Five Isoentangled Mutually Unbiased Bases and Mixed States Designs Praca Licencjacka Na Kierunku ﬁzyka Do´Swiadczalna I Teoretyczna

UNIWERSYTET JAGIELLONSKIW´ KRAKOWIE WYDZIAŁ FIZYKI,ASTRONOMIII INFORMATYKI STOSOWANEJ

Jakub Czartowski Nr albumu: 1125510

Five isoentangled mutually unbiased bases and mixed states designs Praca licencjacka Na kierunku ﬁzyka do´swiadczalna i teoretyczna

Praca wykonana pod kierunkiem Prof. Karola Zyczkowskiego˙ Instytut Fizyki

Kraków, 24 maja 2018 JAGIELLONIAN UNIVERSITYIN CRACOW FACULTY OF PHYSICS,ASTRONOMYAND APPLIED COMPUTER SCIENCE

Jakub Czartowski Student number: 1125510

Five isoentangled mutually unbiased bases and mixed states designs Bachelor thesis on a programme Fizyka do´swiadczalnai teoretyczna

Prepared under supervision of Prof. Karol Zyczkowski˙ Institute of Physics

Kraków, 24 May 2018 Contents

Abstract 5

Acknowledgements5

1 Introduction6

2 Space of pure quantum states7

3 Space of mixed quantum states7

4 Complex projective t-designs9 4.1 Symmetric Informationally Complete (SIC) POVM ...... 10 4.2 Mutually Unbiased Bases (MUB) ...... 10

5 Mixed states designs 10 5.1 Mean traces on the space of mixed states ...... 12

6 Notable examples 12 6.1 t = 1 - Positive Operator Valued Measure ...... 12 6.2 N=2, t=2, M=8 - Isoentangled SIC POVM for d 4...... 13 = 6.3 Short interlude - Numerical search for isoentangled MUB ...... 13 6.4 N=2, t=2, M=20 - Isoentangled MUB ...... 15 6.5 N=2, t=2, M=10 - The standard set of 5 MUB in H4 ...... 16 6.6 N=2, t=3, M=10 - Hoggar example number 24 ...... 17 6.7 N=2, t=3, M=20(40) - Witting polytope ...... 18 6.8 Interlude - Summary for N 2 ...... 18 = 6.9 N=3, t=2, M=90 - MUB ...... 19 6.9.1 Standard representation ...... 19 6.9.2 Isoentangled MUB - a "should", a "might" or a "can’t"? ...... 20

7 Minimal arrangements of points in the probability simplex 21 7.1 Connection with numerical integration ...... 21 7.2 N 2 - the easiest case ...... 22 = 7.3 Short detour - the ﬂat measure on the interval ...... 25 7.4 Interlude - comparison of polynomials ...... 26 7.5 N 3 - Potential problems for qutrits ...... 26 = 8 Conclusions 29

9 Open problems 30

Appendix A: Isoentangled 2-qubit MUBs 31

Appendix B: Search for isoentangled MUB in H4 - code 32

3 Appendix C: Welch bound saturation for MUB and SIC 34 SIC-POVM ...... 34 MUB ...... 34

4 Abstract

EN: Some basic notions of quantum theory including generalised measurements and complex projective t-designs are reviewed. We show that mutually unbiased bases and symmetric informationally complete POVM are examples of projective 2-designs. Subsequently, we introduce the notion of mixed states designs and show a constructive method to obtain them. We analyse certain explicit examples, construct analytically isoentangled mutually unbiased bases for two qubits and analyse results of numerical search for analogous conﬁguration for two qutrits. Finally we study minimal point arrangements reﬂecting probability distributions in the space of eigenvalues for qubit and for qutrit, showing possible applications of such constellations for approximate numerical integration.

PL: W niniejszej pracy przypominamy niektóre podstawowe poj˛eciateorii kwantowej, w tym uogólnione pomiary oraz zespolone projektywne t-desenie. Udowadniamy, ze˙ wzajemnie nieobci ˛azone˙ bazy oraz symetryczne kompletnie symetryczne POVM’y stanowi ˛a przykłady 2-deseni. Nast˛epnie wprowadzamy poj˛eciedeseni dla stanów mieszanych i pokazujemy konstruk- tywn ˛ametod˛eich otrzymywania. Analizujemy konkretne jawne przykłady, konstruujemy analitycznie jednakowo spl ˛atane wzajemnie nieobci ˛azone˙ bazy dla dwóch qubitów oraz analizujemy poszukiwania numeryczne analogicznej struktury dla dwóch qutrytów. Na ko´ncu przedstawiamy analiz˛eminimalnych układów punktów rekonstruuj ˛acych rozkłady prawdopodobie´nstwa w przestrzeni warto´sciwłasnych dla qubitu i qutrytu, przed- stawiaj ˛acmozliwe˙ zastosowanie takich układów do przyblizonego˙ całkowania numerycznego.

Acknowledgements

I would like to thank all the people that helped me prepare the following work. In particular, I am grateful to my supervisor, prof. Karol Zyczkowski,˙ who provided innumerable comments and corrections as well as guidance for the past 2.5 years. Special gratitude goes to Dardo Goyeneche, whose comments allowed to push this work forward and with whom I had an opportunity to work for about 1.5 years on another project. I am also grateful to Markus Grassl, who provided analytic solution to isoentangled MUB problem based on computer simulation conducted for this work. Finally, I would like to express my deepest gratitude to all the people that supported me throughout the whole 3 years of university studies - I would have never succeeded if not for their continued support.

5 1 Introduction

Mathematical notion of complex projective t-designs, introduced by Neumaier[1] and further studied by Hoggar[2] is useful in the field of quantum mechanics and quantum information. For instance, an important class of generalised quantum measurements called tight informationally complete positive operator valued measurement (IC - POVM) form t-designs - see Scott[3]. These notions are useful in quantum tomography [3], quantum cryptography [4], quantum fingerprinting [5] or entanglement detection [6]. They can be also generalised to conical t-designs [7] and related notion of unitary t-designs [8]. In particular, one may single out two outstanding families of complex projective 2-designs, namely symmetric IC-POVM (SIC-POVM) and mutually unbiased bases (MUB) studied ex- tensively in numerous works[9, 10, 11, 12]. A very special configuration of isoentangled of SIC-POVM for 2 qubits which consists of 16 pure states of the same degree of entanglement was analysed by Zhu et. al [9]. The result is interesting in itself and leads to the question whether similar configuration may be found for mutually unbiased bases for 2 qubits. Numerical simulations suggested existence of such a constellation, but no clear structure could be found. However, as the Bloch representation of points corresponding to partial traces of each of pure states of two-qubit system revealed, with use of a very special arrangement inside the Bloch ball related to a platonic solid it was possible to pursue a heuristic approach, which in turn yielded an analytic solution of a set of 5 4 20 states, · = which have the same degree of entanglement and form five unbiased bases in H4[13]. Since this approach worked, the question appeared whether a new concept of mixed states t-designs defined for the space of mixed states may prove itself useful like its coun- terparts. The consideration of isoentangled structures lead to yet another question - what are the arrangements of points that resemble distributions over the space of eigenvalues? In the following work we attempt to introduce and analyse the idea of mixed states t- designs and provide examples based on already known structures as well as a new, previously unknown set of isoentangled MUB for two qubits. This work is organised as follows. In sections 2-4 we recall the basic notions used throughout the work. In section 5 we introduce elementary definition of mixed states t-designs and provide an easy scheme for obtaining such ensembles. In section 6 we provide a study of a few examples of potential mixed states t-designs implied by previously known complex projective t-designs in larger dimensions. In section 7 we explore arrangements of point arrangements resembling properties of t-designs defined for flat Lebesgue measure and Hilbert-Schmidt measure for eigenvalues of density matrices for qubit and qutrit. Finally, sections 8 and 9 provide problems left open within this work and general conclusions. Additionally, in Appendix A the code for numerical search for isoentangled set of mutually unbiased bases in H4 is given. In Appendix B analytic form of the first basis of isoentangled MUB is provided. In Appendix C proofs of MUB and SIC-POVMs saturating Welch bound are provided.

6 2 Space of pure quantum states

Here we introduce the notion of the space of pure states Ξd belonging to the complex Hilbert space Hd of dimension d. States in this space are represented as vectors, ¯ ® denoted as ¯ψ Hd in the Dirac notation and its dual is ¯ ∈ denoted by ψ¯. The important notion is that the states are normalised ¯ ® and two distinguishable states i and ¯j are orthogonal. | 〉 ¯ ® j i¯j δ . (2.1) = i Final ingredient is the probabilistic interpretation. We may consider a decomposition of state in an orthonormal basis - a full set of distinguishable states: Figure 2.1: Space of pure states Ξ2 - 2- d d ¯ ® X ¯ ® X sphere. ¯ψ i i¯ψ ci i (2.2) = i 1 | 〉 = i 1 | 〉 = = ¯ ® where in general c C. Probabilistic interpretation says that the probability of ﬁnding ¯ψ i ∈ in a state i is equal to absolute value of the coefﬁcient c squared, | 〉 i 2 ¯ ¯ ®¯2 p c ¯ i¯ψ ¯ . (2.3) i = | i | = As the probabilities sum to unity, P p 1, the normalisation condition follows: i i = ¯ ® ψ¯ψ 1. (2.4) = Making use of the probabilistic assumption (2.3) and the normalisation requirement (2.4) ¯ ® ¯ ® one can identify of states ¯ψ and λ¯ψ for any λ C, which leads to the conclusion that ∈ d 1 the space of pure quantum states is in fact complex projective space Ξ CP − [14]. d =

3 Space of mixed quantum states

In the previous section we discussed pure states associated to the elements of the Hilbert space Hd . However, one may imagine two common characters in quantum information - Alice and Bob - having a spin 1/2 particle each. Both particles live separately in H2 each, but together these spaces form a composite Hilbert space, H H H . 2A ⊗ 2B = 4 Here we may distinguish between product states and non-product states. Former are deﬁned as set of states that can be represented as

¯ ® ¯ ® ¯ ® ¯ψ ¯ψ ¯ψ (3.1) prod = A ⊗ B ¯ ® ¯ ® where ¯ψ H and ¯ψ H . Thus Alice and Bob have their systems in particular A ∈ 2A B ∈ 2B states.

7 Non-product states are deﬁned as those that cannot be put in form (3.1). In order to answer the question in what state Alice will measure her particle if the entire system is in a non-product state we need to introduce the notion density matrix ρψ associated with the pure state ψ,

¯ ® ¯ X iµ¡ ¯ ®¢¡ ¯ ¢ ρψ ρ ¯ψ ψ¯ ρ jν i ¯µ j¯ ν (3.2) ≡ = = i, j,µ,ν | 〉 ⊗ ⊗ 〈 | for bi-partite systems. The deﬁnition can be generalised further. The operations from previous section can be translated to the language of the density matrices

¯ ® ψ¯ψ 1 Trρ 1, (3.3) = ⇒ ψ = ¯ ¯ ®¯2 ¡ ¢ p ¯ ψ¯i ¯ p Tr ρ ρ . (3.4) i = ⇒ i = ψ i Now we need to deﬁne the notion of partial trace which describes averaging over all states of a single subsystem. The outcome state is left in the so called mixed state. This is done by contracting one pair of the indices of the bi-partite state ρ,

X iµ ¯ ® ¯ ρ A TrB ρ ρ jµ ¯i j¯, (3.5) = ≡ i, j,µ X iµ ¯ ® ¯ ρB TrA ρ ρiν ¯µ ν¯. (3.6) = ≡ i,ν,µ

In order to gain some intuition let us consider matrix in H H H with a block 4 = 2A ⊗ 2B structure

µAB ¶ M (3.7) = CD and write its partial traces,

Tr M A D, (3.8) A = + µTr A TrB ¶ Tr M . (3.9) B = TrC TrD

We can see that partial trace has quite intuitive interpretation when viewed from this stand- point and can be generalised for higher dimensional systems. Finally, let us recall a possible measure of quantum entanglement applicable for a pure ¯ ® state of a bipartite system ¯ψ H H called linear entropy and deﬁned as one minus ∈ A ⊗ A trace of squared partial trace of the density matrix

· d 1¸ Entρ 1 Tr¡ρ2 ¢ 0, − . (3.10) = − A ∈ d

8 Taking all the above into the account we may deﬁne set of mixed states. Let ΩN denote the space of dimension N, which contains positive hermitian matrices of unit trace n o Ω ρ : Trρ 1, ρ ρ†, ρ 0, ρ : H H , (3.11) N = = = ≥ N → N the structure of which was studied in [14].

4 Complex projective t-designs

Complex projective t-designs have been introduced by Neumaier [1] and prove them- selves to be useful in many applications in quantum information. Intuitively one can say that a projective t-design is a set of pure states that simulates the unitarily invariant Fubini- Study measure on the space of pure states up to t-th moment. Stricter deﬁnition which we will use follows: ©¯ ®ªM DEFINITION 1. Any ensemble ¯ψi i 1 of pure states in Hd is called a projective t-design = if for any polynomial ft of degree at most t in both components of the states and their conjugates the average over the ensemble coincides with the average over the space of pure states d 1 Ξd , equivalent to CP − :

M Z 1 X ¡ ¢ ¡ ¢ ft ψi ft ψ dψFS . (4.1) M i 1 = Ξd = Here we need to mention the Welch bound, which holds for any Hilbert space:

THEOREM 1. For any ﬁnite collection of M (not necessarily normalised) vectors { x }M in | i 〉 i 1 Hilbert space H and any integer t 1the following inequality holds (see theorem 4.1= from d ≥ [10])

µ ¶ Ã !2 d t 1 X¯ ¯ ®¯2t X t + − ¯ xi ¯x j ¯ xi xi (4.2) t i, j ≥ i 〈 | 〉

For normalised vectors, which can represent pure states, the bound is saturated only if the given collection of vectors forms a t-design [3]. Notable property of t-designs for t 2 is that any state in dimension d may be decom- ≥ posed with respect to such an ensemble according to the formula

M (d 1)M X ρ + pi Πi 1d (4.3) = d i 1 − = d ¯ ® ¯ ¡ ¢ where Π ¯ψ ψ ¯ and p Tr ρΠ [3]. i = M i i i = i Collections of states forming a 2-designs have been shown to optimise the quantum tomography [15]. Moreover, 2-designs have been shown to be useful in areas such as cryptography [4] or ﬁngerprinting [5]. There are two notable classes of 2-designs known: mutually unbiased bases (MUB) and symmetric informationally complete POVM (SIC-POVM).

9 4.1 Symmetric Informationally Complete (SIC) POVM Symmetric informationally complete POVM in dimension d is deﬁned as a set of vectors ¯ ® d2 {¯ψi }i 1 in Hd such that = j ¯ ¯ ®¯2 δi d 1 ¯ ψi ¯ψj ¯ + (4.4) = d 1 + Direct inspection of saturation of (4.2) for t 2 indicates that SIC-POVM provide a 2- = design whenever they exist (see Appendix C). Moreover, they provide the minimal arrangement unique up to unitary transformations which forms a 2-design [10]. SIC-POVM have been shown to exist for all d 21 [11, 12, 16, 17, 18, 20] and some other ≤ dimensions up to 124 as well as numerically for some dimensions up to 844[19, 20]. An outstanding example of isoentangled SIC-POVM containing d 2 bi-partite pure states of the same degree of entanglement has been provided by Zhu et al[9].

4.2 Mutually Unbiased Bases (MUB) Another notable example of 2-design is formed by a full set of d 1 mutually unbiased bases n¯ + Eo ©¯ µ®ª ¯ ν (MUB) in Hd . We say that orthonormal bases ¯ψi and ¯φj are mutually unbiased when ¡ ¢ ¯D µ¯ E¯2 δµνδi j d 1 δµν ¯ ψ ¯φν ¯ + − . (4.5) ¯ i ¯ j ¯ = d Three or more orthonormal bases are all mutually unbiased if they are pairwise mutually unbiased. It is known [21] that there exist at most d 1 mutually unbiased bases in H . If + d the maximal set exists, it provides a 2-design on the space. This can be demonstrated by showing that the Welch bound (4.2) becomes saturated - see Appendix C. However, the full set of d 1 MUBs are known only for prime and prime power dimen- + sions d pn, for which an explicit construction has been given [22]. Interestingly, even the = lowest composite dimension d 6 poses signiﬁcant difﬁculties [23] and the problem of the = existence of the maximal number of MUBs in H6 remains open.

5 Mixed states designs

In this section we pose the question whether we can find structures similar to complex projective t-designs within the set ΩN of mixed states, propose a definition of such mixed states designs and show a constructive way to get such constellations. Let us recall the definition of the Hilbert-Schmidt distance between two states dHS q 2 = Tr¡ρ σ¢ . This notion allows one to define the Hilbert-Schmidt measure: any ball in − sense of HS distance of a fixed radius has the same volume. It can be shown that the spectrum (λ1,...,λN ) of a random state ρ distributed according to this measure in the set ΩN of density matrices of order N is given by the following joint probability distribution [24]:

10 Ã N ! N X Y ¡ ¢2 PHS(λ1,...,λN ) CN δ 1 λi λj λk , (5.1) = − i 1 j k − = < where CN is a proper normalisation constant depending on the dimension N. Having de- ﬁned this measure we may proceed with the deﬁnition of mixed states t-designs. M DEFINITION 2. Any ensemble of M density matrices {ρi : ρi ΩN }i 1 is called a mixed states ∈ = t-design if for any polynomial gt of degree t in the eigenvalues λj of the state ρ the average over the ensemble is equal to the mean value over the space of mixed states ΩN with respect to the Hilbert-Schmidt measure dρHS:

M Z 1 X ¡ ¢ ¡ ¢ gt ρi gt ρ dρHS . (5.2) M i 1 = ΩN = There exists a direct method to generate such sets of points. ©¯ ®ªM PROPOSITION 1. Any complex projective t-design ¯ψi i 1 in the composite Hilbert space 2 = H A HB of dimension d N induces, by partial trace, a mixed quantum states design © ª⊗M =¯ ® ¯ ρi i 1 in ΩN with ρi TrB ¯ψi ψi ¯. The same property holds also for the dual set {ρi0 : ρi0 ¯ = ® ¯ = = TrA ¯ψi ψi ¯}. Proof. It is known that the Fubini-Study measure in the space of pure states of order N 2, related to the Haar measure on the group U(N 2), induces by partial trace the Hilbert-Schmidt measure on the reduced space of mixed states [24]. ¯ ® ¯ The density matrix corresponding to a pure state ρψ ¯ψ ψ¯ is linear in both vector = ¯ ® ¯ coordinates and their conjugates. Also its reduction ρ A TrB ¯ψ ψ¯ retains this property. = © ªN It is useful to think of the matrix ρ A as decomposed in the canonical basis ei j i,j 1 in the space of matrices with some coefﬁcients, =

N X i j ρ A a ei j . (5.3) = i,j 1 = The Schmidt decomposition of a bipartite state

N ¯ ® X q ¯ ® ¯ ® ¯ψ λj ¯j 0 ¯j 00 , (5.4) = j 1 ⊗ = which provides the eigenvalues λi of the partial trace ρ A, may be viewed as a decomposition in a certain basis. Therefore each eigenvalue λi can be represented as N X kl kl λi Λi a , (5.5) = k,l 1 = kl where Λi is a transition matrix for the change of basis. The above shows that every λi is linear with respect to entries of the reduced matrix, which leads to conclusion that it is ¯ ® linear with respect to the components of the pure state ¯ψ . Having established the proper class of polynomials gt of eigenvalues of order t and the ﬂat Hilbert-Schmidt measure, We have demonstrated that Proposition 1 holds true.

11 5.1 Mean traces on the space of mixed states Considering deﬁnition (5.2) one ﬁnds that a necessary condition for a set of mixed states to form a t-design is to have moments of order t same as the mean values over the entire set of mixed states. Mean values of traces with respect to the HS measure on the set ΩN are known [24],

2N 5N 2 1 14N 3 10N Trρ2® , Trρ3® + , Trρ4® + . HS = N 2 1 HS = (N 2 1)(N 2 2) HS = (N 2 1)(N 2 2)(N 2 3) + + + + + + Since mixed states t-designs have to reconstruct all the polynomials in eigenvalues of density matrix, in particular we require that any t-designs has the mean moment of degree S consistent with the average TrρS® with respect to the HS measure for any S t. This HS ≤ simple condition is necessary for a collection of M points to form a mixed t-design.

6 Notable examples

Following sections provide examples of mixed states t-designs in the set ΩN of states of size N. Each subsection is denoted by t connected with t-design, dimension of the space N and number M of points forming the constellation.

6.1 t = 1 - Positive Operator Valued Measure Note that a positive operator valued measure (POVM) is deﬁned by any partition of identity operator into a set of k positive operators Ei acting on a Hilbert space Hd , satisfying [14]

k X † Ei 1, Ei Ei and Ei 0, 1...k. (6.1) i 1 = = ≥ = = Deﬁnition (5.2) for t 1 includes only linear combinations of mixed states. In particular = M 1 X 1 ρi 1N (6.2) M i 1 = N = from which an important observation follows.

OBSERVATION 1. Every POVM induces a mixed states 1-design.

The observation is easily shown since for t 1 both deﬁnitions coincides. Hence it is nat- = ural to expect that mixed states 2-designs may be obtained from a special class of POVM-s enjoying some additional symmetries. Furthermore, we can state another observation.

OBSERVATION 2. Tensor product of mixed states 1-designs forms a mixed 1-design.

12 It follows directly from the fact that a tensor product of two POVMs is a POVM. How- ever, this property does not hold for any t 2 which can be seen by noting that the tensor ≥ product of t-designs sets on spaces H A and HB generates only product mixed states and therefore does not provide us with such set on product space H H . A ⊗ B

6.2 N=2, t=2, M=8 - Isoentangled SIC POVM for d 4 =

Figure 6.1: One qubit mixed state 2-designs composed of 8 points, obtained by partial trace 2 ©¯ ®ª16 of 16 pure states H ⊗ ¯ψ forming the isoentangled SIC-POVM for 2 2 3 i i 1 qubits [9]. Both panels a) and b)= present sets of 8 points (each is doubly degenerated) belonging to the sphere of radius 2/5 inside the Bloch ball of radius 1/2, which correspond to both partial traces.

The first significant example of a visible regularity is the design corresponding to the isoentangled symmetric informationally complete positive operator valued measurement [9] ¯ ® forming a cube shown in Fig 6.1. As all pure states ¯ψi H4 have the same degree of en- ∈ ¯ ® ¯ tanglement and the same Schmidt coefficients, their partial traces ρ Tr ¯ψ ψ ¯ have i = B i i the same spectrum (λ,1 λ) and are represented by points belonging to the same sphere of − radius R 2/5 inside the Bloch ball Ω of radius 1/2. = 2 Note that the other reduction ρi0 generates 8 doubly degenerated points placed on a sphere of radius R which do not form a cube.

6.3 Short interlude - Numerical search for isoentangled MUB Up till now it was believed that the set of 5 isoentangled MUBs for a two qubit system might not exist. To ascertain this, numerical simulations have been conducted suggesting that

13 such a constellation of states should exist in a highly regular form. The simulation has been done in a form of a random walk in the space of unitary matrices. Finding an isoentangled set associated with particular set of MUBs is equivalent ¯ ® to ﬁnding unitary transformation U from the set {¯ψi } that provides a set of isoentangled ¯ ® ¯ vectors. For clarity let us set σ Tr ¯ψ ψ ¯. The algorithm proceeds as follows: i = A i i

1. Take random starting point U Ustart in space of unitary transformations and calcu- = ¯ ® ¯ ® late the starting deviation of purity ∆ in the set {¯(ψ ) } {U ¯ψ }. best best i = i 2. Take random unitary step

U ei Ht , step = where H is a random hermitian matrix providing a direction in the space of unitary matrices and t t is the starting size of a step. = 0 ¯ ® ¯ ® 3. Calculate purity deviation ∆ for the set of transformed states, {¯(ψ ) } {U ¯(ψ ) }. step step i = step best i ¯ ® ¯ ® If ∆ ∆ , accept the new best point ∆ ∆ ,{¯(ψ ) } {¯(ψ ) }. step < best best =→ step best i → step i Else reduce the step size t t/2. → 4. Repeat steps 2 and 3 until minimal step size t is reached. If t t , reset step size min = min t t . = 0

5. Repeat steps 2-4 until number of iterations reaches ﬁxed limit nmax.

Figure 6.2: Visualisation of results of a numerical search for 20 vectors forming a full set of 5 isoentangled MUB in H4 and represented by their one-qubit reductions in the Bloch ball. Each of the bases is depicted as a tetrahedron of a different colour. Explicit dodecahedral symmetry can be seen.

Code for the above simulation has been written in Mathematica and is provided in Ap- pendix B. With use of this procedure solution for MUB in H4 characterised by the variance

14 of purity ∆ 0.0026 was obtained, which is presented in the ﬁgure 6.2. The symmetry best = observed has been used by Markus Grassl to derive an analytic solution to the problem, which has been further simpliﬁed in the process. Most importantly, the procedure can be applied to any set of of states in any dimension, providing a reliable way to search for isoentangled variants of already known t-designs.

6.4 N=2, t=2, M=20 - Isoentangled MUB

Figure 6.3: One qubit mixed state 2-designs composed of 20 points, obtained by partial 2 trace of 20 states in H2⊗ forming a set of isoentangled mutually unbiased bases (MUB) for 2 qubits [13] inside the Bloch ball of radius 1/2. All points belong to sphere of radius R 2/5. Both panels show partial traces on both subsystems. =

This solution, obtained from 5 isoentangled MUBs for 2 qubits, has the property that reductions of every state have the same spectrum, which allows one to prepare all the states via local operations performed on a selected 2 qubit pure state possessing the suitable degree of entanglement. From a geometric point of view, reductions of four states stemming from a single basis in H4 form a regular tetrahedron in both reductions, which means that after resizing their Bloch vectors they form a SIC-POVM for a single qubit. All 20 points together form a regular dodecahedron. If we then consider each basis separately as a tetrahedron, we can recognise that all bases form the structure called 5-tetrahedron compound with the same chirality in both reductions. Due to uncanny similarity, one may be tempted to draw a parallel between the dodecahedral conﬁguration of states forming an isoentangled MUB and the dodecahedral arrangement of 3/2-spin states forming the Penrose dodecahedron [25, 26]. However seemingly suitable it may seem at ﬁrst, there is no direct connection - we have 20 states as opposed to 40 states of Penrose dodecahedron, with a different set of the over- ¯ ¯ ®¯2 laps ¯ ψi ¯ψj ¯ . In fact, it is known that Penrose dodecahedron has the same structure as Witting polytope [27], which is studied further in the notes in section 6.7.

15 6.5 N=2, t=2, M=10 - The standard set of 5 MUB in H4

Figure 6.4: One qubit mixed states 2-design obtained by partial trace of the standard set of 2 5 MUBs in H2⊗ consisting of 20 points. Two maximally entangled bases induce a point of weight 8 in the centre of the Bloch ball, while each of the remaining three separable bases induces two antipodal points on the Bloch sphere with weight 2 each.

Here we present an example that provides a reason for considering the weighted designs of mixed states. We may observe platonic regularity in the form of octahedron with vertices of weights 2 each, but the point at the centre has weight 8. Thus, the relative weights of 20 points can be rescaled by a factor of two. The total conﬁguration consists here of 7 points (instead of 8 for SICs) at the expense of weighing the central point as four surface points.

16 6.6 N=2, t=3, M=10 - Hoggar example number 24

Figure 6.5: One qubit mixed states 3-design obtained by partial trace of the projective 3-design consisting of 60 states in H4 provided by Hoggar [2]. The set of points is identical to the one shown in Figure 6.4, which shows that the former example 2 generated by the set of the standard MUBs in H2⊗ yields a mixed 3-design.

This mixed states 3-design is identical with the 2-design obtained from 5 standard MUBs in H4 with respect both to point locations and weights up to a multiplicative constant common to all points. This hints to the possibility of mixed states design with higher t than one would expect from the starting projective t-design. It is important to stress that both reductions are the same for this constellation of pure states and both coincide with the standard MUB conﬁguration.

17 6.7 N=2, t=3, M=20(40) - Witting polytope

Figure 6.6: One qubit mixed states 3-design obtained by partial trace of 40 states in H4 leading to the Witting polytope, consisting of 8 points in reduction A and 14 in reduction B. The points at the poles in reduction A have weight 1 and the remaining 6 points have weight 3 each. In reduction B points at the poles have weight 2 while the remaining 12 points have weight 3 each.

In the case of the Witting polytope (which is equivalent to the Penrose dodecahedron [27]) we have two regular ﬁgures - a parallelepiped (a) and an elongated bipyramid (b) in respec- tive reductions. This could suggest that properly resized regular polytopes could serve as templates for t-designs of different orders.

6.8 Interlude - Summary for N 2 = In Table 6.1 we present a comparison between the average moments computed for selected sets of states of size N 2 and the mean values with respect to the HS measure. = k 2 k 3 k 4 = = = Trρk ® 4/5 0.8 7/10 0.7 22/35 0.629 HS = = ≈ 6.1 0.8 0.7 0.62 6.2 0.8 0.7 0.62 Number of the set 6.4 0.8 0.7 0.65 6.5, 6.6 0.8 0.7 0.65 6.7 0.8 0.7 0.622

1 PM k Table 6.1: Average moments M i 1 Trρi calculated for particular examples of mixed states t-designs compared with= the mean values averaged over the entire set of mixed states Ω2.

As can be seen, the averages diverge from calculations only for the 4-th moment, oth- erwise they agree exactly. This seems to suggest that all of the examples considered form

18 3-designs, but we have an evidence this is the case for examples 6.5-6.7.

6.9 N=3, t=2, M=90 - MUB 6.9.1 Standard representation

(001)

12 12 (100) (010)

Figure 6.7: Mixed states 2-design obtained for a single qutrit obtained by reduction of 90 states forming the standard set of 10 MUBs in H9 [28], visualised in the triangle of eigenvalues of partial traces. Points at the vertices, corresponding to pure states, are obtained from 4 separable bases, while the remaining 6 maximally entangled bases ones yield the centre point representing the maximally mixed state with spectrum (1/3,1/3,1/3). The numbers in red denote weights assigned to each point, adding up to the total 90, the total number of states.

Let us consider now the qutrit case, N 3. Figure 6.7 presents a visualisation of points cor- = responding to 90 states forming the standard MUB set in dimension 9. The visualisation

19 involves taking the eigenvalues of reduced density matrix and plotting them in an equilat- eral triangle representing the simplex of eigenvalues, λ λ λ 1. 1 + 2 + 3 =

6.9.2 Isoentangled MUB - a "should", a "might" or a "can’t"?

(001)

(100) (010)

Figure 6.8: Mixed states 2-design for one qutrit obtained by reduction of the set of 10 MUBs optimised to minimise the variance of entanglement, visualised by 90 points in the triangle of eigenvalues. The green arcs of radius p2/15 represents a family of minimal sets of states forming a 2-design for eigenvalues.

The numerical simulations aimed at ﬁnding an isoentangled 9-dimensional MUB were not entirely successful. Hence we could not reach any conclusion about whether such an isoentangled constellation for two qutrits exists. Whereas for 5 MUBs in d 4 we could ﬁnd nu- = 3 merically a solution with the variance of purity of partial traces at the level of 10− , for d 9 = the analogous achieves the minimum amount of 0.04 0.05. − The distribution of states that make up the MUBs (red points) seems to resemble the distribution over the space with respect to the Hilbert-Schmidt measure. It is important

20 to note that the points are clustered around the circle representing the set of states which provides a 2-design-like structure in the space of eigenvalues for density matrices of size 3.

7 Minimal arrangements of points in the probability simplex

Consider first a measure µ(x) defined on the interval [0,1]. We wish to find a minimal se- quence of M points {x : x [0,1]}M which satisfy the condition analogous to t-design, i i ∈ i 1 with respect to a selected integration= measure µ(x):

M Z 1 1 X t t xi x µ(x)dx (7.1) M i 1 = 0 = In particular, if we choose the measure equivalent to HS measure for a qubit, such a set of points yields the minimal set of radii on which a projective t-design in Ω2 are located. In higher-dimensional spaces they may serve as a starting point for search of isoentangled conﬁgurations.

7.1 Connection with numerical integration We pose a simple question - can the structures devised above be used for numerical integration? Let us consider any differentiable function f (x) and its deﬁnite integral with respect to the ﬂat Lebesgue measure.

Z 1 f (x)dx (7.2) 0

Using the Taylor expansion we may approximate this function around any point x0

i X∞ (x x0) f (x) − , (7.3) = i 0 i! = so the integral becomes

Z 1 Z 1 X∞ 1 i f (x)dx (x x0) dx . (7.4) 0 = i 0 i! 0 − = However, making use of the deﬁnition (7.1) we may write

i Z 1 Ã t M i ! X∞ 1 d f (x)¯ X X 1 d f (x)¯ ¯ (x x )i dx ¯ (x x )t O(xt 1) (7.5) i ¯ 0 i ¯ j 0 + Mi! x x0 − = i! x x0 − + i 0 dx = 0 i 0 j 1 dx = = = = Hence such structures can provide a good method of numerical integration as long as the function we consider is well approximated by a polynomial of order t. Similar argu- ment holds when we consider a proper set of points for different measures and analogous integration over higher dimensional sets as well.

21 7.2 N 2 - the easiest case = The Hilbert-Schmidt measure deﬁned in (5.1) for one qubit systems takes form

P (λ ,λ ) C δ(1 λ λ )(λ λ )2 HS 1 2 = 2 − 1 − 2 1 − 2 where delta ensures that we can set λ 1 λ . Thus, the probability distribution depends 2 = − 1 only on a single variable λ x and reads 1 = P (x) 3(2x 1)2. (7.6) HS = − The constant has been set such that the distribution is normalised on [0,1]. From here we may proceed to calculate moments of the distribution,

Z 1 Z 1 1 2® 2 2 x HS xPHS(x)dx , x HS x PHS(x)dx , (7.7) 〈 〉 = 0 = 2 = 0 = 5 Z 1 Z 1 3® 3 7 4® 4 11 x HS x PHS(x)dx , x HS x PHS(x)dx , (7.8) = 0 = 20 = 0 = 35 Z 1 5® 5 2 x HS x PHS(x)dx . (7.9) = 0 = 7

M Having the above expected values we proceed to ﬁnd ensembles of points {xi }i 1 such that they satisfy the condition deﬁning the t-designs, namely that the mean value of= a particular variable coincides with the average over the entire space

M 1 X Z 1 f (xi ) f (x)PHS(x)dx (7.10) M i 1 = 0 = such that the number of points M is as small as possible. The trivial case is for the average x to be satisﬁed - we are satisﬁed with just one point with x 1/2. The second smallest arrangement of points is such that we introduce a constant 1 = a (0,1/2) and set ∈

t 1, M 2 : x 1/2 a, x 1/2 a. (7.11) = = 1 = + 2 = − Now, if we want to satisfy both x and x2® simultaneously, one point will not suf- 〈 〉HS HS ﬁce. However, using the family of solutions speciﬁed above, we can get two points with the desired average by setting a p3/20, which yields =

1 p15 1 p15 t 2, M 2 : x1 , x2 . (7.12) = = = 2 + 10 = 2 − 10

It is important to note that x 1 x , so that both points have the same value of x2 1 = − 2 i + (1 x )2 4/5, which is exactly what we need to have for radius of the sphere inside the − i =

22 Bloch ball which contains points forming a one-qubit mixed 2-design residing on a single radius within the Bloch ball. An interesting thing one may encounter is that the values of xi speciﬁed in (7.12) also satisfy the condition for the third moment. 1 P2 ¡x3¢ x3® , seemingly pointing to the 2 i 1 i = HS possibility that isoentangled 3-designs may exist= for qubits. However, they fail to do so for 4® the fourth moment x HS. Next arrangement should consist of 3 points satisfying all moments up to third moment, 3® x HS. Such an arrangement exists,

1 1 3 1 3 t 3, M 3 : x1 , x2 , x3 . (7.13) = = = 2 = 2 + 2p10 = 2 − 2p10 This choice, however, does not satisfy the condition for the 4-th moment. As the last set, 4® we provide the four numbers fulﬁlling all the averages up to x HS

1 1 q 1 1 q t 4, M 4 : x1 735 70p21, x2 735 70p21, (7.14) = = = 2 − 70 − = 2 + 70 − 1 1 q 1 1 q x3 735 70p21, x4 735 70p21. (7.15) = 2 − 70 + = 2 − 70 + and an interesting observational fact - the set satisﬁes the 5-th moment condition as well. As a ﬁnal example, there exists analytic an solution for 5 points satisfying all moment 5® conditions up to x HS, but it is not real, and as such it does not satisfy our conditions. The very same situation occurs for t M 6 and we are unable to show it holds for arbitrary = = M,t 5. ≥ It may be useful to see how the arrangements look like when plotted - see Figure 7.1:

t=5, M=4 t=3, M=3 t=3, M=2 t=1, M=1

0 1/2 1

Figure 7.1: Sets of points xi forming a t-design in the interval [0,1] with respect to the HS measure (7.6) The dots have been shifted up in order to provide a legible picture of each set, so that any vertical structure is accidental.

As a ﬁnal note, one may be interested in knowing whether any solutions exist for partic- t ® ular moment x HS and particular number of vectors M, and is it even worthwhile to look for them. By the use of Groebner basis one may check the existence of solution and the number of parameters it allows. The results of an analysis for up to t 9 and M 9 have = = been presented in the Table 7.1.

23 M 1 2 3 4 5 6 7 8 9 t 1 0 1 2 3 4 5 6 7 8 2 - 0 1 2 3 4 5 6 7 3 - 0 0 1 2 3 4 5 6 4 - - - 0 1 2 3 4 5 5 - - - 0 0 1 2 3 4 6 - - - - - 0 1 2 3 7 - - - - - 0 0 1 2 8 ------0 1 9 ------0 0

M Table 7.1: The number of parameters which are allowed for sets {xi }i 1 of M points recreat- ing all t moments on [0,1] interval with respect to either by= HS or ﬂat measure.

Finally, we provide a list of polynomials for which solutions are the points shown before. For simpliﬁcation we shift the axis and consider the symmetric interval, [ 1/2,1/2] which − assures simpler form of the polynomials.

3 t 1, M 1 : x 0, t 3, M 2 : x2 0, (7.16) = = = = = − 20 = µ 9 ¶ 3x2 51 t 3, M 3 : x x2 0, t 5, M 4 : x4 0. (7.17) = = − 40 = = = − 10 + 2800 =

24 7.3 Short detour - the ﬂat measure on the interval It is interesting to see that for any t,M 5 the minimal set of M points satisfying the condi- > tion (7.1) with respect to the HS measure (7.6) contains also complex points and thus does not yield a real design. The question arises - how a solution of the analogous problem looks like for the ﬂat measure? The answer is twofold. Firstly, we need to note that table 7.1 stays the same - we still maintain the property that any minimal arrangement for any even t is also a minimal arrangement for t 1. + On the other hand in this case we can go with t up to at least t 7 with use of non-military = grade equipment and presumably to any t we want, given we have proper computational power - see Figure 7.2. The structure becomes wider and wider, whereas the last arrangement of points for HS measure is more narrow than the previous one.

t=7, M=7 t=7, M=6 t=5, M=5 t=5, M=4 t=3, M=3 t=3, M=2 t=1, M=1

0 1/2 1

Figure 7.2: Set of M points xi forming a t-design in [0,1] with respect to the ﬂat Lebesgue measure for t 1,...,7. For larger t the structure becomes wider as the design = covers entire interval.

Let us write down analytic values of points for the ﬁrst 5 designs.

1 t 1, M 1 : x1 , (7.18) = = = 2 1 p3 1 p3 t 3, M 2 : x1 , x2 , (7.19) = = = 2 − 6 = 2 + 6 1 p2 1 1 p2 t 3, M 3 : x1 , x2 , x3 , (7.20) = = = 2 + 4 = 2 = 2 + 4 1 1 q 1 1 q t 5, M 4 : x1 75 30p5, x2 75 30p5, = = = 2 − 30 − = 2 + 30 − 1 1 q 1 1 q x3 75 30p5, x4 75 30p5, (7.21) = 2 − 30 + = 2 + 30 +

25 1 1 q 1 1 q 1 t 5, M 5 : x1 15 3p11, x2 15 3p11, x3 , = = = 2 − 12 + = 2 − 12 − = 2 1 1 q 1 1 q x4 15 3p11, x5 15 3p11. (7.22) = 2 + 12 − = 2 + 12 + The last thing to be done is to write down some characteristic polynomials that provide xi as their roots. In order to simplify the things we shift the axis and consider the interval [ 1/2,1/2]. −

1 t 1, M 1 : x 0 t 3, M 2 : x2 0 (7.23) = = = = = − 12 = µ 1¶ x2 13 t 3, M 3 : x x2 0 t 5, M 4 : x4 0 (7.24) = = − 8 = = = − 6 + 3600 = µ 5x2 7 ¶ t 5, M 5 : x x4 0 (7.25) = = − 24 + 1152 =

7.4 Interlude - comparison of polynomials For the sake of clarity, we present below a side-by-side comparison of polynomials defining the points that resemble the flat distribution and HS distribution on the line. They have been rescaled so that they correspond to the interval [ 1,1] as well as all the coefficients − have been set to be integers.

HS measure Lebesgue measure t 3, M 2 : 5x2 3 0 3x2 1 0 = = − = − = t 3, M 3 : x¡10x2 9¢ 0 x¡2x2 1¢ 0 = = − = − = t 5, M 4 : 175x4 210x2 51 0 45x4 30x2 1 0 = = − + = − + = t 5, M 5 : x¡72x4 60x2 7¢ 0 = = − + =

7.5 N 3 - Potential problems for qutrits = As for the qutrits, the HS distribution (5.1) of eigenvalues inside the simplex P3 λ 1 i 1 i = reads: =

P ¡x, y¢ 1680(x y)2(2x y 1)2(2y x 1)2. (7.26) HS = − + − + − where for simplicity we denote λ x, λ y and λ 1 x y. 1 = 2 = 3 = − − Here the tricky part seems to be that if we want to mimic the properties of t-designs, we need to retrieve all the different possible polynomials of x and y of proper order. The expected values are written down below

26 1 x y® , (7.27) 〈 〉HS = HS = 3 1 1 x2® y2® , x y® , (7.28) HS = HS = 5 HS = 15 23 1 x3® y3® , x2 y® x y2® , (7.29) HS = HS = 165 HS = HS = 33 17 1 1 x4® y4® , x3 y® x y3® , x2 y2® . (7.30) HS = HS = 165 HS = HS = 55 HS = 99 Here we shall use vector notation

µx¶ ~x . (7.31) 1 = y As previously, we may say that the simplest way to reconstruct averages (7.27) is to take one point sitting in the center of the simplex,

µ 1 ¶ 3 t 1, M 1 : ~11 1 (7.32) = = = 3 and create a two-parameter family of pairs of points

µ 1 ¶ µ 1 ¶ 3 3 t 1, M 2 : ~x1 1 ~a ~x2 1 ~a (7.33) = = = 3 + = 3 − where~a is any two reals such that~x1 and~x2 stay within the accepted range. Again, if we are to satisfy anything more, we need to fix the vectors~a to a certain value. There are two distinct choices of a vector such that the first of conditions in (7.28) is satis- fied:

2p5µ1¶ 2p5µ 1 ¶ ~a1 , ~a2 . (7.34) = 15 1 = 15 1 − However, they do not fulﬁl the second condition, yielding 1/5 and 1/45 respectively. More- over, in the ﬁrst case one of the points lives outside the allowed space. Thus, we should shift to a larger family - a triplet of vectors satisfying (7.27). They are similar to the ones presented in (7.33), but include another constant vector~b:

µ 1 ¶ µ 1 ¶ µ 1 ¶ 3 3 ~ 3 ~ t 1, M 3 : ~x1 1 ~a ~x2 1 b ~x3 1 ~a b (7.35) = = = 3 + = 3 + = 3 − −

Using the above one can derive families of one-parameter vectors satisfying (7.27) and (7.28), one of which is shown below

27  q  Ã 1 ! 1 8 2 φ 1 φ 3 3φ t 2, M 3 : ~x q + ~x − 2 15 − − 2 + 3 1 1 8 2 φ 1 2  ³p ´ = = = 3φ = 1 120 675φ2 15φ 10 − 2 15 − − 2 + 3 30 − − + q 2 Ã 1 q 8 φ 1 ! 2 3φ2 1 1 ³ ´ 1 ³ ´ 5 ~x3 2 15 − − 2 + 3 φ ( , 5 3p5 ) ( 3p5 5 , ) = φ 1 ∈ −3 30 − ∪ 30 + 3 + 3 (7.36)

It is curious that the minimal set here - again - fulfil the condition that every ~xi vector represents a point at a fixed circle inside the triangle, that is it suffers from having x2 y2 + + (1 x y)2 3/5. This configuration is precisely what we need to have when looking for a − − = set of isoentangled states in H H . 3 ⊗ 3 Note that the entire family lives on a circle of radius r p2/15 centred within the triangle = of eigenvalues. Moreover, each triplet of states forms a regular triangle, see Figure 7.3.

2 r= 15

0 1/2 1

Figure 7.3: Sets of three points forming a 2-design in the probability simplex for N 3 with = respect to the HS measure (7.26). They describe possible spectra of partial traces of states forming an isoentangled 2-design in H H H . Two points in 9 = 3 ⊗ 3 green, given in (7.34), satisfy conditions for both x2® and y2® , but fail to ® HS HS satisfy x y HS. Red-blue circle depicts a continuous family of triplets (triangles) satisfying all conditions for t 2. =

Since we already now all this, we might simplify things a notch. First, note that positions within the triangle are translated back and forth to λ eigenvalues by the relation, Ã ! Ã ! µx ¶ p3 µx¶ p3 0 0 2 2 y 1 y . (7.37) y0 = 1 y = x 2 + 2

28 Position of any of the points within a triplet can be parametrised by an angle r 2 µsinθ¶ 1µ1¶ ~r , (7.38) = 15 cosθ + 3 1 But what values can θ have? It seems easiest to consider that, once the origin is aligned with the center of the circle, the lower side of the triangle is deﬁned by the line with y0 = p3/6. Thus it is easy to see that we need to have − Ã ! r5 θ arccos . (7.39) = − 8 This number is all that we need, because the allowed angles can be depicted always as

2π θ [θ0 ∆,θ0 ∆], θ0 0, (7.40) ∈ − + = ± 3 Thus we conclude that we found the phase ∆ as ¯ Ã r ! ¯ ¯ 5 2π¯ ¯ ¯ ∆ ¯arccos ¯ (7.41) = ¯ − 8 − 3 ¯ and the state triplets have nice parametrisation, since there are always three of them with angular shifts of 0,2π/3,4π/3. The continuous family suggests signiﬁcant difﬁculties in obtaining analytic form of isoentangled MUB for 2 qutrits as we do not have any compelling hints as to how should the 30 triplets be distributed across this continuous set. Thus further research is necessary.

8 Conclusions

In this thesis we have provided a novel structure in the space of mixed states ΩN - mixed states t-designs - bearing resemblance to already known complex projective t-designs. We provided means to produce mixed states designs by partial trace of known projective t-designs containing pure states in the extended space H H and analysed selected N ⊗ N examples of such conﬁgurations. Observed regularities along with the numerical simulations provided a hint to derive an isoentangled structure of MUB in H4. Numerical simulations have been conducted in search for isoentangled MUB in H9 but no analytic results have been found yet. The algorithm provided may serve to search for isoentangled structures based on already known complex projective t-designs. Finally a short study of design-like arrangements of points on a line and in a simplex according to ﬂat and HS measures is provided, possibly the allowed Schmidt vectors of pure states in the extended space which may form an isoentangled 2-design. Some further hints on the search for the set of 10 isoentangled MUB in H9 are formulated basing on the continuous family of triplets reconstructing momenta of the HS distribution in the 3- simplex.

29 9 Open problems

1. Find a practical application of the Bloch dodecahedron - the new structure of ﬁve isoentangled MUBs in H4.

2. Find sufﬁcient condition for mixed states designs providing constructive criteria to ascertain whether a given collection of points in ΩN forms a mixed t-design or not.

3. Establish whether there exist 90 isoentangled pure states in H H which form a 3 ⊗ 3 complete set of 10 MUBs in H9. Continuous family of triplets of eigenvalues in Ω3 provided in section 7 may be useful.

4. Structures introduced in section 7 reconstruct particular properties of the space, so they provide means to conduct approximate integration over the space. Extending the list of such structures may prove to be a valuable addition to the methods of approximate integration.

30 Appendix A: Isoentangled 2-qubit MUBs

Any pure state of a qubit can be represented using spherical coordinates on a Bloch sphere:

µ θ ¶ ¯ ¡ ¢® cos 2 ¯ψ θ,φ iφ θ . (9.1) = e sin 2 Using the above parametrisation we are able to write down each pure state in H H as 2 ⊗ 2 its Schmidt decomposition:

iλ ¯ ¡ ¢® ¯ ¡ ¢® iλ ¯ ¡ ¢® ¯ ¡ ¢® Ψ e 1 a ¯ψ θ ,φ ¯ψ θ ,φ e 2 b ¯ψ˜ θ ,φ ¯ψ˜ θ ,φ , (9.2) | 〉 = A A A ⊗ B B B + A A A ⊗ B B B ¯ ® ¯ ® where ¯ψ and ¯ψ are antipodal, and constants a and b are such that the state has the proper degree of entanglement:

s s 5 p15 5 p15 a + , b − . (9.3) = 10 = 10

The coefﬁcients of the decomposition for the vectors in 1-st basis are listed in Table 9.1, with constants:

Ã ! µ 1 ³ ´¶ r1³ ´ α arctan 12 5p15 , β arccos 4 p15 , = 11 − = 8 − Ã ! Ã ! r1 r 1 γ 2arccos , ∆ arccos . (9.4) = − 3 = − 10

λ1 λ2 θA φA θB φB ¯ ® ¯ψ1 0 α 0 - 0 - ¯ ® π ¯ψ2 π β 4 γ 0 γ ∆ ¯ ® 3π − π 2π 2π ¯ψ3 2 β 4 γ 3 γ ∆ 3 ¯ ® +3π 2π − 2π ¯ψ 0 β γ γ ∆ 4 + 4 − 3 + 3 Table 9.1: Coefficients of the Schmidt decomposition as specified in equation (9.2) for four vectors of the first basis in the set of isoentangled MUBs in H4.

¯ ® Note that the ﬁrst vector ¯ψ1 is distinct from the other three and the remaining three are connected by local transformation speciﬁed by matrix

1 0 0 0 i 2π Ã ! Ã ! 0 e 3 0 0 1 0 1 0 U   2π 2π .  i 2π  i i = 0 0 e− 3 0 = 0 e 3 ⊗ 0 e− 3 0 0 0 1

31 Important fact is that the basis maintains proper entanglement solely by the fact that the components of the Schmidt decomposition are taken with coefficients’ absolute values as a and b. Although the values of parameters α, β and ∆ are obtained by „educated guesses”, the vectors are shown to be analytically orthonormal. Note that λ1 coefficients are non-zero for two of the vectors and they cannot be taken out of the picture - without these coefficients the change of basis does not provide a change from the standard MUB to an isoentangled one. Thus, in order to get the isoentangle- ment we need to introduce overall phase to each vector separately. These have not been yet shown analytically but there is strong evidence based on applying the unitary transformation induced from this basis to the whole set of MUBs that they are correct. The set can be seen visually on the Figure 9.1.

Figure 9.1: Graphical representation of analytic representation of 20 one-qubit mixed states forming representations for both reductions ρ A and ρB (for comparison put into the same Bloch ball) as defined by constants in Table 9.1 and the reconstruction formula (9.2). Configuration from subsystem A occurs to be shifted by phase ∆ with respect to the complementary configuration from reduction B.

Appendix B: Search for isoentangled MUB in H4 - code

Needs["qi ‘"]; SetDirectory[NotebookDirectory []]; StartMUB = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1/2, 1/2, 1/2, 1/2}, {1/2, (1/2) , 1/2 , (1/2)} , {1/ − − 2 , (1/2) , (1/2), 1/2}, {1/2, 1/2, (1/2) , (1/2)}, {1/2, I/2, − − − − I /2 , (1/2)}, {1/2, ( I /2) , ( I /2) , (1/2)}, {1/2, I/2, ( I /2) , − − − − − 1/2} , {1/2 , (I/2), I/2, 1/2}, {1/2, 1/2, (I/2), I/2}, {1/ − −

32 2 , (1/2), I/2, I/2}, {1/2, 1/2, I/2, ( I / 2 ) } , {1/ − − 2 , (1/2) , ( I /2) , (I/2)}, {1/2, (I/2), 1/2, I/2}, {1/2, I/ − − − − 2 , (1/2), I/2}, {1/2, I/2, 1/2, ( I / 2 ) } , {1/ − − 2 , ( I /2) , (1/2) , (I/2)}} // N; − − − MyEntanglementQuicker = Compile[{{vec, _Complex, 1}}, (Abs[vec[[1]]]^2 + Abs[vec[[3]]]^2)^2 + (Abs[vec[[2]]]^2 + Abs[vec[[4]]]^2)^2 + 2*(Conjugate[vec[[2]]] * vec [ [ 1 ] ] + Conjugate[vec[[4]]] * vec [ [ 3 ] ] ) * (Conjugate[vec[[1]]] * vec [ [ 2 ] ] + Conjugate[vec[[3]]] vec[[4]]) , Parallelization > True , * − CompilationOptions > {"ExpressionOptimization" > True } ] ; − − MyMeanDevEntanglement = Compile[{{vecset , _Complex, 2}}, MeanDeviation[MyEntanglementQuicker /@ vecset ] , Parallelization > True , − CompilationOptions > {"ExpressionOptimization" > True } ] ; − − MyRandomHermitian = Compile[{} , Module [ { a } , a = RandomReal[1 , {4 , 4 } ] + I RandomReal[1 , {4 , 4 } ] ; a + ConjugateTranspose[a]] , Parallelization > True , − CompilationOptions > {"ExpressionOptimization" > True } ] ; − − MyRandomUnitary = Compile[{{t, _Real}}, Module[{a}, a = MatrixExp[I * t *MyRandomHermitian [ ] ] ] , Parallelization > True , − CompilationOptions > {"ExpressionOptimization" > True } ] ; − − nmax = 100; tmin = 2^ 17; − imax = 50; jmax = 1000;

Monitor[Do[ spoint = MyRandomUnitary [ 1 ] ; currentmindev = MyMeanDevEntanglement[ spoint .# & /@ StartMUB ] ;

n = 0; nmax += 100; time = Timing[While[n <= nmax, t = 2^3; While[t > tmin, i = 0;

33 While[i <= imax, point = MyRandomUnitary[t ].spoint; I f [ MyMeanDevEntanglement[ point .# & /@ StartMUB] < currentmindev , spoint = point; currentmindev = MyMeanDevEntanglement[ spoint .# & /@ StartMUB ] ; i = 0 , i + + ] ] ; t = t/2;]; n++;]][[1]];

Parallelize[Save["data.dat", spoint]; Save["minimas.dat", currentmindev]; Save["times.dat", time]] , {j, 1, jmax}], {currentmindev , nmax n } ] − Appendix C: Welch bound saturation for MUB and SIC

SIC-POVM Right-hand side of (4.2) comes as a straight-forward consequence of d 2 vectors in SIC- POVM

2 Ã !2 Ã d2 ! X t X 4 R xi xi 1 d . = i 〈 | 〉 = i 1 = = Left-hand side comes out as a consequence of deﬁning relation (4.4)

µ ¶ d2 µ ¶2 d 1 X¯ ¯ ®¯4 X 1 (d 1)δi j L + ¯ xi ¯x j ¯ + − = 2 i, j = i j d Ã ¡ ¢! d(d 1) d 2 d 2 1 + d 2 − = 2 + (d 1)2 + d(d 1) 2d 3(d 1) + + d 4, = 2 (d 1)2 = + thus both sides are equal, which proves SIC-POVM to be 2-design on any space where full set exists.

MUB Let us consider the right-hand side of (4.2) for t 2 ﬁrst. Full set of MUB consists in d(d 1) = + vectors, thus it yields

Ã !2 Ãd(d 1) !2 X t X+ 2 2 R xi xi 1 d (d 1) . = i 〈 | 〉 = i 1 = + =

34 On the left-hand side we consider all the entries to the Gramm matrix raised to a proper power. when we remember expression (4.5), we see that the double sum splits into two ingredients

µ ¶ d d 1 Ã ¡ ¢!2 d 1 X¯ ¯ ®¯4 d(d 1) X X+ δµνδi j d 1 δµν L + ¯ xi ¯x j ¯ + + − = 2 i, j = 2 i,j µ,ν d d(d 1)µ d 3(d 1)¶ + d(d 1) + d 2(d 1)2. = 2 + + d 2 = + Thus we prove that whenever a complete set of d 1 MUBs in H exists, it saturates + d Welch bound and provide a projective 2-design for the space.

References

[1] A. Neumaier, “Combinatorial conﬁgurations in terms of distances,” Dept. of Mathe- matics Memorandum 81-09 (Eindhoven University of Technology, 1981).

[2] S. G. Hoggar, t-Designs in Projective Spaces, Europ. J. Combinatorics (1982) 3, 233-254, example 24

[3] A. J. Scott, Tight informationally complete quantum measurements, J. Phys. A 39, 13507 (2006)

[4] J. Renes, Equiangular spherical codes in quantum cryptography, Quantum Inf. Com- put. 5, 81 (2005).

[5] A. J. Scott, J. Walgate, B. Sanders, Optimal ﬁngerprinting strategies with one-sided er- ror, Quantum Inf. Comput. 7, 243 (2007).

[6] J. Bae, B. C. Hiesmayr, D. McNulty, Linking Entanglement Detection and State Tomog- raphy via Quantum 2-Designs, arXiv:1803.02708 [quant-ph]

[7] M. A. Graydon, D. M. Appleby, Quantum Conical Designs, J. Phys. A: Math. Theor. 49 (2016) 085301

[8] A. Roy, A. J. Scott, Unitary designs and codes, Des. Codes Cryptogr. 53, 13-31 (2009)

[9] H. Zhu, Y. S. Teo, B. G. Englert, Structure of Two-qubit Symmetric Informationally Complete POVMs, Phys. Rev. A 82, 042308 (2010)

[10] A. Belovs, Welch bounds and quantum state tomography. MSc thesis, Univer- sity of Waterloo, Canada (2008). Available at https://uwspace.uwaterloo.ca/ bitstream/handle/10012/4159/ABelovs.pdf

[11] J. M. Renes, R. Blume-Kohout, A. J. Scott, C. Caves. Symmetric informationally complete quantum measurements, J. Math. Phys 45, 2171 (2004).

35 [12] M. Grassl, On SIC-POVMs and MUBs in dimension 6, quant-ph/0406175.

[13] M. Grassl, private communication

[14] I. Bengtsson, K. Zyczkowski,˙ "Geometry of Quantum States, an introduction to quantum entanglement"

[15] A. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41,055308 (2008).

[16] G. Zauner: Quantendesigns. Grundzuge einer nichtkommutativen Designtheorie, PhD thesis ,20 Univ. Wien 1999. Available in English translation in Int. J. Quant. Inf. 9 (2011) 445.

[17] M. Appleby, T.-Y. Chien, S. Flammia, and S. Waldron, Constructing exact symmetric informationally complete measurements from numerical solutions, arXiv:1703.05981.

[18] D. M. Appleby, SIC-POVMs and the extended Clifford group, J. Math. Phys. 46, 052107 (2005).

[19] A. Scott, SICs: Extending the list of solutions, arXiv:1703.03993

[20] M. Grassl, A. Scott, Fibonacci-Lucas SIC-POVMs, J. Math. Phys. 58, 12, 122201 (2017).

[21] W. Wootters, B. Fields, Optimal State-Determination by Mutually Unbiased Measure- ments, Ann. Phys. 191, 363 (1989)

[22] S. Bandyopadhyay, P.Boykin, V. Roychowdhury et al. Algorithmica (2002) 34: 512.

[23] L. Chen, L. Yu, Mutually unbiased bases in dimension six containing a product-vector basis, arXiv:1706.00311 [quant-ph]

[24]K. Zyczkowski,˙ H. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A: Math. Gen. 37 (2004) 8457–8466

[25] J. E. Massad and P.K. Aravind, The Penrose dodecahedron revisited, American Journal of Physics 67, 631 (1999)

[26] R. Penrose: On Bell Non-locality without Probability: some Curious Geometry. Pub- lished in J.Ellis and D.Amati (Eds.), Quantum Reﬂections (Cambridge University Press 2000)

[27] M. Waegell, P. K. Aravind, The Penrose dodecahedron and the Witting polytope are identical in CP3, Physics Letters A 381, 1853-1857 (2017)

[28] MUB for 2 qutrits calculated using software provided by Wojciech Bruzda