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Mathematical Challenges in Quantum Information

Cambridge University

5
Followers
6
Plays
Mathematical Challenges in Quantum Information

Mathematical Challenges in Quantum Information

Cambridge University

5
Followers
6
Plays
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About Us

Quantum information is currently one of the most dynamic and exciting areas of science and technology. Its breadth of significance ranges from deep fundamental issues of the ultimate physical limits of information processing and foundations of quantum mechanics, to the technological exploitation of quantum physics for exponentially enhanced computing power and novel possibilities for communication and information security. It is a highly cross-disciplinary subject with essential inputs from computer science, information theory, mathematics, quantum physics, engineering and others. In view of the central role of information processing and communication in most aspects of modern society, government and daily life, the transformative potential of Quantum Information for 21st century technology is immense.Read more at http://www.newton.ac.uk/programmes/MQI/

Latest Episodes

Correlations, area laws, and stability of open and thermal many-body quantum systems

Investigating scaling laws of correlations and entanglement, stability and simulatability of quantum states on spin lattice systems is a central topic in Hamiltonian complexity theory. In this talk, we discuss open systems and thermal analogues of features of ground states of quantum many-body systems, using proof tools inspired by ideas of quantum information theory. For open systems, we establish a connection between mixing times - either captured by Liouvillian gaps or Log-Sobolev-constants independent of the system size - and the clustering of correlations and area laws. For Gibbs states, we prove that above a universal critical temperature only depending on local properties of the Hamiltonian's interaction hypergraph, thermal quantum states of local Hamiltonians are stable against distant Hamiltonian perturbations. As a consequence, local expectation values can be approximated in polynomial time. The stability theorem also provides a definition of temperature as a local quantit...

64 min2013 NOV 4
Comments
Correlations, area laws, and stability of open and thermal many-body quantum systems

Inequalities for the Ranks of Quantum States

We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other \alpha-R\'enyi entropies for α∈(0,1)∪(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α∈(0,1) is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of \alpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., \alp...

65 min2013 NOV 4
Comments
Inequalities for the Ranks of Quantum States
the END

Latest Episodes

Correlations, area laws, and stability of open and thermal many-body quantum systems

Investigating scaling laws of correlations and entanglement, stability and simulatability of quantum states on spin lattice systems is a central topic in Hamiltonian complexity theory. In this talk, we discuss open systems and thermal analogues of features of ground states of quantum many-body systems, using proof tools inspired by ideas of quantum information theory. For open systems, we establish a connection between mixing times - either captured by Liouvillian gaps or Log-Sobolev-constants independent of the system size - and the clustering of correlations and area laws. For Gibbs states, we prove that above a universal critical temperature only depending on local properties of the Hamiltonian's interaction hypergraph, thermal quantum states of local Hamiltonians are stable against distant Hamiltonian perturbations. As a consequence, local expectation values can be approximated in polynomial time. The stability theorem also provides a definition of temperature as a local quantit...

64 min2013 NOV 4
Comments
Correlations, area laws, and stability of open and thermal many-body quantum systems

Inequalities for the Ranks of Quantum States

We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other \alpha-R\'enyi entropies for α∈(0,1)∪(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α∈(0,1) is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of \alpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., \alp...

65 min2013 NOV 4
Comments
Inequalities for the Ranks of Quantum States
the END
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