Level 5 · Advanced undergraduate
Objective. Explain what can be reconstructed from repeated measurements.
Before this lesson. L5-05, L3-08
Hands-on lab. Qubits and the Bloch sphere
Learn¶
The von Neumann entropy measures mixedness through the eigenvalues of a density operator. Use logarithm base two to express entropy in bits. A pure state has zero entropy. A maximally mixed qubit has one bit of entropy.
Quantum tomography estimates a state from measurements of many equally prepared systems. For a qubit, measurements of X, Y, and Z determine a Bloch vector. A physical Bloch vector has length at most one. Statistical estimates can violate this condition before constrained fitting.
A full density matrix grows rapidly with system size. Tomography can therefore require many settings and samples. State assumptions can reduce this cost but also restrict the conclusion. A neural estimator does not remove the need for an identifiable measurement design.
Practice¶
Solution
The eigenvalues are one and zero, so the entropy is zero. This is a pure state. No number of samples can reveal a physically irrelevant global phase.
Check your understanding¶
Which three means specify a qubit density operator?
A. X, Y, and Z expectation values
B. Only three repeated Z outcomes
C. Only its entropy
Show the answer
A. X, Y, and Z expectation values
The three Pauli means specify the Bloch vector. Entropy or one measurement basis alone does not determine all state components.
Sources¶
This lesson draws on Watrous, 2018Preskill, n.d..
Next: L5-10 · Noise, encoding, and error correction
- 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/