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arXiv:2607.28486

Hayden--Preskill recovery at finite temperature on a quantum processor: dynamics and initial state from the SYK model

Jeongho Bang, Moongul Byun, Kyoungho Cho, Keun-Young Kim, Hyeonsoo Lee

hep-thquant-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations
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Abstract

In the original Hayden--Preskill recovery, the post-injection scrambler and initial state are {\it not related}. We extend this setup in two ways: by using a SWAP gate so that the scrambler and initial state are {\it related}, and by considering recovery at {\it finite} temperature. For this modified protocol, we show that the information is successfully recovered in the sense that the postselection probability is non-negligible and the conditional fidelity is large. We find that both the postselection probability and the conditional fidelity are proportional to temperature, reflecting the reduced entanglement of the initial state at lower temperatures. We also derive their late-time analytic estimates under the assumption of uniform operator spreading and show that they agree well with the numerical results. This demonstrates that strong scrambling is important for successful information recovery. Implementing the protocol on an IBM superconducting processor using a binary sparse SYK Hamiltonian with $N = 8$ Majoranas, we observe that the data retain the qualitative recovery dynamics and that a SWAP-based error-mitigation scheme improves both the postselection probability and the conditional fidelity.

Comments
21 pages, 12 figures

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In the original Hayden--Preskill recovery, the post-injection scrambler and initial state are {\it not related}. We extend this setup in two ways: by using a SWAP gate so that the scrambler and initial state are {\it related}, and by considering recovery at {\it finite} temperature. For this modified protocol, we show that the information is successfully recovered in the sense that the postselection probability is non-negligible and the conditional fidelity is large. We find that both the postselection probability and the conditional fidelity are proportional to temperature, reflecting the reduced entanglement of the initial state at lower temperatures. We also derive their late-time analytic estimates under the assumption of uniform operator spreading and show that they agree well with the numerical results. This demonstrates that strong scrambling is important for successful information recovery. Implementing the protocol on an IBM superconducting processor using a binary sparse SYK Hamiltonian with N=8N = 8N=8 Majoranas, we observe that the data retain the qualitative recovery dynamics and that a SWAP-based error-mitigation scheme improves both the postselection probability and the conditional fidelity.
We introduce and study the problem of scrambler hacking, which is the procedure of quantum information extraction from and installation on a quantum scrambler given only partial access. This problem necessarily emerges from a central topic in contemporary physics - information recovery from systems undergoing scrambling dynamics, such as the Hayden-Preskill protocol in black hole studies - because one must replace quantum data with another when extracting it due to the no-cloning theorem. For large scramblers, we supply analytical formulas for the optimal hacking fidelity, a quantitative measure of the effectiveness of scrambler hacking with limited access. In the two-user scenario where Bob attempts to hack Alice's data, we find that the optimal fidelity converges to 64/(9π2)≈0.7264/(9π^2)\approx0.7264/(9π2)≈0.72 with increasing Bob's hacking space relative to Alice's user space. We applied our results to the black hole information problem and showed that the limited hacking fidelity implies the reflectivity decay of a black hole as an information mirror, which questions the solvability of the black hole information paradox through the Hayden-Preskill type protocol.
We study properties of the Petz recovery map in chaotic systems, such as the Hayden-Preskill setup for evaporating black holes and the SYK model. Since these systems exhibit the phenomenon called scrambling, we expect that the expression of the recovery channel R\mathcal{R}R gets simplified, given by just the adjoint N†\mathcal{N}^{\dagger}N† of the original channel N\mathcal{N}N which defines the time evolution of the states in the code subspace embedded into the physical Hilbert space. We check this phenomenon in two examples. The first one is the Hayden-Preskill setup described by Haar random unitaries. We compute the relative entropy S(R[N[ρ]]∣∣ρ)S(\mathcal{R}\left[\mathcal{N}[ρ]\right] ||ρ)S(R[N[ρ]]∣∣ρ) and show that it vanishes when the decoupling is archived. We further show that the simplified recovery map is equivalent to the protocol proposed by Yoshida and Kitaev. The second example is the SYK model where the two dimensional code subspace is defined by an insertion of a fermionic operator, and the system is evolved by the SYK Hamiltonian. We check the recovery phenomenon by relating some matrix elements of an output density matrix ⟨T∣R[N[ρ]]∣T′⟩\langle T|\mathcal{R}[\mathcal{N}[ρ]]|T' \rangle⟨T∣R[N[ρ]]∣T′⟩ to Rényi-two modular flowed correlators, and show that they coincide with the elements for the input density matrix with small error after twice the scrambling time.
The Hayden-Preskill protocol probes the capability of information recovery from local subsystems after unitary dynamics. As such it resolves the capability of quantum many-body systems to dynamically implement a quantum error-correcting code. The transition to coding behavior has been mostly discussed using effective approaches, such as entanglement membrane theory. Here, we present exact results on the use of Hayden-Preskill recovery as a dynamical probe of scrambling in local quantum many-body systems. We investigate certain classes of unitary circuit models, both structured Floquet (dual-unitary) and Haar-random circuits. We discuss different dynamical signatures corresponding to information transport or scrambling, respectively, that go beyond effective approaches. Surprisingly, certain chaotic circuits transport information with perfect fidelity. In integrable dual-unitary circuits, we relate the information transmission to the propagation and scattering of quasiparticles. Using numerical and analytical insights, we argue that the qualitative features of information recovery extend away from these solvable points. Our results suggest that information recovery protocols can serve to distinguish chaotic and integrable behavior, and that they are sensitive to characteristic dynamical features, such as long-lived quasiparticles or dual-unitarity.
In the Hayden--Preskill protocol, the decoder is usually assumed to know the microscopic identity of the collected output qubits. We study what happens when these labels are unavailable and only the relative order of the received qubits is preserved. The resulting order-preserving deletion channel maps an nnn-qubit scrambled register to a subsequence of length ℓ\ellℓ. For a diary of fixed size kkk, we prove that the optimal entanglement fidelity converges to the no-output value 4−k4^{-k}4−k when ℓ=o(n2/3)\ell=o(n^{2/3})ℓ=o(n2/3), uniformly over the scrambling unitary. For a Haar-random scrambling unitary, it converges to one when ℓ=ω(n2/3)\ell=ω(n^{2/3})ℓ=ω(n2/3) and ℓ=o(n)\ell=o(n)ℓ=o(n). Monotonicity then gives the fixed-error recovery scale ℓrec=Θ(n2/3)\ell_{\mathrm{rec}}=Θ(n^{2/3})ℓrec​=Θ(n2/3). We also consider partial position information obtained by dividing the register into BBB consecutive blocks and revealing the block of origin of each received qubit. The recovery scale becomes ℓrec(n,B)≍n2/3B−1/3\ell_{\mathrm{rec}}(n,B)\asymp n^{2/3}B^{-1/3}ℓrec​(n,B)≍n2/3B−1/3 for B≤nB\leq\sqrt nB≤n​ and ℓrec(n,B)≍n/B\ell_{\mathrm{rec}}(n,B)\asymp n/Bℓrec​(n,B)≍n/B for B≥nB\geq\sqrt nB≥n​. The exponent 2/32/32/3 is traced to rank-aligned coincidences between random subsequences, which control both the converse and the recovery argument. Thus, even purely classical information about the origin of the received subsystems can change the amount of quantum output required for Hayden--Preskill recovery.
Abstract We study properties of the Petz recovery map in chaotic systems, such as the Hayden–Preskill setup for evaporating black holes and the Sachdev–Ye–Kitaev (SYK) model. Since these systems exhibit the phenomenon called scrambling, we expect that the expression of the recovery channel R\mathcal {R}R gets simplified, given by just the adjoint N†\mathcal {N}^{\dagger }N† of the original channel N\mathcal {N}N which defines the time evolution of the states in the code subspace embedded into the physical Hilbert space. We check this phenomenon in two examples. The first one is the Hayden–Preskill setup described by Haar random unitaries. We compute the relative entropy S(R[N[ρ]]∣∣ρ)S(\mathcal {R}\left[\mathcal {N}[\rho ]\right] ||\rho )S(R[N[ρ]]∣∣ρ) and show that it vanishes when the decoupling is archived. We further show that the simplified recovery map is equivalent to the protocol proposed by Yoshida and Kitaev. The second example is the SYK model where the 2D code subspace is defined by an insertion of a fermionic operator, and the system is evolved by the SYK Hamiltonian. We check the recovery phenomenon by relating some matrix elements of an output density matrix ⟨T∣R[N[ρ]]∣T′⟩\langle{T}|\mathcal {R}[\mathcal {N}[\rho ]]|{T^{\prime }}\rangle⟨T∣R[N[ρ]]∣T′⟩ to Rényi-two modular flowed correlators, and show that they coincide with the elements for the input density matrix with small error after twice the scrambling time.
We study the Hayden-Preskill thought experiment at finite temperature and obtain the decoupling condition that the information thrown into an old black hole can be extracted by decoding the Hawking radiation. We then consider the decoding Hayden-Preskill protocol at finite temperature assuming the observer outside the black hole who has the access to the full radiation and the unitary dynamics of the black hole. We also consider the cases when the Hawking radiation has noise and decoherence in the storage. The decoding probabilities and the corresponding fidelities are calculated. It is shown that for all the three cases we have considered, the decoding fidelities are less than unity in general. This result indicates that at finite temperature, the decoding strategy and the recovery algorithm are harder to realize than that at infinite temperature.