It is known that mutually unbiased bases (MUBs), whenever they exist, are optimal in an information theoretic sense for the determination of the unknown state of a quantum ensemble. Such bases may not exist for most dimensions. The present paper deals with information gain in some generalizations and approximations of MUBs. We give estimates of the information loss (relative to MUBs) in these suboptimal choice of bases. For some generalization of MUBs we give exact calculations for the information gain. It is calculated directly in terms of transition probabilities among the measurement bases. We also give the formal solutions for the problem of quantum state tomography in these casesSome of the results in this paper was reported in the proceedings of EQIS 2005. A large part of this work was done at CQCT, Macquarie University, Sydney..
It is known that mutually unbiased bases (MUBs), whenever they exist, are optimal in an information theoretic sense for the determination of the unknown state of a quantum ensemble. Such bases may not exist for most dimensions. The present paper deals with information gain in some generalizations and approximations of MUBs. We give estimates of the information loss (relative to MUBs) in these suboptimal choice of bases. For some generalization of MUBs we give exact calculations for the information gain. It is calculated directly in terms of transition probabilities among the measurement bases. We also give the formal solutions for the problem of quantum state tomography in these casesSome of the results in this paper was reported in the proceedings of EQIS 2005. A large part of this work was done at CQCT, Macquarie University, Sydney..
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
It is known that mutually unbiased bases, whenever they exist, are optimal in an information theoretic sense for the determination of unknown state of a quantum ensemble. These bases may not exist in most dimensions and some suboptimal choices have to be made. The present paper deals with estimates of the information loss in suboptimal choice of bases. The information is calculated directly in terms of transition probabilities. I give estimates for the information content of measurement in some approximate MUBs proposed recently.
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d -dimensional Hilbert space, one can construct a complete set of mutually unbiased measurements. Besides their intrinsic link to MUB, we show that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy a non-trivial entropic uncertainty relation similar to MUB.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′ the scalar product b⋅b′ has modulus d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cd when measured in the mutually unbiased bases B_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when d is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension d. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the d-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for d even or odd number, and to recover the known result of existence of d+1 MUB's for d a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse 1/d, with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=pm using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.
Quantum measurements based on mutually unbiased bases (MUBs) play crucial roles in foundational studies and quantum information processing. It is known that there exist inequivalent MUBs, but little is known about their operational distinctions, not to say experimental demonstration. In this Letter, by virtue of a simple estimation problem, we experimentally demonstrate the operational distinctions between inequivalent triples of MUBs in dimension 4 based on high-precision photonic systems. The experimental estimation fidelities coincide well with the theoretical predictions with only 0.16% average deviation, which is 25 times less than the difference (4.1%) between the maximum estimation fidelity and the minimum estimation fidelity. Our experiments clearly demonstrate that inequivalent MUBs have different information extraction capabilities and different merits for quantum information processing.