Matthew Hastings

Last updated
Matthew Hastings
Alma mater Massachusetts Institute of Technology
Scientific career
Fields Physics
Mathematics
Institutions Microsoft
Duke University
Los Alamos National Laboratory

Matthew Hastings is an American physicist, currently a Principal Researcher at Microsoft. Previously, he was a professor at Duke University and a research scientist at the Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory. He received his PhD in physics at MIT, in 1997, under Leonid Levitov. [1]

Contents

While Hastings primarily works in quantum information science, he has made contributions to a range of topics in physics and related fields.

He proved an extension of the Lieb-Schultz-Mattis theorem (see Lieb-Robinson bounds) to dimensions greater than one, [2] providing foundational mathematical insights into topological quantum computing.

He disproved the additivity conjecture for the classical capacity of quantum channels, a long standing open problem in quantum Shannon theory. [3]

He and Michael Freedman formulated the NLTS conjecture, a precursor to a quantum PCP theorem (qPCP). [4]

Awards and honours

He is invited to speak at the 2022 International Congress of Mathematicians in St. Petersburg in the mathematical physics section. [5]

Publications

Related Research Articles

<span class="mw-page-title-main">Nonlinear Schrödinger equation</span> Nonlinear form of the Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in quantum mechanics, quantum entanglement is an effect by which certain widely separated events can be correlated in ways that, at first glance, suggest the possibility of communication faster-than-light. The no-communication theorem gives conditions under which such transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics, such as the EPR paradox, or violations of local realism obtained in tests of Bell's theorem. In these experiments, the no-communication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance".

<span class="mw-page-title-main">Elliott H. Lieb</span> American mathematical physicist

Elliott Hershel Lieb is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.

In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is

In theoretical condensed matter physics and quantum field theory, bosonization is a mathematical procedure by which a system of interacting fermions in (1+1) dimensions can be transformed to a system of massless, non-interacting bosons. The method of bosonization was conceived independently by particle physicists Sidney Coleman and Stanley Mandelstam; and condensed matter physicists Daniel C. Mattis and Alan Luther in 1975.

In physics, the no-broadcasting theorem is a result of quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem. The no-cloning theorem for pure states says that it is impossible to create two copies of an unknown state given a single copy of the state. Since quantum states cannot be copied in general, they cannot be broadcast. Here, the word "broadcast" is used in the sense of conveying the state to two or more recipients. For multiple recipients to each receive the state, there must be, in some sense, a way of duplicating the state. The no-broadcast theorem generalizes the no-cloning theorem for mixed states.

In quantum physics, the quantum inverse scattering method (QISM) or the algebraic Bethe ansatz is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979.

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In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. The mass gap is the spectral gap between the vacuum and the lightest particle. A Hamiltonian with a spectral gap is called a gapped Hamiltonian, and those that do not are called gapless.

In many-body physics, most commonly within condensed-matter physics, a gapped Hamiltonian is a Hamiltonian for an infinitely large many-body system where there is a finite energy gap separating the ground space from the first excited states. A Hamiltonian that is not gapped is called gapless.

In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015, Alexandre Eremenko gave a simplified proof of Stahl's theorem.

Jan Philip Solovej is a Danish mathematician and mathematical physicist working on the mathematical theory of quantum mechanics. He is a professor at University of Copenhagen.

Zohar Komargodski is an Israeli theoretical physicist who works on quantum field theory, including conformal field theories, gauge theories and supersymmetry.

In physics, stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 1967–1968, but a shorter and more conceptual proof was found later by Elliott Lieb and Walter Thirring in 1975.

In quantum information theory, the no low-energy trivial state (NLTS) conjecture is a precursor to a quantum PCP theorem (qPCP) and posits the existence of families of Hamiltonians with all low-energy states of non-trivial complexity. It was formulated by Michael Freedman and Matthew Hastings in 2013. An NLTS proof would be a consequence of one aspect of qPCP problems – the inability to certify an approximation of local Hamiltonians via NP completeness. In other words, an NLTS proof would be one consequence of the QMA complexity of qPCP problems. On a high level, if proved, NLTS would be one property of the non-Newtonian complexity of quantum computation. NLTS and qPCP conjectures posit the near-infinite complexity involved in predicting the outcome of quantum systems with many interacting states. These calculations of complexity would have implications for quantum computing such as the stability of entangled states at higher temperatures, and the occurrence of entanglement in natural systems. There is currently a proof of NLTS conjecture published in preprint.

<span class="mw-page-title-main">Nikolas Breuckmann</span> German mathematical physicist

Nikolas P. Breuckmann is a German mathematical physicist affiliated with the University of Bristol, England. His research focuses on quantum information theory, in particular quantum error correction and quantum complexity theory. He is known for his work on proving the NLTS conjecture, a famous open problem in quantum information theory.

References

  1. Hastings, Matthew B. "Curriculum Vitae" (PDF). Center for Nonlinear Studies. Los Alamos National Laboratory. Retrieved 13 June 2022.
  2. Hastings, M. B. (2004). "Lieb-Schultz-Mattis in Higher Dimensions". Phys. Rev. B. 69 (10): 104431. arXiv: cond-mat/0305505 . Bibcode:2004PhRvB..69j4431H. doi:10.1103/physrevb.69.104431. S2CID   119610203.
  3. Hastings, M. B. (2009). "A Counterexample to Additivity of Minimum Output Entropy". Nature Physics. 5: 255. arXiv: 0809.3972 . doi: 10.1038/nphys1224 .
  4. Freedman, Michael H.; Hastings, Matthew B. (January 2014). "Quantum Systems on Non-$k$-Hyperfinite Complexes: a generalization of classical statistical mechanics on expander graphs". Quantum Information and Computation. 14 (1&2): 144–180. arXiv: 1301.1363 . doi:10.26421/qic14.1-2-9. ISSN   1533-7146. S2CID   10850329.
  5. "ICM Section 11. Mathematical Physics".