L3-03 · Matrices as transformations
Apply a matrix to a vector and test whether a simple transformation preserves norm.
Level 3 · High-school bridge
Objective. Apply a matrix to a vector and test whether a simple transformation preserves norm.
Before this lesson. L3-02
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A matrix is a rectangular array of numbers. A matrix can transform a vector. For each output entry, multiply one matrix row by the input column, then add the products. The number of matrix columns must equal the number of input entries.
A square complex matrix U is unitary when its conjugate transpose times itself equals the identity matrix. The conjugate transpose changes rows into columns and conjugates every entry. A unitary matrix preserves inner products and norms. Its inverse is its conjugate transpose.
Closed quantum systems use unitary time evolution. Measurements and noise need more general descriptions. A matrix with real positive entries is not necessarily a quantum gate. Also, a stochastic transition matrix preserves total probability under its chosen convention, which differs from preserving an amplitude vector’s norm.
Practice¶
Solution
The first result is (1/√2,1/√2). The second result is (1,0), because the second output entry cancels. Both output vectors have norm one.
Check your understanding¶
Which property must a unitary matrix preserve?
A. Every individual vector entry
B. The norm of every input vector
C. The sign of every real entry
Show the answer
B. The norm of every input vector
The identity U†U = I makes the squared output norm equal the squared input norm for every vector.
Sources¶
This lesson draws on Watrous, 2018Basics of quantum information, n.d..
Next: L3-04 · Tensor products and joint systems
- Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
- Basics of quantum information. (n.d.). https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information