Positive Maps in Quantum Information Theory
by Seung-Hyeok Kye
Lecture Notes, Fall semester 2022, Seoul National University
Positive linear functionals and positive linear maps had been playing important roles
in the theory of operator algebras, which reflect noncommutative order
structures. Such order structures provide basic mathematical frameworks
for current quantum information theory. The main purpose of this lecture note is to introduce
basic notions like separability/entanglement and Schmidt numbers from quantum information theory
in terms of positive maps between matrix algebras.
Basic tools are Choi matrices and duality arising from bilinear pairing between matrices.
We begin with concrete examples of positive maps $\ad_s$ which sends $x$ to $s^*xs$,
and define separability/entanglement and Schmidt numbers in terms of Choi matrices. We also use
duality to introduce various kinds of positivity, like $k$-positivity and complete positivity.
Positive maps which are not completely positive are indispensable tools to detect entanglement
through the duality.
In Chapter 1, we introduce the above notions and exhibit nontrivial examples of positive maps.
We also provide a unified argument to recover various known criteria through ampliation. We will
focus in Chapter 2 on the issue how positive maps detect entanglement. Through the discussion,
exposed faces of the convex cones of all positive maps play important roles. We exhibit three
classes of positive maps, by Choi, Woronowicz and Robertson in 1970's and 1980's
which generate exposed extreme rays of the convex cone of all positive linear maps.
This is the collection of lecture notes during the fall semester of
2022, at Seoul National University, Seoul, Korea. The author tried to minimize preliminaries,
requiring only undergraduate linear algebra.
The author is grateful to all the audiences for their
feedbacks on the notes. Special thanks are due to Kyung Hoon Han for his careful reading of the drafts.
Nevertheless, any faults in this lecture notes are responsibility of the author.
February 2023
CHAPTER 1: Positive Maps and Bi-partite States
CHAPTER 2: Detecting Entanglement by Positive Maps
1.1. Preliminaries
1.2. Positive maps
1.3. Positive maps between 2 x 2 matrices
1.4. Choi matrices and separable states
1.5. Duality and completely positive maps
1.6. Nontrivial examples of positive maps
1.7. Isotropic states and Werner states
1.8. Mapping cones and tensor products
1.9. Historical Remarks
2.1. Exposed faces
2.2. The Choi map revisited
2.3. Maximal faces
2.4. Entanglement detected by positive maps
2.5. Positive maps of Choi type
2.6. Exposed positive maps by Woronowicz
2.7. Exposed positive maps by Robertson