In this thesis for the degree of Master of Science from Stockholm University we explore the ideas of Symmetric Informationally Complete Positive Valued Measures (SIC-POVMs; commonly just SICs). This is an emerging concept in quantum information theory with ambitious claims, such as being a candidate for standard measurements [23] and perhaps being of importance to error correcting universal quantum computing [32]. While the definition of a SIC is exceedingly simple they have proven notoriously hard to find. This thesis explores new approaches to finding SICs. It is our ambition that this thesis shall provide the reader unfamiliar with SICs with a thorough introduction to the subject along with both the necessary quantum theory and group theory. We also hope to intrigue the reader already attuned to SICs by establishing a link between how close to a SIC a state is and how close to a MUS (Minimum Uncertainty State) it is. This is the main result of this thesis and we leave the reader with several open questions relating to this discovery to provoke further scrutiny of the matter. The thesis is divided into two parts: the first part provides the necessary background and theory; while the second part presents our results. There are also three appendices attached to this thesis where we delve into a discussion about computing power and also present some of the code used. Being appendices these are not essential to the thesis per se – they are rather supplied as a reference for the curious reader who might be interested in recreating some of our results.
In this thesis for the degree of Master of Science from Stockholm University we explore the ideas of Symmetric Informationally Complete Positive Valued Measures (SIC-POVMs; commonly just SICs). This is an emerging concept in quantum information theory with ambitious claims, such as being a candidate for standard measurements [23] and perhaps being of importance to error correcting universal quantum computing [32]. While the definition of a SIC is exceedingly simple they have proven notoriously hard to find. This thesis explores new approaches to finding SICs. It is our ambition that this thesis shall provide the reader unfamiliar with SICs with a thorough introduction to the subject along with both the necessary quantum theory and group theory. We also hope to intrigue the reader already attuned to SICs by establishing a link between how close to a SIC a state is and how close to a MUS (Minimum Uncertainty State) it is. This is the main result of this thesis and we leave the reader with several open questions relating to this discovery to provoke further scrutiny of the matter. The thesis is divided into two parts: the first part provides the necessary background and theory; while the second part presents our results. There are also three appendices attached to this thesis where we delve into a discussion about computing power and also present some of the code used. Being appendices these are not essential to the thesis per se – they are rather supplied as a reference for the curious reader who might be interested in recreating some of our results.
In this work, we generalise an object that has proved interesting to both the pure and applied mathematician alike. Specifically, symmetric informationally complete positive operator valued measures (SIC-POVMs) were first encountered in the study finite tight frames, where they provide examples of complex equiangular tight frames. Then, the quantum information theory community discovered their own utility for these objects, for they provided a particularly useful example of a generalised measurement of a quantum system, a type of positive operator valued measure. In particular, they have proven useful to quantum state tomography and for probing quantum foundations. SIC-POVMs are collections of rank-one orthogonal projections acting on finite-dimensional Hilbert spaces and operational SIC-POVMs are their arbitrary rank counterparts. In this thesis, we cover necessary background on the standard quantum formalism, define SIC-POVMs and outline how they have been constructed up until this point via the orbit of the Weyl-Heisenberg group, which motivates Zauner’s conjecture. We go on to discuss generalised operational SIC-POVMs and give some new theorems that classify them. We then show that an infinite family of operational SIC-POVMs can be obtained via the representation theory of the symmetric group. Possibly shedding light on the unproven Zauner’s conjecture, we prove that an element of this infinite family cannot be generated via an orbit of the Weyl-Heisenberg group. We finish by describing the relation between these operational SIC-POVMs and Bell inequalities.
Although symmetric informationally complete positive operator valued measures (SIC POVMs, or SICs for short) have been constructed in every dimension up to 67, a general existence proof remains elusive. The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems. Although it is still not clear whether these reformulations of the problem will make it more tractable, we believe that the fact that SICs have these connections to other areas of mathematics is of some intrinsic interest. Specifically, we reformulate the SIC problem in terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result being a greatly strengthened version of one previously obtained by Appleby, Flammia and Fuchs). The connection between these three reformulations is non-trivial: It is not easy to demonstrate their equivalence directly, without appealing to their common equivalence to SIC existence. In the course of our analysis we obtain a number of other results which may be of some independent interest.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
In the stabilizer formalism of quantum computation, the Gottesman–Knill theorem shows that universal fault-tolerant quantum computation requires the resource called magic (nonstabilizerness). Thus stabilizer states serve as “classical states,” and states beyond them are necessary for genuine quantum computation. Characterization, detection, and quantification of magic states are basic issues in this context. In the paradigm of quantum measurement, symmetric informationally complete positive operator valued measures (SIC-POVMs, further abbreviated as SICs) play a prominent role due to their structural symmetry and remarkable features. However, their existence in all dimensions, although strongly supported by extensive theoretical and numerical evidence, remains an elusive open problem (Zauner’s conjecture). A standard method for constructing SICs is via the orbit of the Heisenberg–Weyl group on a fiducial state, and most known SICs arise in this way. A natural question arises regarding the relation between stabilizer states and fiducial states. In this paper, we connect them by showing that they are on two extremes with respect to the p -norms of characteristic functions of quantum states. This not only reveals a simple path from stabilizer states to SIC fiducial states, showing quantitatively that they are as far away as possible from each other, but also provides a simple reformulation of Zauner’s conjecture in terms of extremals for the p -norms of characteristic functions. A convenient criterion for magic states and some interesting open problems are also presented.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such n2 vectors in Cn for any n .But the existence is only proved when n≤15 and n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 ρ∈Mn(C) is called a density matrix (or state) if ρ≥0 and ′b(ρ)=1.For any density matrix ρ , we can define a state ρ^ by ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t (≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators {Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if Pi≥0(1≤i≤k) and ∑i=1kP=I.If P1 is a projection, then {Pi}i=1k is called a projection valued measure (PVM) .
Although symmetric informationally complete positive operator valued measures (SIC POVMs, or SICs for short) have been constructed in every dimension up to 67, a general existence proof remains elusive. The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems. Although it is still not clear whether these reformulations of the problem will make it more tractable, we believe that the fact that SICs have these connections to other areas of mathematics is of some intrinsic interest. Specifically, we reformulate the SIC problem in terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result being a greatly strengthened version of one previously obtained by Appleby, Flammia and Fuchs). The connection between these three reformulations is non-trivial: It is not easy to demonstrate their equivalence directly, without appealing to their common equivalence to SIC existence. In the course of our analysis we obtain a number of other results which may be of some independent interest.