Norms and Cones in the Theory of Quantum Entanglement
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as “more” or “less” positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively.
Rather complementary to the various cones that arise in entanglement theory are norms. The trace norm (or operator norm, depending on context) for operators and the diamond norm (or completely bounded norm) for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces.
Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
- Nathaniel Johnston
- Local copy of thesis – PDF, zip file containing TeX source and figures
- Official thesis at the University of Guelph Electronic Theses website (PDF only)
- Copy of thesis from arXiv:1207.1479 [quant-ph]
- Defence presentation slideshow – PDF, TeX
- N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012.
- A Family of Norms With Applications in Quantum Information Theory – the norms that are the main focus of the thesis were introduced here
- A Family of Norms With Applications in Quantum Information Theory II – they were further studied here
- Characterizing Operations Preserving Separability Measures via Linear Preserver Problems – much of chapter 3 deals with preserver problems and results from this paper
- Quantum Gate Fidelity in Terms of Choi Matrices – a result in chapter 5 about minimum gate fidelity was originally proved here
- Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory – the first half of chapter 6 contains (and elaborates on) the results originally proved here
- Mapping Cones are Operator Systems – the latter half of chapter 6 is largely based on this paper
- Partially Entanglement Breaking Maps and Right CP-Invariant Cones – an unpublished preprint that contains some results about right CP-invariant cones, which play a role in the thesis