Level 5 · Advanced undergraduate
Objective. Calculate purity and explain a partial trace.
Before this lesson. L5-04, L3-05
Hands-on lab. Noise and quantum channels
Learn¶
A density operator describes both pure states and statistical mixtures. It is positive semidefinite and has trace one. For a pure state, form the outer product |psi><psi|. A mixture is a weighted sum of these outer products.
Purity is the trace of the squared density operator. A pure state has purity one. A mixed qubit can have purity as low as one half. Low local purity can arise from classical uncertainty or from entanglement with another system.
The partial trace gives local predictions from a joint state. It is an information reduction, not an assumption that the other system was physically destroyed. Two different joint states can have the same reduced state. Local data then cannot distinguish those joint preparations.
Practice¶
Solution
Square both diagonal entries and add them. The result is 0.36 + 0.16 = 0.52. Since this is below one, the state is mixed.
Check your understanding¶
What does a reduced density operator determine?
A. Every joint correlation
B. Only the global preparation label
C. All measurement probabilities on that subsystem
Show the answer
C. All measurement probabilities on that subsystem
The reduced operator determines local probabilities. It does not retain the full joint correlation structure or uniquely identify the global preparation.
Sources¶
This lesson draws on Watrous, 2018Preskill, n.d..
Next: L5-06 · Quantum channels and complete positivity
- Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
- Preskill, J. (n.d.). Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). California Institute of Technology course page with lecture notes. https://www.preskill.caltech.edu/ph219/