Quantum Information Problem Map

Quantum information problems are usually finite-dimensional, but they are unforgiving about tensor-factor order, basis conventions, and operational meaning.

Route

Stage	Problem Family	Canonical Page	Skill
1	Qubit geometry	Bloch Vector and Bloch Sphere	translate states into Pauli coordinates
2	Tensor products	Tensor Product Exercises	product bases and local operators
3	Bell states	Bell States	entanglement and correlations
4	Partial trace	Partial Trace Exercises	reduced states and local statistics
5	Entropy and distance	Entropy Identities, Trace Distance, and Fidelity	quantify information and distinguishability
6	Gates	Quantum Gates	unitary maps and basis order
7	Stabilizers	Stabilizer Identities and Stabilizer Circuit	commuting Pauli groups and syndrome logic
8	Channels	Depolarizing Channel and Amplitude-Damping Channel	completely positive trace-preserving maps

Representative Problem IDs

ID	Prompt Type	Watch For
QM-PROB-QI101	Convert a qubit density matrix to a Bloch vector	positivity requires $\lVert\mathbf r\rVert\le1$
QM-PROB-QI201	Compute one-qubit reductions of a Bell state	off-diagonal terms vanish under trace over the other qubit
QM-PROB-QI301	Distinguish a Bell state from a classical mixture	same local states, different global correlations
QM-PROB-QI401	Verify a Kraus map is trace preserving	$\sum_a K_a^\dagger K_a=I$
QM-PROB-QI501	Build a stabilizer projector	stabilizer group must be abelian and exclude $-I$
QM-PROB-QI601	Compare two states by trace distance or fidelity	convention for squared versus unsquared fidelity

Common Mistakes

• Omitting identity factors in local operators.
• Changing qubit order between gate and state-vector notation.
• Treating local reduced states as complete descriptions of global entanglement.
• Assuming every mixed state is an ignorance mixture over a unique ensemble.
• Forgetting that channel representations depend on the chosen input-output basis.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

Exercises

1. Two states have the same one-qubit reduced density matrices. Does that prove the two-qubit states are the same?

Solution

No. The Bell state $\lvert\Phi^+\rangle$ and the classical mixture of $\lvert00\rangle$ and $\lvert11\rangle$ have the same one-qubit reduced states but different global density matrices and correlations.