Opening block — instructor guide

Teaching the four opening lectures

The opening block asks students to follow the same information through several descriptions: a global state, the observables an observer can measure, an encoding, and the state restricted to a recoverable algebra. The small models are the main calculations. Their holographic interpretations become questions whose missing physical ingredients students can name.

The AQFT course is the reference for depth and independent study. These notes use its classroom/self-study distinction. The allocations below are plans for four meetings of 120 minutes, not results of a rehearsal. They occupy the first two weeks of the four-hour-per-week lecture schedule; exercise sessions are additional.

Before the first meeting

Students should be able to expand a tensor-product vector, use the Born rule, multiply small matrices, and diagonalize a positive matrix. They need no prior von Neumann algebra course.

A useful preparation question is to compare the pure Bell state with the classical mixture of $|00⟩$ and $|11⟩$. Ask for the probability of equal outcomes when both observers measure $Z$, then when both measure $X$. The first experiment does not distinguish the preparations; the second does. This diagnoses whether students already separate classical correlations, coherence, and local mixedness.

Put the following distinctions on a reusable board or handout:

Object	Question it answers
State of the complete system	How are all allowed measurements distributed?
Accessible algebra	Which measurements and operations are being retained?
Restricted state	What statistics survive that restriction?
Encoding isometry	How are logical preparations represented physically?
Recovery map	How can the retained information be extracted?

Introduce symbols through examples. Students should encounter the Bell calculation before the general definition of restriction, and the explicit decoder before a general recovery theorem.

Lecture 1: access determines what can be distinguished

Use Lecture 1.

Time	Board calculation or activity	Evidence of understanding
0–15 min	Prepare the two Bell states and compute the $XX$ expectation	Students locate the distinguishing measurement in a joint correlation
15–35 min	Calculate both partial traces, including an off-diagonal term	Students can explain why every measurement confined to $A$ gives identical statistics
35–65 min	Define an algebra and a restricted state; compare the full qubit algebra with the diagonal algebra	Students distinguish changing the state from changing the available observables
65–85 min	Prove no-signaling with Kraus operators; discuss conditioning on a reported outcome	Students identify where the classical record is needed
85–105 min	Compute the commutant and center in the tensor-factor and diagonal examples	Students explain the difference between an algebra and its center
105–120 min	Checkpoints and exit calculation	Students justify a conclusion using an expectation, not only a slogan

Board discipline. Keep one fixed ordering of the two factors. When taking a partial trace, write ${\mathrm{Tr}}_B(|i,j⟩⟨k,l|)={\delta }_{jl}|i⟩⟨k|$ once and use it visibly. Do not begin with a catalogue of algebra types.

Likely difficulty. “Alice sees a mixed state, therefore Alice and Bob are entangled.” Return to the classical mixture, which has the same local states. The entropy of a reduction certifies entanglement for a pure bipartite global state; the purity assumption matters.

If discussion needs more time. Teach the commutants by checking the matrix examples. Leave the general matrix-unit proof in Section 7 for self-study. Keep the no-signaling/conditional-state distinction.

Exit question. Can a unitary on Bob’s system change Alice’s unconditional measurement statistics? Give the one-line calculation.

Expected answer. No: cyclicity of the partial trace over $B$ gives ${\mathrm{Tr}}_B[(I\otimes U)\rho (I\otimes U^{†})]={\rho }_A$. This does not forbid different conditional states after Bob communicates an outcome.

After class. Assign Problems 1–3 as the core set. Problems 4–6 consolidate channels and commutants. The QFT extension is an orientation question and does not require proving type-III structure.

Transition. Local invisibility is only the first requirement of an encoding. Next ask whether the surviving correlations contain enough information to recover an arbitrary quantum input.

Lecture 2: recovery is a calculation on every input

Use Lecture 2.

Time	Board calculation or activity	Evidence of understanding
0–10 min	Erase one register in the repetition encoding	Students show exactly which coherence disappears
10–25 min	Write all three qutrit codewords and check orthogonality	Students can translate the modular-arithmetic formula into vectors
25–50 min	Trace a diagonal and an off-diagonal encoded matrix unit	Students see why checking basis-state probabilities alone is insufficient
50–80 min	Derive and apply the two-share permutation decoder	Students recover a superposition with a complex phase
80–105 min	Check logical $X,Z$ representatives and the disjoint-recovery contradiction	Students distinguish agreement on the code from equality of physical operators
105–120 min	Worked input, questions, and exit statement	Students can say what the decoder proves about an unknown state

Board discipline. Use $i$ for the logical label and $r$ for the summed physical label. Circle the instruction “modulo three.” For the decoder, show both its inverse and its action on an encoded basis vector. The relabeling $s=r+2i$ is the decisive step: the auxiliary state loses its dependence on the logical input.

Likely difficulty. Students may mistake the two overlapping reconstructions for two independent copies. Draw the three physical shares in a row and mark the authorized pairs. The pairs overlap. Then use commuting representatives on genuinely disjoint regions to explain the obstruction.

A second difficulty. The equation $V^{†}{x}_{1}V=cI$ does not say that $x_1$ preserves the code. Contrast it with $O_{12}V=VO$. Ask which equation is needed to compose reconstructed operators.

If discussion needs more time. Work one pair in class and leave the cyclic pairs to the exercise session. Defer the reference-system derivation and entropy table; retain their conclusions as targets the self-study section will prove. Lecture 4 begins by recalling the entropy calculation.

Exit question. What extra fact must hold beyond “the three encoded basis states can be decoded”?

Expected answer. The same linear decoder must preserve all off-diagonal matrix units, hence every superposition and every correlation with an external reference. Here the factorization identity supplies that fact.

After class. Assign Problems 1–4. Use the reference-system problem as the central self-study check. The noise problem is optional and must state that the lost position is known.

Transition. An entangled auxiliary state appeared without being chosen for each input. Next calculate the operator structure carried by a fixed entangled state.

Lecture 3: modular structure before the abstract theorem

Use Lecture 3.

Time	Board calculation or activity	Evidence of understanding
0–20 min	Schmidt weights, matrix units, cyclic and separating	Students identify the role of nonzero weights
20–50 min	Compute $S_T$, $J$, and $\Delta$ on basis vectors	Students keep the Tomita map antilinear
50–75 min	Work the unequal-qubit example and rotate $X$	Students can distinguish $K_A$ from $-\log \Delta$
75–95 min	Substitute a Gibbs state into the modular flow	Students obtain physical time $s=\beta t$ with the chosen sign
95–110 min	Normalize the oscillator thermofield double and take its partial trace	Students recognize a thermal state from its spectrum
110–120 min	Oscillator counterexample, questions, and exit check	Students do not infer an algebraic type from one modular spectrum

This divides the oscillator allocation in the lecture into 15 minutes of calculation and 10 minutes of interpretation and questions.

Board discipline. Compute $S_T|i,j⟩$ first. Do not guess $\Delta$ from an analogy with a density matrix. Keep the three equations $\Delta ={\rho }_A\otimes {\rho }_B^{-1}$, $K_A=-\log {\rho }_A$, and $\hat{K}=K_A-K_B$ visible with tensor identities where needed.

Likely difficulty. Students may silently treat $S_T$ as linear. Apply it to $i|i,j⟩$ before computing its square. Another useful check is $\Delta \Omega =\Omega$, which a one-sided density matrix generally fails.

KMS placement. In class, explain that the condition compares two orderings at an imaginary-time separation. The analytic matrix proof in Section 6 is self-study. If the students already know thermal Green functions, use the qubit matrix-unit example in place of part of the oscillator discussion.

If discussion needs more time. Normalize the oscillator state and display the reduced probabilities. Defer the entropy sum. Preserve the point that an unbounded modular operator is compatible with a type-I oscillator algebra.

Exit question. A state is Gibbs for $H$. What is the relation between its modular parameter and Heisenberg time, and does every modular flow define the physical Hamiltonian of the system?

Expected answer. With the declared convention, ${\sigma }_t={\alpha }_{\beta t}$. For a general faithful state one may define $-\log \rho$, but identifying it with a prescribed physical Hamiltonian is additional input.

After class. Assign Problems 1–4. Use Problems 5–7 to test the KMS sign and oscillator limits. Treat the spacetime comparison as a reading exercise, not as a theorem established by the finite calculation.

Transition. The logarithm of a state gives a generator, while its expectation gives entropy. Return to encodings and calculate how both change when a fixed entangled factor is added.

Lecture 4: an entropy formula and the meaning of its terms

Use Lecture 4.

Time	Board calculation or activity	Evidence of understanding
0–10 min	Recall the decoded qutrit pair and its spectrum	Students derive $S({\rho }_{12})=S(\rho )+\log 3$
10–30 min	Build the factor code and reconstruct the two commuting logical algebras	Students identify what each physical region can recover
30–60 min	Derive entropy additivity and work the four-qubit example	Students separate variable logical information from fixed auxiliary entropy
60–95 min	Teach the explicit two-sector, three-level example in Section 8	Students calculate three distinct entropies for the same preparation
95–110 min	State the general central-operator formula and its gravitational comparison	Students identify the extra geometric assumptions still needed
110–120 min	Questions and exit explanation	Students avoid equating the restricted algebra’s entropy with the full logical entropy

Teach the center by example. Section 8 can precede Section 7. Compute the physical spectrum $\{p,(1-p)/2,(1-p)/2\}$ before introducing a general direct sum. Then identify the common sector measurement. A phase between the two sectors is present globally but absent from either local algebra.

Likely difficulty. “The global input is pure, so the bulk/algebra entropy term must vanish.” At $p=1/2$, compare the three numbers explicitly: full logical entropy $0$, restricted diagonal-algebra entropy $\log 2$, physical-region entropy $3\log 2/2$. Ask which observables each entropy refers to.

If discussion needs more time. Defer the full direct-sum derivation and the relative-entropy/modular calculations. Keep the explicit center example; it supplies the algebraic content that a constant auxiliary term alone would miss.

Exit question. Which step in the finite calculation proves that ${\mathcal{L}}_A$ is the area of an extremal surface?

Expected answer. None. The calculation proves an entropy identity with a central operator fixed by the encoding. A metric, a gravitational regime, and an extremal-surface prescription are additional ingredients.

After class. Assign Problems 1–4. The self-study section develops relative entropy, compression, and the first law; Problems 5–7 check distinguishability and entropy accounting. Students beginning a course project can compare the factor and central examples before reading a general code theorem.

What students should be able to reconstruct after the block

Give a preparation, an encoding, and a specified accessible region. Ask students to:

1. identify the accessible algebra and compute the restricted state;
2. show whether the input can be recovered, using one map on every logical state;
3. calculate the entropy terms and identify which ones are fixed by the encoding;
4. distinguish exact model statements from proposed geometric interpretations.

A useful ten-minute written check is to revisit the central code at $p=1/2$ with a variable phase. Students should calculate the local spectrum, explain the phase’s invisibility, and state why both regions can recover the central label without cloning a qubit.

Lectures 5–7 add finite constrained algebras, small tensor networks, and a deformation that spoils recovery before passing to continuum QFT. Their purpose is to expose exactly where tensor-factor reasoning ceases to be adequate. All 32 lecture drafts are now written. The syllabus gives the full sequence, and the complete instructor guide supplies the continuation routes.