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Painless Reconstruction from Magnitudes of Frame Coefficients
 J FOURIER ANAL APPL (2009) 15: 488–501
, 2009
"... The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present line ..."
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Cited by 49 (10 self)
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The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2designs in finitedimensional real or complex Hilbert spaces. Examples of such frames are twouniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a ddimensional Hilbert space.
Constructions of Mutually Unbiased Bases
 in Proc. 7th Int. Conf. on finite fields and applications, Lecture
"... Abstract. Two orthonormal bases B and B ′ of a ddimensional complex innerproduct space are called mutually unbiased if and only if 〈bb ′ 〉  2 = 1/d holds for all b ∈ B and b ′ ∈ B ′. The size of any set containing pairwise mutually unbiased bases of C d cannot exceed d + 1. If d is a power of ..."
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Cited by 32 (1 self)
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Abstract. Two orthonormal bases B and B ′ of a ddimensional complex innerproduct space are called mutually unbiased if and only if 〈bb ′ 〉  2 = 1/d holds for all b ∈ B and b ′ ∈ 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.
Mutually Unbiased Bases are Complex Projective 2Designs
 PROC. 2005 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY
, 2005
"... 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 nonpri ..."
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Cited by 31 (0 self)
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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 nonprime 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 2designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2designs with angle set {1/(d+1)}.
On SICPOVMs and MUBs in dimension 6
 Proc. ERATO Conf. on Quant. Inf. Science (EQUIS 2004
, 2004
"... We provide a partial solution to the problem of constructing mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SICPOVMs) in nonprimepower dimensions. An algebraic description of a SICPOVM in dimension six is given. Furthermore it is shown that several sets of three mutu ..."
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Cited by 25 (2 self)
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We provide a partial solution to the problem of constructing mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SICPOVMs) in nonprimepower dimensions. An algebraic description of a SICPOVM in dimension six is given. Furthermore it is shown that several sets of three mutually unbiased bases in dimension six are maximal, i.e., cannot be extended. Keywords: Mutually unbiased bases, SICPOVMs, finite geometry 1
Reconstruction of signals from magnitudes of redundant representations
"... Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm ..."
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Cited by 12 (6 self)
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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm that finds the leastsquare solution and is robust in the presence of noise. We analyze its numerical performance by comparing it to two versions of the CramerRao lower bound. 1.
Complementary reductions for two qubits
 J. MATH. PHYS. 48, 012107, 2007.
, 2007
"... Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of ..."
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Cited by 11 (4 self)
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Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A1; A2; : : : ; Ak are pairwise complementary (or quasiorthogonal) subalgebras of the algebra M4(C) of all 4 4 matrices and they are isomorphic to M2(C), then k 4. The proof is based on a Cartan decomposition of SU(4). In the way to the main result, contributions are made to the understanding of the structure of complementary reductions.
On estimating the state of a finite level quantum system
 Infinite Dimensional Analysis, Quantum Probability and Related Topics
"... Summary: We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a dlevel quantum system, first studied by W. K. Wootters and B. D. fields[7] in 1989 and later investigated by S. Bandyopadhyay et al [3] in 2001 and A. O. Pittenger and M. H. Rubin [ ..."
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Summary: We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a dlevel quantum system, first studied by W. K. Wootters and B. D. fields[7] in 1989 and later investigated by S. Bandyopadhyay et al [3] in 2001 and A. O. Pittenger and M. H. Rubin [6] in 2003. Our approach is based directly on the Weyl operators in the L 2space over a finite field when d = p r is the power of a prime. When d is not a prime power we sacrifice a bit of optimality and construct a recovery operator for reconstructing the unknown state from the probabilities of elementary events in different measurements.
Frames for Linear Reconstruction without Phase
"... Abstract — The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communication ..."
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Cited by 7 (1 self)
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Abstract — The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiberoptical transmissions. The algorithms discussed here rely on suitable rankone operator valued frames defined on finitedimensional real or complex Hilbert spaces. Examples of such frames are the rankone Hermitian operators associated with vectors from maximal sets of equiangular lines and maximal sets of mutually unbiased bases. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct up to one lost frame coefficient occurring in each basis except for one without loss. I.
State discrimination with postmeasurement information
 IEEE Transactions on Information Theory
"... We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. In particular, the following special case plays an important role in quantum cryptographic protocols in the bou ..."
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Cited by 5 (2 self)
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We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. In particular, the following special case plays an important role in quantum cryptographic protocols in the bounded storage model: Given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases, you may perform any measurement, but then store at most q qubits of quantum information. Later on, you learn which basis was used. How well can you compute a function f(x) of x, given the initial measurement outcome, the q qubits and the additional basis information? We first show a lower bound on the success probability for any balanced function, and any number of mutually unbiased bases, beating the naive strategy of simply guessing the basis. We then show that for two bases, any Boolean function f(x) can be computed perfectly if you are allowed to store just a single qubit, independent of the number of possible input strings x. We then investigate how much advantage the additional basis information can give for a Boolean function. To this end, we prove optimal bounds for the success probability for the AND and the XOR function for up to three mutually unbiased