Level 3 · High-school bridge
Objective. Normalize a vector and calculate its overlap with a basis vector.
Before this lesson. L3-01
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A vector is an ordered list of numbers. In this lesson, its entries can be complex. The order matters because each position names a basis direction. A two-entry vector can describe an arrow in a plane, or the amplitudes of a two-outcome quantum system.
The inner product compares two vectors. Conjugate the entries of the first vector, multiply corresponding entries, then add. The inner product of a vector with itself is its squared norm. Divide a nonzero vector by its norm to obtain a normalized vector.
Orthogonal vectors have inner product zero. For a normalized quantum state, the squared magnitude of its inner product with a normalized measurement basis vector gives that outcome’s probability. This statement requires an orthonormal measurement basis. An arbitrary coordinate entry is not automatically a physical probability.
Practice¶
Solution
The norm is √2. The normalized vector is (1/√2,1/√2). Its overlap with e1 is 1/√2, and the squared overlap is 1/2.
Check your understanding¶
What does an inner product of zero mean?
A. The vectors have equal length
B. The vectors must both be zero
C. The vectors are orthogonal
Show the answer
C. The vectors are orthogonal
Orthogonality is defined by a zero inner product. Nonzero vectors can be orthogonal even when their lengths differ.
Sources¶
This lesson draws on Watrous, 2018Preskill, n.d..
Next: L3-03 · Matrices as transformations
- 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/