Computational Notebooks

This page is the notebook index and reproducibility contract for composite systems and entanglement. It specifies what each notebook should compute, which conventions it must state, and which checks must pass before the notebook should be treated as a reliable teaching or reference artifact.

At this draft stage, the entries below are specifications rather than downloadable notebook links. When notebook files are added, they should live under a stable path such as notebooks/composite-systems/, preserve these validation checks, and link back to the canonical explanatory pages.

What This Page Owns

This page owns the computational contract for small, transparent notebooks in this volume. It does not own the derivations of tensor products, partial traces, Schmidt decomposition, entanglement entropy, or second quantization. Those stay in their canonical pages and are linked below.

The notebook suite should emphasize reproducible understanding:

• state the basis ordering before constructing arrays;
• print dimensions of vectors, matrices, and reshaped tensors;
• check normalization, Hermiticity, trace preservation, and eigenvalue sanity;
• compare numerical outputs to analytic formulas from the text;
• avoid hidden high-level library calls for the core operation being taught;
• expose small parameters that readers can change without rewriting the notebook.

Shared Conventions

Unless a notebook explicitly states otherwise, two-qubit vectors use the product basis

$$\lvert00\rangle,\quad \lvert01\rangle,\quad \lvert10\rangle,\quad \lvert11\rangle,$$

with subsystem $A$ written first and subsystem $B$ second. If $a,b\in\{0,1\}$, the array index is

$$\operatorname{index}(ab) = 2a+b.$$

For $n$ qubits in this convention,

$$\operatorname{index}(a_1a_2\cdots a_n) = \sum_{j=1}^{n} a_j\,2^{n-j}.$$

This is a big-endian convention for displayed bit strings. Other conventions are legitimate, but a notebook must state the convention before using compact labels.

Density matrices are represented as square arrays in the same product-basis order. A two-qubit density matrix can be reshaped as

$$\rho_{a b,a' b'} \quad \longleftrightarrow \quad \rho[a,b,a',b'].$$

The partial trace over $B$ is then the index contraction

$$(\rho_A)_{a a'} = \sum_b \rho_{a b,a' b}.$$

Required Validation Checks

Every notebook that constructs a state or density operator should include the relevant subset of these checks:

$$\langle\psi\vert\psi\rangle = 1, \qquad \operatorname{Tr}\rho = 1, \qquad \rho^\dagger = \rho.$$

Density-operator eigenvalues should be nonnegative up to numerical tolerance:

$$\lambda_{\min}(\rho) \ge -\epsilon,$$

where $\epsilon$ is a stated roundoff tolerance, not an unexamined escape hatch.

Reduced states should preserve trace:

$$\operatorname{Tr}\rho_A = \operatorname{Tr}\rho_{AB}.$$

Pure-state Schmidt coefficients should satisfy

$$\sum_r s_r^2 = 1, \qquad s_r\ge0.$$

For fermionic notebooks, the creation and annihilation matrices should satisfy

$$\{c_i,c_j^\dagger\} = \delta_{ij}I, \qquad \{c_i,c_j\} = 0, \qquad \{c_i^\dagger,c_j^\dagger\} = 0.$$

The notebook should print the maximum residual norm for these identities. A result that is “visually close” in a matrix display is not enough.

Core Notebook Suite

Notebook specification	Canonical filename	Main computation	Required checks	Canonical pages
Tensor-product basis ordering	tensor-product-basis-ordering.ipynb	Build two- and three-qubit bases; construct local Pauli operators	Dimensions, index map, local-operator commutators	Product Bases
Partial trace for two qubits	partial-trace-two-qubits.ipynb	Reduce product, Bell, and classically correlated states	Trace preservation, Hermiticity, local expectation values	Partial Trace
Schmidt decomposition by SVD	schmidt-decomposition-svd.ipynb	Reshape a pure-state vector into a coefficient matrix and compute singular values	Reconstruction error, normalized singular values, entropy comparison	Schmidt Decomposition
Bell-state correlations	bell-states-correlations.ipynb	Compute Pauli correlation matrices and reduced states for Bell states	Orthonormality, one-qubit reductions, correlation signs	Bell States
Small-system entanglement entropy	entanglement-entropy-spin-chain-small.ipynb	Diagonalize small spin chains and compute subsystem entropies	Normalization, trace preservation, entropy benchmarks	Entanglement in Many-Body Physics
Fermionic signs and Jordan-Wigner preview	fermionic-signs-jordan-wigner-preview.ipynb	Build occupation basis and fermionic ladder operators	Anticommutators, sign conventions, number eigenvalues	Fermionic Anticommutation Relations
Small second-quantized Hamiltonian	second-quantized-hamiltonian-small-basis.ipynb	Build a two-site Hubbard-style Hamiltonian in fixed particle sectors	Hermiticity, particle-number conservation, sector dimensions	Many-Particle Hamiltonians

Tensor-Product Basis Ordering Notebook

This notebook should begin with arrays, not abstractions. For two qubits, list the basis states, array indices, and subsystem labels in a table:

$$\begin{array}{c|c|c} \text{index} & \text{state} & (a,b)\\ \hline 0 & \lvert00\rangle & (0,0)\\ 1 & \lvert01\rangle & (0,1)\\ 2 & \lvert10\rangle & (1,0)\\ 3 & \lvert11\rangle & (1,1) \end{array}$$

Then construct local Pauli operators as

$$X_A=X\otimes I, \qquad X_B=I\otimes X, \qquad Z_A=Z\otimes I, \qquad Z_B=I\otimes Z.$$

The notebook should verify that operators on different subsystems commute:

$$[X_A,Z_B] = 0, \qquad [Z_A,X_B] = 0.$$

It should also show one failure mode: if the basis ordering is silently changed, an array with the same numerical entries can represent a different physical operator.

Partial-Trace Notebook

The partial-trace notebook should compute reduced states in three examples:

1. a product pure state;
2. a Bell state;
3. a classically correlated separable mixed state.

For the plus Bell state,

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

the notebook should obtain

$$\rho_A = \rho_B = \frac12 I.$$

For the classically correlated state

$$\rho_{\mathrm{cc}} = \frac12\lvert00\rangle\langle00\rvert + \frac12\lvert11\rangle\langle11\rvert,$$

the same local reductions occur, but the joint state is separable. This is a mandatory comparison: it prevents the common mistake of using mixed marginals as a mixed-state entanglement test.

The notebook should verify local statistics by comparing

$$\operatorname{Tr}_{AB} \bigl[ \rho_{AB}(O_A\otimes I_B) \bigr]$$

with

$$\operatorname{Tr}_A(\rho_A O_A)$$

for several one-qubit observables $O_A$.

Schmidt-SVD Notebook

For a pure vector

$$\lvert\psi\rangle = \sum_{ij} C_{ij}\, \lvert i\rangle_A\lvert j\rangle_B,$$

the notebook should reshape the state into the coefficient matrix $C$ and compute its singular-value decomposition:

$$C = U\Sigma V^\dagger.$$

The diagonal entries of $\Sigma$ are the Schmidt coefficients. The notebook should reconstruct the state and report the residual

$$\left\| C-U\Sigma V^\dagger \right\|.$$

It should compare the entropy computed from singular values,

$$S = - \sum_r s_r^2\log s_r^2,$$

with the entropy computed from eigenvalues of the reduced density operator $\rho_A=CC^\dagger$.

Useful test cases:

• a product state, with one singular value equal to one;
• a Bell state, with two singular values equal to $1/\sqrt2$;
• a random normalized two-qubit state, with results checked by reconstruction.

Bell-Correlations Notebook

The Bell-correlation notebook should compute the matrix

$$T_{ij} = \langle \sigma_i\otimes\sigma_j \rangle, \qquad i,j\in\{x,y,z\},$$

for each Bell state. It should also compute the one-qubit Bloch vectors and show that they vanish for all Bell states:

$$\langle \sigma_i\otimes I\rangle = 0, \qquad \langle I\otimes\sigma_i\rangle = 0.$$

This makes the distinction between local randomness and joint correlation explicit. It also gives a numerical bridge to Bell measurements, correlation tensors, witnesses, and foundations examples.

Small-System Entanglement-Entropy Notebook

The spin-chain entropy notebook should stay small enough that a reader can inspect the arrays. A good first target is a few qubits with a Hamiltonian such as

$$H = -J\sum_{i=1}^{L-1} Z_iZ_{i+1} -h\sum_{i=1}^{L} X_i,$$

with open boundary conditions and $L$ small. The notebook should diagonalize $H$, choose a low-energy state, trace out part of the chain, and compute

$$S_A = - \operatorname{Tr}(\rho_A\log\rho_A).$$

Validation should include benchmark states whose entropy is known before diagonalization:

• a product state, with $S_A=0$ for every cut;
• a Bell pair crossing the cut, with $S_A=\log2$;
• a GHZ state, with one bit or one nat depending on logarithm base;
• a classically correlated mixed state, used only to show why mixed-state entropy is not automatically entanglement entropy.

The notebook should state the logarithm base and label entropy units.

Fermionic-Signs Notebook

The fermionic-sign notebook should build an occupation basis for $M$ modes, such as

$$\lvert n_1n_2\cdots n_M\rangle, \qquad n_i\in\{0,1\}.$$

With canonical mode order $1,2,\ldots,M$, the annihilation operator acts as

$$c_i \lvert n_1\cdots n_i\cdots n_M\rangle = (-1)^{S_i} n_i \lvert n_1\cdots 0_i\cdots n_M\rangle,$$

where

$$S_i = \sum_{k<i} n_k.$$

The creation operator is similar, with $n_i$ replaced by $1-n_i$. The notebook should verify anticommutation relations as matrix identities and show how Jordan-Wigner strings implement the same signs in a qubit representation.

This notebook should not present the Jordan-Wigner transformation as a full many-body algorithm. It is a sign-convention preview that prepares readers for later computational and many-body treatments.

Second-Quantized Hamiltonian Notebook

The small-Hamiltonian notebook should construct a finite basis and a Hamiltonian such as a two-site spinful Hubbard model:

$$H = -t \sum_{\sigma} \left( c_{1\sigma}^\dagger c_{2\sigma} + c_{2\sigma}^\dagger c_{1\sigma} \right) + U \sum_{i=1}^{2} n_{i\uparrow}n_{i\downarrow}.$$

It should build the Hamiltonian from creation and annihilation matrices, verify Hermiticity, and check particle-number conservation:

$$[H,N] = 0.$$

For small sectors, the notebook should print the basis states explicitly and diagonalize each sector separately. This keeps the connection between first-quantized intuition, occupation notation, and matrix mechanics visible.

Optional Advanced Notebooks

Optional notebooks should be added only after the core suite is stable:

• negativity-ppt-two-qubits.ipynb: partial transpose, eigenvalues, and negativity for simple two-qubit mixed states.
• ghz-w-entanglement.ipynb: reduced states of GHZ and W states for different traced-out subsystems.
• occupation-number-bosons-two-modes.ipynb: bosonic occupation basis, ladder-operator square roots, and two-mode examples.
• two-mode-squeezed-state-truncation.ipynb: truncated continuous-variable calculations and normalization errors.
• jordan-wigner-preview.ipynb: a slightly deeper spin-chain mapping after the sign notebook is mature.

Each optional notebook should have a small analytic benchmark. A notebook that only produces plausible plots is not ready.

Common Numerical Issues

• Silent basis-ordering changes. The same array can mean different states under different bit conventions.
• Using rounded matrices as exact evidence. Always report norms or tolerances.
• Forgetting to renormalize after truncation. This is especially dangerous for continuous-variable and Fock-space examples.
• Treating a negative eigenvalue from roundoff as physical. Compare it with the stated tolerance and with Hermiticity error.
• Using local entropy as mixed-state entanglement. Entropy is an entanglement measure only in the pure bipartite setting.
• Guessing fermionic signs. Derive them from the mode order or verify the anticommutators.
• Hiding the central operation in a library call. A notebook teaching the partial trace should show the contraction at least once.

Cross-Links

• Formula Sheet
• Common Composite States
• Entanglement Diagnostic Table
• Tensor Product Exercises
• Partial Trace Exercises
• Fock Space Exercises
• Tensor Product Ordering
• Product Bases
• Partial Trace
• Schmidt Decomposition
• Bell States
• Entanglement Entropy
• Fermionic Anticommutation Relations
• Many-Particle Hamiltonians

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain”, Annals of Physics 16, 407-466, 1961, doi:10.1016/0003-4916(61)90115-4.
• P. Jordan and E. Wigner, “Uber das Paulische Aquivalenzverbot”, Zeitschrift fur Physik 47, 631-651, 1928.
• J. Hubbard, “Electron correlations in narrow energy bands”, Proceedings of the Royal Society of London A 276, 238-257, 1963, doi:10.1098/rspa.1963.0204.
• U. Schollwoeck, “The density-matrix renormalization group in the age of matrix product states”, Annals of Physics 326, 96-192, 2011, doi:10.1016/j.aop.2010.09.012.

Exercises

1. Basis index check. In the stated two-qubit convention, what are the array indices of $\lvert01\rangle$ and $\lvert10\rangle$? Why does this matter for local operators?

Solution

The convention is

$$\operatorname{index}(ab) = 2a+b.$$

Thus $\lvert01\rangle$ has index $1$ and $\lvert10\rangle$ has index $2$. This matters because an operator such as $X\otimes I$ flips the first bit, while $I\otimes X$ flips the second bit. If the basis order is changed silently, the same numerical matrix can act on the wrong subsystem.

2. Trace preservation. A notebook computes $\rho_A=\operatorname{Tr}_B\rho_{AB}$ for a normalized two-qubit density operator. What scalar check should be printed immediately?

Solution

It should print

$$\operatorname{Tr}\rho_A - \operatorname{Tr}\rho_{AB}.$$

For a normalized density operator this should be zero up to numerical tolerance. Since $\operatorname{Tr}\rho_{AB}=1$, the reduced state should also have trace one.

3. SVD entropy check. A normalized two-qubit pure state has Schmidt coefficients $s_1=s_2=1/\sqrt2$. What entropy should both the SVD route and the reduced-density-matrix route produce?

Solution

The probabilities are $p_1=p_2=1/2$. Therefore

$$S = - \sum_{r=1}^{2} p_r\log p_r = - 2 \left( \frac12\log\frac12 \right) = \log2.$$

Both computational routes should give the same value, apart from the chosen logarithm base.

4. Fermionic sign. In a three-mode occupation basis with order $1,2,3$, compute the sign factor for $c_3$ acting on $\lvert110\rangle$ and on $\lvert101\rangle$.

Solution

For $c_3$, the sign exponent is

$$S_3 = n_1+n_2.$$

For $\lvert110\rangle$, $n_3=0$, so the annihilation operator gives zero regardless of the sign. The formal sign factor would be $(-1)^{2}=+1$.

For $\lvert101\rangle$, $n_3=1$ and $S_3=1+0=1$, so the sign factor is $-1$:

$$c_3\lvert101\rangle = -\lvert100\rangle.$$