Math Needed for Quantum Information

Quantum information starts with a small set of ideas: complex state vectors, unitary gates, probabilities, and tensor products. You can learn a first two-qubit circuit with these tools, then add density matrices, entropy, and optimization when you need noise or information measures. This route leads into Quantum Information and Computation.

Qubits, gates, and two-qubit states

1. Read finite-dimensional Hilbert spaces and orthonormal bases until you can normalize a complex column and compute an overlap.
2. Study unitary operators and Pauli matrices to apply gates. Bloch-sphere geometry helps picture a single qubit.
3. Use tensor products to write two-qubit states in a declared basis order, then try single-qubit gates and controlled operations.

Keep measurement probabilities tied to a specified measurement basis. A unitary preserves norms but can change those probabilities; a global phase of the whole state changes neither.

Add tools when the question needs them

Your question	Mathematics to learn	Continue with
Can this pure state factor into separate subsystems?	SVD and Schmidt decomposition	Entanglement measures
What does one subsystem look like?	Partial trace and positive trace-one density operators	Density operators for quantum information
How does noise change a state?	Linear maps on matrices, complete positivity, and trace preservation	Common noise models and the open-systems route
How much information or uncertainty is present?	Entropy, relative entropy, and conditional probability	Classical information review
How does a Fourier-based algorithm work?	Roots of unity, geometric sums, and finite Fourier transforms	Quantum Fourier transform and phase estimation
How can errors be detected and corrected?	Pauli strings, projectors, binary vector spaces, and parity checks	Stabilizer formalism

The quantum Fourier transform acts on amplitudes in a quantum register. A classical FFT computes a list of Fourier coefficients. Their related mathematics does not mean that a quantum circuit outputs the full transformed list efficiently.

Try the first circuit

Start with $\lvert00\rangle$. A Hadamard gate on the first qubit gives

$$(H\otimes I)\lvert00\rangle=\frac{\lvert00\rangle+\lvert10\rangle}{\sqrt2}.$$

A controlled-NOT with the first qubit as control maps $\lvert10\rangle$ to $\lvert11\rangle$ and leaves $\lvert00\rangle$ fixed, producing

$$\lvert\Phi^+\rangle=\frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2}.$$

The coefficient matrix is $C=I_2/\sqrt2$, which has rank two. It cannot equal an outer product of two columns, so the state is entangled. This calculation uses only basis order, matrix action, and rank; Bell states develops the physical consequences.

Exercise: local uncertainty and joint purity

Find the reduced density operator of either qubit in $\lvert\Phi^+\rangle$. Compute its von Neumann entropy $S(\rho)=-\operatorname{Tr}(\rho\log_2\rho)$, with $0\log_2 0=0$, and compare it with the entropy of the joint state.

Solution

Tracing out the other qubit removes the cross terms because $\langle0\vert1\rangle=0$. Thus $\rho_A=\rho_B=I_2/2$, whose eigenvalues are $1/2,1/2$. Each marginal has entropy one bit. The joint state is pure, with eigenvalues $1,0,0,0$, so its entropy is zero. Mixed marginals do not imply that the joint state is a classical mixture.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018. Author’s text.