Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden $\mathfrak{so}(4)$ generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength $2/3$. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.
Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden so(4) generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength 2/3. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.
All the distinctive features of quantum mechanics versus its classical counterpart crucially depends on the non-commutativity of observables. For instance, the existence of disjoint sets of mutually commuting operators implies quantum complementarity, in the sense that precise measurements on one set implies that possible outcomes of measurements on the other set are equally probable. The maximal number of such mutually disjoint sets (also called mutually unbiased bases) and their structure relates on concepts of finite projective geometry 1 and group theory, that we developed in the past five years 2. Among such mutually commuting sets, those sharing a base of entangled states are at the origin of paradoxes named Kochen-Specker or Bell's theorems, that recognize the contradiction between the algebraic structure of eigenvalues/ measurements and the corresponding expressions for the eigenstates 3. We investigated the zoology and geometry of all mutually unbiased bases for spaces of prime dimension 4 or composite dimension 5. Inhabitants in quantum computing world are qubits and tensor products of them, their houses are the mutually commuting sets, finite geometries are the villages and towns. I discovered a large city with 696 729 600 apartments that is, in mathematics, isomorphic to the largest crystallographic group 6, named W(E8). Groups of three inhabitants/states in the villages are in general highly connected/entangled, their contract is of the type GHZ (true tripartite union), W (three bipartite unions), CPT (a compromise), chain state (a pair can get a divorce) and so on. The population of the Pauli group Pn is 4n+1, meaning that at most n people are connected (by a tensor product). The whole unitary world is infinite, but people usually travel in a finite part of it named the Clifford group Cn. The Clifford group divides into males/females and thus into two dipolar groups denoted C±n . Transsexual people do exist in Cn, e.g. for n = 2, |P2| = 64, |C2| = 92160 and C+ 2 ∩ C−2 = 8 (as described in 7.). The group C+ 3 is the largest maximal subgroup of W(E8) and indeed the aforementioned contracts are work contracts. Following Mermin's great intuition about quantum paradoxes, I have discovered two real twoqubit matrices controling every action in Cn, the first is a braiding matrix R (of a topological character) and the second is a CPT matrix S (related to charge conjugation C, parity P and time reversal T ). The dipolar group of a male type C+ n is labeled by S and the dipolar group C−n of female type is labeled by R. The geometry of the octahedron is associated to the group R, S generated by R and S, as already anticipated in the XIXth century 8 and thoroughly developed in the context of self-dual codes 9. By the way, the CPT invariance is expressed, up to an isomorphism, by the single qubit Pauli group P1 10 and is the kernel of the threequbit representation of W(E8), the full Dirac group of gamma matrices is the two-qubit Pauli group P2 and is generated in W(E8) by matrices sustaining quantum states both of the GHZ and CPT type. Further type of three-qubit entangled states are carried by larger subgroups such as W(F4), W(H4), W(E6) and W(E7) yet to be investigated in detail, in relation to the quaternionic and octonionic representations. All these topics, and related ones, will be introduced in my series of lectures at ZIF, on August 12 to 14, 2009. 1Levay P, Saniga M and Vrana P 2008 Phys. Rev. D 78 124022. 2Planat M and Saniga M 2008 Quant. Inf. Comp. 8 127. 3Mermin N D 1993 Rev. Mod. Phys. 65 803. 4Planat M and Rosu H 2005 Eur. Phys. J. D 36 133. 5Planat M, Baboin A C and Saniga M 2008 Int. J. Theor. Phys. 47 1127. 6Planat M 2009 Preprint 0904.3691 (quant-ph). 7Planat M and Sol´e P 2008 J. Phys. A: Math. Theor. 42 8Klein F 1956 Lectures on the icosahedron and the solution of equations of the fifth degree (Dover: New York). 9Nebe G, Rains E M and Sloane N J A Self-dual codes and invariant theory (Berlin: Springer). 10Socolovsky M 2004 Int. J. Theor. Phys. 43 1941.
We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of m real mutually unbiased bases existing in dimension N>2 admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are m=k+1 mutually unbiased bases in every dimension N=k2 with k3/2 free parameters, where k is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.
In the Bloch sphere based representation of qudits with dimensions greater than two, the Heisenberg-Weyl operator basis is not preferred because of presence of complex Bloch vector components. We try to address this issue and parametrize a qutrit using the Heisenberg-Weyl operators by identifying eight real parameters and separate them as four weight and four angular parameters each. The four weight parameters correspond to the weights in front of the four mutually unbiased bases sets formed by the eigenbases of Heisenberg-Weyl observables and they form a four-dimensional unit radius Bloch hypersphere. Inside the four-dimensional hypersphere all points do not correspond to a physical qutrit state but still it has several other features which indicate that it is a natural extension of the qubit Bloch sphere. We study the purity, rank of three level systems, orthogonality and mutual unbiasedness conditions and the distance between two qutrit states inside the hypersphere. We also analyze the two and three-dimensional sections centered at the origin which gives a close structure for physical qutrit states inside the hypersphere. Significantly, we have applied our representation to find mutually unbiased bases(MUBs) and to characterize the unital maps in three dimensions. It should also be possible to extend this idea in higher dimensions.
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 = p M , 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.
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. 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. We review the known constructions of MUBs and 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}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.