A Concise Guide to Complex Hadamard Matrices
01 Jun 2006-Open Systems & Information Dynamics (Kluwer Academic Publishers)-Vol. 13, Iss: 2, pp 133-177
TL;DR: Basic properties of complex Hadamard matrices are reviewed and a catalogue of inequivalent cases known for the dimensions N = 2, 16, 12, 14 and 16 are presented.
Abstract: Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,..., 16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12,14 and 16, which we could not find in the existing literature.
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TL;DR: In this paper, a sixmode universal system consisting of a cascade of 15 Mach-Zehnder interferometers with 30 thermo-optic phase shifters integrated into a single photonic chip was demonstrated.
Abstract: Linear optics underpins fundamental tests of quantum mechanics and quantum technologies. We demonstrate a single reprogrammable optical circuit that is sufficient to implement all possible linear optical protocols up to the size of that circuit. Our six-mode universal system consists of a cascade of 15 Mach-Zehnder interferometers with 30 thermo-optic phase shifters integrated into a single photonic chip that is electrically and optically interfaced for arbitrary setting of all phase shifters, input of up to six photons, and their measurement with a 12-single-photon detector system. We programmed this system to implement heralded quantum logic and entangling gates, boson sampling with verification tests, and six-dimensional complex Hadamards. We implemented 100 Haar random unitaries with an average fidelity of 0.999 ± 0.001. Our system can be rapidly reprogrammed to implement these and any other linear optical protocol, pointing the way to applications across fundamental science and quantum technologies.
929 citations
DOI•
01 May 2006
TL;DR: In this article, the space of isospectral 0Hermitian matrices is shown to be the space in which the number 6) and 7) occur twice in the figure, and the discussion between eqs.(5.14) and (5.15) is incorrect.
Abstract: a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)
835 citations
Cites background from "A Concise Guide to Complex Hadamard..."
...10 For complex Hadamard matrices in general, see Tadej and Życzkowski [883], and Szöllősi [880]....
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...Geometry of Quantum States Ingemar Bengtsson and Karol Życzkowski An Introduction to Quantum Entanglement...
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TL;DR: In this paper, the authors 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, and 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.
Abstract: 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.
632 citations
TL;DR: The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.
Abstract: This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.
353 citations
Cites background from "A Concise Guide to Complex Hadamard..."
...If d = 2, 3 or 5 the only dephased Hadamards are Fourier matrices [74]; hence we can conclude that every pair of coherent COS in these dimensions is closed....
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References
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TL;DR: An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given, and optical experiments with any type of radiation exploring higher-dimensional discrete quantum systems become feasible.
Abstract: An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given. Our recursive algorithm factorizes any N\ifmmode\times\else\texttimes\fi{}N unitary matrix into a sequence of two-dimensional beam splitter transformations. The experiment is built from the corresponding devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix. Thus optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems become feasible.
1,699 citations
"A Concise Guide to Complex Hadamard..." refers methods in this paper
...For instance, Hadamard matrices (rescaled by 1/ √ N to achieve unitarity), are known in quantum optics as symmetric multiports [40, 41] (and are sometimes called Zeilinger matrices) and may be used to simulate non–local Hamiltonians [42]....
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TL;DR: Experimental data indicate that CP is nowhere maximally nonconserved and the question of maximal CP nonconservation is discussed.
Abstract: The structure of the quark mass matrices in the standard electroweak model is investigated. The commutator of the quark mass matrices is found to provide a convention-independent measure of $\mathrm{CP}$ nonconservation. The question of maximal $\mathrm{CP}$ nonconservation is discussed. The present experimental data indicate that nowhere is $\mathrm{CP}$ nonconservation maximal.
1,338 citations
"A Concise Guide to Complex Hadamard..." refers background in this paper
...However, in spite of many years of research, the problem of finding all complex Hadamard matrices of a given size N is still open [21, 22]....
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TL;DR: It is shown 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.
1,330 citations
"A Concise Guide to Complex Hadamard..." refers background in this paper
...(2) If the dimension N is prime or a power of prime the number of maximally unbiased bases (MUB) is equal to N + 1 [51, 52], but for other dimensions the answer to this question is still unknown [53, 54, 55]....
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TL;DR: In this paper, the problem of state determination is reconsidered under the assumption that every quantal measurement may give data about the post-measurement state of the inspected ensemble, and 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 state to be effected.
Abstract: 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.
688 citations
"A Concise Guide to Complex Hadamard..." refers background in this paper
...If the dimension N is prime or a power of prime the number of maximally unbiased bases (MUBs) is equal to N + 1 [51, 52], but for other dimensions the answer to this question is still unknown [53, 54, 55]....
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TL;DR: In this paper, the authors established a one-to-one correspondence between quantum teleportation, dense coding, orthonormal bases of maximally entangled vectors, and unitary operators with respect to the Hilbert-Schmidt scalar product.
Abstract: We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert–Schmidt scalar product and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be `tight' in the sense that all Hilbert spaces involved have the same finite dimension d, and the classical channel involved distinguishes d 2 signals. A general construction procedure for orthonormal bases of unitaries, involving Latin squares and complex Hadamard matrices is also presented.
390 citations
"A Concise Guide to Complex Hadamard..." refers background in this paper
...On the other hand, Hadamard matrices find numerous applications in several problems of theoretical physics....
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