L5-07 · Measurements and instruments
Separate an outcome probability from an outcome-conditioned state.
Level 5 · Advanced undergraduate
Objective. Separate an outcome probability from an outcome-conditioned state.
Before this lesson. L5-06
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A positive operator valued measure, or POVM, specifies outcome probabilities. Each effect is positive and the effects sum to the identity. The probability is the trace of an effect times the state. Effects alone do not determine the state after measurement.
A quantum instrument adds the state change for each recorded outcome. Each outcome map is completely positive and trace nonincreasing. Their sum is a channel. Divide the outcome map result by its trace to obtain the normalized conditional state.
This distinction matters for causal experiments. Two apparatuses can have the same immediate outcome probabilities but prepare different states for the next laboratory. An experiment that records only outcomes can miss this difference. Include the operational intervention when specifying the model.
Practice¶
Solution
The final preparation is |+>, independent of the first outcome. A later Z measurement therefore gives 1 with probability one half. The first measurement effect alone would not specify this result.
Check your understanding¶
What extra information does an instrument supply beyond a POVM?
A. The conditional state change
B. A different probability rule
C. A unique hidden variable
Show the answer
A. The conditional state change
A POVM gives the probabilities of recorded outcomes. An instrument also specifies the physical transformation associated with each recorded outcome.
Sources¶
This lesson draws on Watrous, 2018Costa & Shrapnel, 2016.
Next: L5-08 · Bell correlations and the CHSH test
- Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
- Costa, F., & Shrapnel, S. (2016). Quantum causal modelling. New Journal of Physics, 18(6), 063032. 10.1088/1367-2630/18/6/063032