Quantum Information: A Brief Overview and Some Mathematical Aspects
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paper_title: Geometry of entangled states, Bloch spheres and Hopf fibrations

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We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of highdimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S 3 .T heS 2 base space of a suitably oriented S 3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S 7 , which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.
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paper_title: An Algebraic and Geometric Approach To Non-bijective Quadratic Transformations

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This work is concerned with an extension of the so-called Levi-Civita and Kustaanheimo-Stiefel transformations. The extension is achieved along two lines. Firstly, the authors examine the latter transformations in other dimensions than the ones originally considered by, on the one hand, Levi-Civita and, on the other hand, by Kustaanheimo and Stiefel. Secondly, they pass from the compact to the non-compact case. This leads to quadratic non-bijective transformations that they refer to as Hurwitz (or Kustaanheimo-Stiefel-like) and quasiHurwitz (or Levi-Civita-like) transformations. The Hurwitz and quasiHurwitz transformations are introduced and studied in an algebraic framework which relies on the use of (eight-dimensional) Cayley-Dickson algebras. An explicit formulation of the Hurwitz and quasiHurwitz transformations is also given in terms of Clifford algebras. The Hurwitz transformations are investigated from a geometrical viewpoint. Indeed, they are connected to Hopf and 'pseudoHopf' fibrations. Finally, some differential aspects of the Hurwitz and (to a lesser extent) quasiHurwitz transformations are developed in view of future physical applications.
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paper_title: Quantum information and quantum computation

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An inexpensive faucet aerator is provided by three molded parts and a thin metal cup that holds two of the molded parts in assembled relationship. The first molded part is an elongated annulus provided with upstream and downstream recesses separated by an inner ring that helps to break up the liquid flow and serves as an abutment to support a second molded, jet-forming, part in the upstream recess and is arranged to be engaged by the third, molded, part located in the downstream recess and aiding in defining air intake means to the aerator.
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paper_title: Quantum mechanics without probability amplitudes

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First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
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paper_title: Quantum information and quantum computation

paper_content:

An inexpensive faucet aerator is provided by three molded parts and a thin metal cup that holds two of the molded parts in assembled relationship. The first molded part is an elongated annulus provided with upstream and downstream recesses separated by an inner ring that helps to break up the liquid flow and serves as an abutment to support a second molded, jet-forming, part in the upstream recess and is arranged to be engaged by the third, molded, part located in the downstream recess and aiding in defining air intake means to the aerator.
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paper_title: Quantum mechanics without probability amplitudes

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First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
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paper_title: A Wigner-function formulation of finite-state quantum mechanics

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Abstract For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N × N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.
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paper_title: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels

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An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, ``Alice,'' and the receiver, ``Bob,'' must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state \ensuremath{\Vert}\ensuremath{\varphi}〉 which Alice destroyed.
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paper_title: Mutually disjoint, maximally commuting set of physical observables for optimum state determination

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We consider the state determination problem using Mutually Unbiased Bases(MUBs). Analogous to Pauli operators of spin-1/2 system, which are experimentally implementable and correspond to the optimum measurement in characterizing the density matrix, for spin-1 and spin-3/2 systems we describe a procedure to construct an orthonormal set of operators from MUBs that are maximally commuting, can be physically realized, and correspond to physical observables. The method of construction is general enough to allow for extensions to higher-dimensional spin systems and arbitrary density matrices in finite dimensions for which MUBs are known to exist.
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paper_title: Mean king's problem with mutually unbiased bases and orthogonal Latin squares

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The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension d is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6 or d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
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paper_title: Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six

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We study the problem of constructing mutually unbiased bases in dimension 6. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family K(2)6 which we confirm analytically. Also, we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.
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paper_title: New construction of mutually unbiased bases in square dimensions

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We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (k,s)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^1/14.8 for all s but finitely many exceptions. Furthermore, our construction gives more mutually orthogonal bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.
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paper_title: Generation of all sets of mutually unbiased bases for three-qubit systems

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We propose a new method of finding the mutually unbiased bases for three qubits. The key element is the construction of the table of striation-generating curves in the discrete phase space. We derive a system of equations in the Galois field GF(8) and show that the solutions of these equations are sufficient for the construction of the general sets of complete mutually unbiased bases. A few examples are presented in order to show how our algorithm works in the cases: striation table with three, two axes, and one and no axis in the discrete phase space.
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paper_title: Mutually unbiased bases are complex projective 2-designs

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Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}
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paper_title: ANGULAR MOMENTUM AND MUTUALLY UNBIASED BASES

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The Lie algebra of the group SU2 is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J2, Ur} scheme, an alternative to the familiar {J2, Jz} quantization scheme corresponding to common eigenvectors of the Casimir operator J2 and the Cartan operator Jz. A connection is established between the eigenvectors of the complete set of commuting operators {J2, Ur} and mutually unbiased bases in spaces of constant angular momentum.
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paper_title: Graph-state formalism for mutually unbiased bases

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A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis with an arbitrary element of the other basis coincide. In case the dimension, $d$, of the considered Hilbert space is a power of a prime number, complete sets of $d+1$ mutually unbiased bases (MUBs) exist. Here, we present a novel method based on the graph-state formalism to construct such sets of MUBs. We show that for $n$ $p$-level systems, with $p$ being prime, one particular graph suffices to easily construct a set of $p^n+1$ MUBs. In fact, we show that a single $n$-dimensional vector, which is associated with this graph, can be used to generate a complete set of MUBs and demonstrate that this vector can be easily determined. Finally, we discuss some advantages of our formalism regarding the analysis of entanglement structures in MUBs, as well as experimental realizations.
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paper_title: About mutually unbiased bases in even and odd prime power dimensions

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Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well known that in prime power dimensions N = pm (with p prime and m a positive integer), there exists a maximal set of N + 1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic (1981 J. Phys. A: Math. Gen. 14 3241) in odd prime dimensions, and Wootters and Fields (1989 Ann. Phys. 191 363) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalized Pauli group in terms of the additive characters of the field, and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized Pauli group, renormalized by a well-chosen phase-factor.
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paper_title: Feynman's path integral and mutually unbiased bases

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Our previous work on quantum mechanics in Hilbert spaces of finite dimension N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N × N approximation of quantum mechanics in applied to our finite-dimensional analogue of a free quantum particle.
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paper_title: On quasi-orthogonal signatures for CDMA systems

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Sum capacity optimal signatures in synchronous code-division multiple-access (CDMA) systems are functions of the codebook length as well as the number of active users. A new signature set must be assigned every time the number of active users changes. This correspondence considers signature sets that are less sensitive to changes in the number of active users. Equiangular signature sequences are proven to solve a certain max-min signal-to-interference-plus-noise problem, which results from their interference invariance. Unions of orthonormal bases have subsets that come close to satisfying the Welch bound. Bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided. Connections are made between these signature design problems, Grassmannian line packing, frame theory, and algebraic geometry
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paper_title: ABOUT THE MEAN KING'S PROBLEM AND DISCRETE WIGNER DISTRIBUTIONS

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When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations. These Bell states are in one to one correspondence with the N2 elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function. Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N2 self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.
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paper_title: Solution to the King’s Problem in Prime Power Dimensions

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It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1 - 5 (2001)].
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paper_title: Wigner functions and separability for finite systems

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A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions, which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation, it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p = 2) and our technique fails to establish the separability property for more than two qubits. Finally, we give a brief discussion of Hamiltonian dynamics in the language developed in the paper.
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paper_title: The isotropic lines of Z 2 d

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We show that the isotropic lines in the lattice are the Lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice. The set of isotropic lines decomposes into orbits under the action of . We give an explicit description of those orbits as well as their number and respective cardinalities. We also develop two group actions on the group related to the topic.
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paper_title: Variations on a theme of Heisenberg, Pauli and Weyl

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The parentage between Weyl pairs, the generalized Pauli group and the unitary group is investigated in detail. We start from an abstract definition of the Heisenberg–Weyl group on the field and then switch to the discrete Heisenberg–Weyl group or generalized Pauli group on a finite ring . The main characteristics of the latter group, an abstract group of order d3 noted Pd, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d3 associated with Pd). Leaving the abstract sector, a set of Weyl pairs in dimension d is derived from a polar decomposition of SU(2) closely connected to angular momentum theory. Then, a realization of the generalized Pauli group Pd and the construction of generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs. Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra of the Lie algebra associated with Pd. This leads to a development of the Lie algebra of U(d) in a basis consisting of d2 generalized Pauli matrices. In the case where d is a power of a prime integer, the Lie algebra of SU(d) can be decomposed into d − 1 Cartan subalgebras.
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paper_title: Mutually unbiased bases and trinary operator sets for N qutrits

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A compete orthonormal basis of N-qutrit unitary operators drawn from the Pauli group consists of the identity and 9{sup N}-1 traceless operators. The traceless ones partition into 3{sup N}+1 maximally commuting subsets (MCS's) of 3{sup N}-1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3{sup N} are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of 16 GHZ-like bases must occur. Every basis state is described by an N-digit trinary number consisting of the eigenvalues of N observables constructed from the corresponding MCS.
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paper_title: A New Proof for the Existence of Mutually Unbiased Bases

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We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions which are power of a prime is presented. It is also proved that in any dimension d the number of mutually unbiased bases is at most d+1. An explicit representation of mutually unbiased observables in terms of Pauli matrices are provided for d=2^m.
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paper_title: Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions

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Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of N+1 mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of SL(2,Z_N) on finite phase space Z_N x Z_N implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime N, Z_N is also a finite field.
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paper_title: Mutually unbiased bases and orthogonal decompositions of Lie algebras

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We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. ::: It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.
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paper_title: Geometrical approach to mutually unbiased bases

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We propose a unifying phase-space approach to the construction of mutually unbiased bases for a two-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the Bloch-sphere principal axes. We suggest how to generalize the method to systems in dimensions that are powers of a prime.
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paper_title: An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

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The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
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paper_title: On mutually unbiased bases

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Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = pM, with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.
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paper_title: Constructions of Mutually Unbiased Bases

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Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if | |^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.
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paper_title: Optimal quantum state reconstruction for cold trapped ions

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We study the physical implementation of an optimal tomographic reconstruction scheme for the case of determining the state of a multi-qubit system, where trapped ions are used for defining qubits. The protocol is based on the use of mutually unbiased measurements and on the physical information described in H. H\"{a}ffner \emph{et. al} [Nature \textbf{438}, 643-646 (2005)]. We introduce the concept of physical complexity for different types of unbiased measurements and analyze their generation in terms of one and two qubit gates for trapped ions.
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paper_title: Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem

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We analyze and further develop a method to represent the quantum state of a system of n qubits in a phase-space grid of NxN points (where N=2{sup n}). The method, which was recently proposed by Wootters and co-workers (Gibbons et al., Phys. Rev. A 70, 062101 (2004).), is based on the use of the elements of the finite field GF(2{sup n}) to label the phase-space axes. We present a self-contained overview of the method, we give insights into some of its features, and we apply it to investigate problems which are of interest for quantum-information theory: We analyze the phase-space representation of stabilizer states and quantum error-correction codes and present a phase-space solution to the so-called mean king problem.
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paper_title: A Wigner-function formulation of finite-state quantum mechanics

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Abstract For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N × N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.
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paper_title: QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY

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This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (S...
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paper_title: Optimal state-determination by mutually unbiased measurements

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For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called “mutual unbiasedness,” which we show plays a special role in the problem of state determination. We define two bases {|vi〉} and {|wj〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν1|wj|=1√N for all i, j. Two non-degenerate measurements are defined to be mutually unbiased if the bases comprising their eigenstates are mutually unbiased. We show that if one can find N + 1 mutually unbiased bases for a complex vector space of N dimensions, then the measurements corresponding to these bases provide an optimal means of determining the density matrix of an ensemble of systems having N orthogonal states, in the sense that the effects of statistical error are minimized. We show further that the number of mutually unbiased bases one may find for a given N is at most N + 1. Finally, we show that N + 1 mutually unbiased bases do exist whenever N is a power of a prime, and we construct such bases explicitly.
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paper_title: Phase operators, phase states and vector phase states for SU(3) and SU(2,1)

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This paper focuses on phase operators, phase states, and vector phase states for the sl3 Lie algebra. We introduce a one-parameter generalized oscillator algebra Aκ(2) which provides a unified scheme for dealing with su3 (for κ 0), and h4⊗h4 (for κ = 0) symmetries. Finite- and infinite-dimensional representations of Aκ(2) are constructed for κ < 0 and κ ⩾ 0, respectively. Phase operators associated with Aκ(2) are defined and temporally stable phase states (as well as vector phase states) are constructed as eigenstates of these operators. Finally, we discuss a relation between quantized phase states and a quadratic discrete Fourier transform and show how to use these states for constructing mutually unbiased bases.
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paper_title: Z4-KERDOCK CODES, ORTHOGONAL SPREADS, AND EXTREMAL EUCLIDEAN LINE-SETS

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When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.
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paper_title: The mean king ’ s problem : Prime degrees of freedom

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We show how one can ascertain the values of a complete set of mutually complementary observables of a prime degree of freedom.
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paper_title: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels

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An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, ``Alice,'' and the receiver, ``Bob,'' must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state \ensuremath{\Vert}\ensuremath{\varphi}〉 which Alice destroyed.
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paper_title: Tomography of quantum states in small dimensions

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Abstract We consider the problem of determining the state of a finite dimensional quantum system by a finite set of different measurements in an optimal way. The measurements can either be projective von Neumann measurements or generalized measurements (POVMs). While optimal solutions for projective measurements are only known for prime power dimensions, based on numerical solutions it is conjectured that solutions for POVMs exist in any dimension. We support this conjecture by constructing explicit algebraic solutions in small dimensions d , in particular d = 12.
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paper_title: Discrete Phase Space Based on Finite Fields

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The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2Nx2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our NxN phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.
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paper_title: Feynman's Integral is About Mutually Unbiased Bases

paper_content:

The Feynman integral can be seen as an attempt to relate, under certain circumstances, the quantum-information-theoretic separateness of mutually unbiased bases to causal proximity of the measuring processes.
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paper_title: Aspects of mutually unbiased bases in odd-prime-power dimensions

paper_content:

We rephrase the Wootters-Fields construction [W. K. Wootters and B. C. Fields, Ann. Phys. 191, 363 (1989)] of a full set of mutually unbiased bases in a complex vector space of dimensions N=p r , where p is an odd prime, in terms of the character vectors of the cyclic group G of order p. This form may be useful in explicitly writing down mutually unbiased bases for N=p r .
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paper_title: Mutually unbiased bases and discrete Wigner functions

paper_content:

Mutually unbiased bases and discrete Wigner functions are closely, but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime $N=d^n$, which describes a composite system of $n$ qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension $N=8=2^3$. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators consist of rotations around two orthogonal axes, acting on the individual qubits only.
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paper_title: Quantum cryptography: Public key distribution and coin tossing

paper_content:

When elementary quantum systems, such as polarized photons, are used to transmit digital information, the uncertainty principle gives rise to novel cryptographic phenomena unachievable with traditional transmission media, e.g. a communications channel on which it is impossible in principle to eavesdrop without a high probability of disturbing the transmission in such a way as to be detected. Such a quantum channel can be used in conjunction with ordinary insecure classical channels to distribute random key information between two users with the assurance that it remains unknown to anyone else, even when the users share no secret information initially. We also present a protocol for coin-tossing by exchange of quantum messages, which is secure against traditional kinds of cheating, even by an opponent with unlimited computing power, but ironically can be subverted by use of a still subtler quantum phenomenon, the Einstein-Podolsky-Rosen paradox.
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paper_title: Geometrical description of quantal state determination

paper_content:

Under the assumption that every quantal measurement may give data about the post-measurement state of the inspected ensemble, the problem of the state determination is reconsidered. It is shown that orthogonal decomposition of the set of complex, n*n, Hermitian matrices into the commutative subsets allows operators to be found such that post-measurement information on these observables allows a partial (in some cases total) determination of the pre-measurement state to be effected.
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paper_title: Quantum mechanics without probability amplitudes

paper_content:

First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
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paper_title: A Wigner-function formulation of finite-state quantum mechanics

paper_content:

Abstract For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N × N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.
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paper_title: Galois quantum systems, irreducible polynomials and Riemann surfaces

paper_content:

Finite quantum systems in which the position and momentum take values in the Galois field GF(pl), are studied. Ideas from the subject of field extension are transferred in the context of quantum mechanics. The Frobenius automorphisms in Galois fields lead naturally to the “Frobenius formalism” in a quantum context. The Hilbert space splits into “Frobenius subspaces” which are labeled with the irreducible polynomials associated with the ypl−y. The Frobenius maps transform unitarily the states of a Galois quantum system and leave fixed all states in some of its Galois subsystems (where the position and momentum take values in subfields of GF(pl)). An analytic representation of these systems in the l-sheeted complex plane shows deeper links between Galois theory and Riemann surfaces.
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paper_title: Quantum systems with finite Hilbert space

paper_content:

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.
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paper_title: Optimal state-determination by mutually unbiased measurements

paper_content:

For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called “mutual unbiasedness,” which we show plays a special role in the problem of state determination. We define two bases {|vi〉} and {|wj〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν1|wj|=1√N for all i, j. Two non-degenerate measurements are defined to be mutually unbiased if the bases comprising their eigenstates are mutually unbiased. We show that if one can find N + 1 mutually unbiased bases for a complex vector space of N dimensions, then the measurements corresponding to these bases provide an optimal means of determining the density matrix of an ensemble of systems having N orthogonal states, in the sense that the effects of statistical error are minimized. We show further that the number of mutually unbiased bases one may find for a given N is at most N + 1. Finally, we show that N + 1 mutually unbiased bases do exist whenever N is a power of a prime, and we construct such bases explicitly.
---
paper_title: Geometrical description of quantal state determination

paper_content:

Under the assumption that every quantal measurement may give data about the post-measurement state of the inspected ensemble, the problem of the state determination is reconsidered. It is shown that orthogonal decomposition of the set of complex, n*n, Hermitian matrices into the commutative subsets allows operators to be found such that post-measurement information on these observables allows a partial (in some cases total) determination of the pre-measurement state to be effected.
---
paper_title: An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

paper_content:

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
---
paper_title: Orthogonal Decompositions and Integral Lattices

paper_content:

Provides an exposition of orthoganal decompositions of complex simple Lie algebras and associated invariant Euclidean lattices. The text is divided into two parts. In Part I, the main result is the complete classification of complex simple Lie algebras admitting orthogonal decompositions with irredu
---
paper_title: Orthogonal Decompositions and Integral Lattices

paper_content:

Provides an exposition of orthoganal decompositions of complex simple Lie algebras and associated invariant Euclidean lattices. The text is divided into two parts. In Part I, the main result is the complete classification of complex simple Lie algebras admitting orthogonal decompositions with irredu
---
paper_title: Orthogonal Decompositions and Integral Lattices

paper_content:

Provides an exposition of orthoganal decompositions of complex simple Lie algebras and associated invariant Euclidean lattices. The text is divided into two parts. In Part I, the main result is the complete classification of complex simple Lie algebras admitting orthogonal decompositions with irredu
---
paper_title: The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type An−1

paper_content:

Properties of the Lie algebra gl(n,C) are described for a basis which is a generalization of the 2×2 Pauli matrices. The 3×3 case is described in detail. The remarkable properties of that basis are the grading of the Lie algebra it offers (each grading subspace is one dimensional) and the matrix group it generates [it is a finite group with the center of SL(n,C) as its commutator group].
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paper_title: Mutually unbiased bases and orthogonal decompositions of Lie algebras

paper_content:

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. ::: It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.
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paper_title: An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

paper_content:

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
---
paper_title: Orthogonal Decompositions and Integral Lattices

paper_content:

Provides an exposition of orthoganal decompositions of complex simple Lie algebras and associated invariant Euclidean lattices. The text is divided into two parts. In Part I, the main result is the complete classification of complex simple Lie algebras admitting orthogonal decompositions with irredu
---
paper_title: Constructions of Mutually Unbiased Bases

paper_content:

Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if | |^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.
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paper_title: Z4-KERDOCK CODES, ORTHOGONAL SPREADS, AND EXTREMAL EUCLIDEAN LINE-SETS

paper_content:

When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.
---
paper_title: Orthogonal Decompositions and Integral Lattices

paper_content:

Provides an exposition of orthoganal decompositions of complex simple Lie algebras and associated invariant Euclidean lattices. The text is divided into two parts. In Part I, the main result is the complete classification of complex simple Lie algebras admitting orthogonal decompositions with irredu
---

Title: Quantum Information: A Brief Overview and Some Mathematical Aspects

Section 1: Introduction
Description 1: Provide an overview of the emergence and significance of the field of quantum information and quantum computing, including historical context and fundamental principles.

Section 2: Qubits
Description 2: Define and explain the concept of qubits, their mathematical representation, properties, and significance in quantum computing.

Section 3: Qudits
Description 3: Discuss the generalization of qubits to qudits, including their mathematical framework and relevance to higher-dimensional quantum systems.

Section 4: Physical Realizations of Qubits
Description 4: Describe various physical systems that can be used to realize qubits, highlighting different technologies and their implementations.

Section 5: Generalities
Description 5: Introduce the basic notions of entanglement and its significance in quantum information theory, contrasting it with classical physics.

Section 6: Entanglement of Qubits
Description 6: Explore the concept of entanglement in multi-qubit systems, providing examples and discussing its implications for quantum information.

Section 7: One-Qubit Gates
Description 7: Explain what one-qubit gates are, their mathematical representations, and their role in quantum computing.

Section 8: Quantum Computing Algorithms
Description 8: Provide an overview of key quantum computing algorithms, including the Deutsch-Jozsa algorithm, the Shor algorithm, and the Grover algorithm.

Section 9: No-Cloning Theorem
Description 9: Discuss the no-cloning theorem, its proof, and its implications for quantum information theory.

Section 10: Quantum Teleportation
Description 10: Describe the process and significance of quantum teleportation, including the necessary conditions and the role of entanglement.

Section 11: Generalities about MUBs
Description 11: Introduce Mutually Unbiased Bases (MUBs), their importance, and applications in quantum information theory.

Section 12: Definition of MUBs
Description 12: Provide a formal definition of MUBs and explain their mathematical properties.

Section 13: Well-Known Results about MUBs
Description 13: Summarize known results and findings related to MUBs and their applications.

Section 14: Interests of MUBs
Description 14: Discuss the relevance and significance of MUBs in advanced quantum mechanics and quantum measurements.

Section 15: Standard Basis for SU(2)
Description 15: Explain the standard basis for the SU(2) group and its significance in the context of quantum information.

Section 16: Nonstandard Bases for SU(2)
Description 16: Introduce and explain nonstandard bases for SU(2) and their relevance.

Section 17: Bases in Quantum Information
Description 17: Discuss how different bases are used in quantum information theory and their practical applications.

Section 18: Shift and Phase Operators
Description 18: Describe shift and phase operators, their mathematical formulation, and importance in quantum computations.

Section 19: Generalized Pauli Matrices
Description 19: Discuss generalized Pauli matrices, their definitions, properties, and applications in quantum information theory.

Section 20: MUBs and the Special Linear Group
Description 20: Explore the relationship between MUBs and the special linear group, including mathematical insights.

Section 21: Galois Field Approach to MUBs
Description 21: Introduce the Galois Field approach for constructing MUBs, including mathematical formulations and proofs.

Section 22: Computational Basis
Description 22: Define the computational basis in the context of Galois Fields and its application in quantum computing.

Section 23: Shift and Phase Operators for GF(p^m)
Description 23: Explain shift and phase operators within Galois Fields and their relevance to quantum information theory.

Section 24: Bases in the Frame of GF(p^m)
Description 24: Discuss the construction of bases using Galois Fields and their importance in forming MUBs.

Section 25: Galois Ring Approach to MUBs
Description 25: Introduce the Galois Ring approach to constructing MUBs in certain dimensions and its mathematical implications.

Section 26: One-Qubit System
Description 26: Discuss the specifics of one-qubit systems, including their mathematical representation and basis construction.