Level 6 · Graduate
Objective. Check why positivity alone is insufficient for a process.
Before this lesson. L6-02, L6-04
Hands-on lab. Confounding and intervention
Learn¶
A process matrix is an operator that assigns probabilities to local quantum instruments without first choosing one global causal order. Local laboratories still obey quantum mechanics. The global operator must satisfy positivity and normalization constraints for every permitted set of deterministic local operations.
In a compatible Choi convention, joint probabilities are a trace pairing of the process with local instrument Choi operators. This simple expression hides convention choices. Transposes can be absorbed into definitions. Import both the pairing and its constraints together, rather than mixing definitions across papers.
A positive operator with a convenient trace is not automatically a valid process matrix. The linear normalization conditions exclude operational loops that would give inconsistent probabilities. For a proposed example, test complete positivity and the full affine constraints of its named scenario.
Practice¶
Solution
No. Normalization must hold for every allowed pair of local deterministic maps. The scenario-specific linear constraints encode that universal requirement. One tested pair cannot establish it.
Check your understanding¶
Which condition is required beyond positive semidefiniteness?
A. Normalization for all permitted local deterministic operations
B. A large enough trace for one setting
C. A nonzero off-diagonal entry
Show the answer
A. Normalization for all permitted local deterministic operations
A process must return a normalized probability distribution for any allowed deterministic local operations. Positivity by itself does not impose that universal normalization.
Sources¶
This lesson draws on Araújo et al., 2015Oreshkov et al., 2012Costa & Shrapnel, 2016.
Next: L6-08 · The quantum switch
- Araújo, M., Branciard, C., Costa, F., Feix, A., Giarmatzi, C., & Brukner, Č. (2015). Witnessing causal nonseparability. New Journal of Physics, 17(10), 102001. 10.1088/1367-2630/17/10/102001
- Oreshkov, O., Costa, F., & Brukner, C. (2012). Quantum correlations with no causal order. Nature Communications, 3(1). 10.1038/ncomms2076
- Costa, F., & Shrapnel, S. (2016). Quantum causal modelling. New Journal of Physics, 18(6), 063032. 10.1088/1367-2630/18/6/063032