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L3-04 · Tensor products and joint systems

Construct a joint vector and distinguish a product state from a correlated state.

Level 3 icon Level 3 · High-school bridge
Objective. Construct a joint vector and distinguish a product state from a correlated state.
Before this lesson. L3-02, L3-03
Hands-on lab. Entanglement and Bell tests

Learn

The tensor product builds a joint space from separate spaces. For two two-entry vectors, it produces four entries. Fix the order as 00, 01, 10, 11. Each entry is a product of one entry from the first vector and one from the second.

A product vector factors into one vector for each subsystem. A general joint quantum vector need not have this form. A pure state that cannot factor across a stated partition is entangled. This definition concerns joint amplitudes, rather than merely the presence of matching measurement results.

For classical independent variables, joint probabilities also form a product. Correlated classical variables do not. Thus a four-entry list alone does not tell you whether a model is classical or quantum. State what the entries mean and what operations the model permits before drawing a physical conclusion.

Practice

Solution

The tensor product is (0,0,1,0). The nonzero entry is in position 10, so the first subsystem gives one and the second gives zero.

Check your understanding

How many amplitudes describe a general pure state of three qubits?

Show the answer

C. Eight

Each added qubit doubles the joint basis size. Three qubits therefore use 2³ = 8 complex amplitudes before normalization constraints.

Sources

This lesson draws on Watrous, 2018Preskill, n.d..

Next: L3-05 · Eigenvectors and observables

References
  1. Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
  2. 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/