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Geometry of Quantum States from Symmetric Informationally Complete Probabilities

Gelo Noel M. Tabia

Published Aug 6, 2013• 3 citations
OpenAlex

Abstract

It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.

Related papers

Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

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We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.

arXiv:1304.8075•Physical Review A•Jul 31, 2013•25 citations

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Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.

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Some people hate the very name of statistics, but I find them full of beauty and interest. Sir Francis Galton In quantum mechanics one often encounters sets of non-negative numbers that sum to unity, having a more or less direct interpretation as probabilities. This includes the squared moduli of the coefficients when a pure state is expanded in an orthonormal basis, the eigenvalues of density matrices, and more. Continuous distributions also play a role, even when the Hilbert space is finite dimensional. From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. In classical mechanics the sample space is phase space, which is typically a continuous space. This leads to technical complications but the space of states in classical mechanics does share a major simplifying feature with the discrete case, namely that every state can be expressed as a mixture of pure states in a unique way. This was not so in the case of colour space, nor will it be true for the convex set of all states in quantum mechanics. Majorization and partial order Our first aim is to find ways of describing probability distributions; we want to be able to tell when a probability distribution is ‘more chaotic’ or ‘more uniform’ than another. One way of doing this is provided by the theory of majorization .

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Physicists have become accustomed to the idea that a theory’s content is most transparent when written in coordinate‐free language. But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues. This is particularly so for the Quantum Bayesian, or QBist, approach to quantum theory: One good coordinate system may be worth more than a hundred blue‐in‐the‐face arguments over the meaning of quantum states. The philosophy here is that if quantum theory really is a theory about how to structure one’s degrees of belief, rather than a theory of ontic quantum states, then the theory should become conceptually clearest when written in its native terms—i.e., in a purely probabilistic language, with no quantum states at all as intermediaries. This paper will chronicle the benefits of one particularly nice class of “coordinate systems” (i.e., operator bases) for enacting a map from quantum states on a finite dimensional Hilbert space to an appropriately chosen finite dimensional probability simplex. This is the so‐called Symmetric Informationally Complete measurements or SICs.

AIP conference proceedings•Jan 1, 2011•2 citations
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.
Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schrödinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15—to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS , might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms—the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and applications of the geometric approach. The first four chapters contain the standard mathematics required to understand the rest of the material presented: specific areas in colour theory, set theory, probability theory, differential geometry and projective geometry are all covered with an eye to the material that follows. Chapter 5 starts the first real discussion of quantum theory in GQS and serves as an elegant, succinct introduction to the geometry which underlies quantum theory. This may be the most worthwhile chapter for the casual reader who wants to understand the key ideas in this field. Chapter 6 builds on the discussion in Chapter 5, introducing a group theoretic approach to understand coherent states and Chapter 7 describes a geometric tool in the form of an approach to complex projective geometry called 'the stellar representation'. Chapter 8 returns to a more purely quantum mechanical discussion as the authors turn to study the space of density matrices. This chapter completes the discussion which started in Chapter 5. Chapter 9 begins the part of the book concerned with applications of the geometric approach. From this point on the book aims, specifically, to prepare the reader for the material in Chapter 15 beginning with a discussion on the purification of mixed quantum states. In the succeeding chapters a definite choice has been made to present a geometric approach to certain quantum information problems. For example, Chapter 10 contains an extremely well formulated discussion of measurement and positive operator-valued measures with several well illustrated examples and Chapter 11 reopens the discussion of density matrices. Entropy and majorization are again revisited in Chapter 12 in much greater detail than in previous chapters. Chapters 13 and 14 concern themselves with a discussion of various metrics and their relation to the problem of distinguishing between probability distributions and their suitability as probability measures.
In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\\(^n\\) and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\\(^{D-1}\\). Further-more, the fibre bundle S\\(^{2D-1}\\) ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\\(^{D-1}\\) is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."
Some people hate the very name of statistics, but I find them full of beauty and interest. Sir Francis Galton In quantum mechanics one often encounters sets of non-negative numbers that sum to unity, having a more or less direct interpretation as probabilities. This includes the squared moduli of the coefficients when a pure state is expanded in an orthonormal basis, the eigenvalues of density matrices, and more. Continuous distributions also play a role, even when the Hilbert space is finite dimensional. From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. In classical mechanics the sample space is phase space, which is typically a continuous space. This leads to technical complications but the space of states in classical mechanics does share a major simplifying feature with the discrete case, namely that every state can be expressed as a mixture of pure states in a unique way. This was not so in the case of colour space, nor will it be true for the convex set of all states in quantum mechanics. Majorization and partial order Our first aim is to find ways of describing probability distributions; we want to be able to tell when a probability distribution is ‘more chaotic’ or ‘more uniform’ than another. One way of doing this is provided by the theory of majorization .
Physicists have become accustomed to the idea that a theory’s content is most transparent when written in coordinate‐free language. But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues. This is particularly so for the Quantum Bayesian, or QBist, approach to quantum theory: One good coordinate system may be worth more than a hundred blue‐in‐the‐face arguments over the meaning of quantum states. The philosophy here is that if quantum theory really is a theory about how to structure one’s degrees of belief, rather than a theory of ontic quantum states, then the theory should become conceptually clearest when written in its native terms—i.e., in a purely probabilistic language, with no quantum states at all as intermediaries. This paper will chronicle the benefits of one particularly nice class of “coordinate systems” (i.e., operator bases) for enacting a map from quantum states on a finite dimensional Hilbert space to an appropriately chosen finite dimensional probability simplex. This is the so‐called Symmetric Informationally Complete measurements or SICs.