidefinite matrix whose trace is equal to 1. We call ρ a pure state if it has rank
25
26 Chapter 3. Quantum information theory one, else it is called a mixed state. Whenever we refer to a unit vector ψ ∈ Cd as a state, it should be understood as the pure state ρ = ψψ∗. We exclusively work with column vectors, so the state ρ = ψψ∗ is indeed a d × d density matrix. For two states φ, ψ ∈ Cd we refer to the complex number φ∗ψ as the amplitude of ψ in the state φ. Throughout this thesis we almost exclusively work with pure states.
For infinite-dimensional systems a pure state can be described by a unit vector in a complex separable Hilbert space.
Quantum operations. The postulates of quantum mechanics say that the pure state ψ of a quantum mechanical system can evolve in one of the following two ways. We can apply a unitary U to ψ to obtain the new quantum state U ψ, such evolutions are studied in Chapter 9. Or, we can measure the system.
Definition 3.1 (POVM). A positive operator-valued measurement (POVM) with m possible outcomes is described by a collection of Hermitian positive semidefinite operators E1, . . . , Emthat satisfyP
i∈[m]Ei= I. When measuring the pure state ψ, the probability of observing outcome i ∈ [m] is given by hψ, Eiψi = Tr(Eiψψ∗).
We sometimes refer to a POVM as a measurement device. Notice that the values hψ, Eiψi can indeed be viewed as a probability of observing outcome i: it is a value between 0 and 1 and Pm
i=1hψ, Eiψi = hψ, ψi = 1. Often, each out-come of a measurement is associated to a numerical value. It thus makes sense to talk about the expected outcome of a measurement. To a measurement (POVM) {E1, . . . , Em} whose outcomes are labeled by v1, . . . , vm∈ R we can associate the Hermitian operatorPm
i=1viEi. This operator is called the observable associated to the measurement. It connects a pure state ψ to the expected outcome under the measurement: ψ 7→ hψ, (Pm
i=1viEi)ψi.
A special class of POVMs is formed by those in which all operators Ei are projectors. Such a POVM is called a projective measurement (PVM). For a PVM we can talk about the post-measurement state. If we observe outcome i when we are measuring ψ with a PVM E1, . . . , Em, then ψ collapses to its projection on the range of Ei, i.e., the state Eiψ/phψ, Eiψi.
An important example of a PVM is the measurement in the computational basis, given by {e1e∗1, . . . , ede∗d} where ei ∈ Cd is the ith standard basis vector (i ∈ [d]).
When using this measurement on a state ψ ∈ Cd the probability of observing outcome i equals ψ∗eie∗iψ = |ψi|2.
Quantum states & linear functionals. To a pure state ψ ∈ Cdwe can associate the linear functional τ : Cd×d→ C defined as
A 7→ hψ, Aψi = ψ∗Aψ = Tr(Aψψ∗).
The linear functional τ maps measurement operators E1, . . . , Emto the probability of observing outcome i when using that measurement: τ (Ei) = ψ∗Eiψ. By linearity it maps observables to the expected outcome of the associated measurement on ψ.
In fact, the linear functional τ maps elements from the matrix algebra Cd×d to complex numbers. The infinite-dimensional analogue of a matrix algebra is the