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Is the holographic embezzlement cost related to magic (departure from fixed-area states)?

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Statement

In the group’s holographic embezzlement program, fixed-area states serve as the flat-spectrum reference against which the embezzlement protocol and its capacity are defined. Benedetti–Dabholkar–Dalmonte identify fixed-area states as flat-spectrum, stabilizer-like tensor-network states, and show that physical (cyclic-separating) bulk states acquire non-flat Rényi corrections — i.e. magic — via the modified cosmic-brane prescription. Is the embezzlement cost / capacity of a holographic state quantitatively related to its magic — its departure from the nearest fixed-area (stabilizer) state?

Why It Matters

It would tie the group’s most developed holographic result — generalized entropy as the operational embezzlement capacity — to the newly-imported magic resource, using an object both frameworks already privilege: the fixed-area state. Embezzlement in type III₁ is “costless” in the strict algebraic sense but has a relative-entropy cost operationally; magic measures distance from the flat/fixed-area sector. If the embezzlement cost and the magic are controlled by the same departure-from-fixed-area quantity, then generalized entropy (the group’s capacity candidate) and magic corrections (the cosmic-brane subleading terms) are two views of one bulk quantity — a concrete instance of the Goto et al. call for a holographic resource theory with more than one resource in play.

What We Know

  • Fixed-area states are the shared reference. Flat spectrum, stabilizer-like, and the base of both the cosmic-brane Rényi computation (the paper, via Dong) and the group’s embezzlement construction (holographic-dual-embezzlement-protocol).
  • Magic = non-flat correction. The paper writes any cyclic-separating state in the fixed-area basis and reads its magic off the non-flat subleading Rényi contribution — the same expansion that controls holographic Rényi entropies.
  • Embezzlement cost is a relative entropy. The group measures embezzlement cost in Araki–Uhlmann relative entropy (embezzlement-cost-relative-entropy); relating it to magic needs the magic ↔ relative-entropy link.
  • Type II tensor-network context. The relevant infinite-tensor-network / type II analysis (Chemissany–Gesteau et al. 2025, cited by the paper) is the same machinery behind the crossed-product entropy the group uses.

Possible Approaches

  1. Magic of fixed-area vs. physical states. Compute the anti-flatness of a holographic state expanded in the fixed-area basis (cosmic-brane subleading term) and compare its scaling to the embezzlement cost of the same state.
  2. Capacity ↔ magic dictionary. Test whether the generalized-entropy capacity from embezzlement-capacity-lower-bound and the magic correction share a common bulk quantity (e.g. the cosmic-brane backreaction / area variance).
  3. RT/QES reference. Use RT / QES fixed-area states as the flat baseline and track how bulk matter (which makes the state physical/magical) simultaneously turns on embezzlement cost.
  4. Resource-theory packaging. Frame fixed-area states as the “free states” and magic + embezzlement as two monotones, answering part of holographic-resource-theory / resource-theory-monotones-in-qft with a worked holographic example.

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