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arXiv:2310.18991

The Petz (lite) recovery map for scrambling channel

Nakayama, Yasuaki, Akihiro Miyata, Tomonori Ugajin

Published Oct 29, 2023• 0 citations
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Abstract

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 $\mathcal{R}$ gets simplified, given by just the adjoint $\mathcal{N}^{\dagger}$ of the original channel $\mathcal{N}$ 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 $S(\mathcal{R}\left[\mathcal{N}[ρ]\right] ||ρ)$ 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 $\langle T|\mathcal{R}[\mathcal{N}[ρ]]|T' \rangle$ 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.

DOI
10.48550/arxiv.2310.18991

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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.
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.
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.
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.
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.
The black hole information paradox is one of the important questions of physics in the last few decades. We study the reconstruction of the bulk operators in AdS/CFT when the geometry contains a black hole. The black hole exterior can be mapped to the CFT via a very simple Petz map which coincides with the HKLL map reconstruction of the black hole exterior. For the interior modes of the bulk theory, using the definition of the Petz recovery channel in modular theory, we can find the mapping from the black hole interior to the dual boundary theory. In the case of the evaporating black hole, it is expected that the interior modes map to some operators that have support only on the bath system, the cavity that absorbs the Hawking radiation. The most important observation that we have here is that in the case that we have a typical black hole microstate in the bulk, the CFT dual of the interior modes that we can find using the Petz recovery channel are exactly the operators that so-called "mirror operator " in the Papadodimas-Raju proposal. The main idea of the PR proposal is to focus on a code subspace of the CFT theory, which is created by acting with a small algebra on the corresponding pure state and then fnd the CFT description of the interior operator in a state-dependent manner just in the chosen code subs. Therefore, we can interpret Papadodimas- Raju proposal as an example of the Petz map reconstruction. It may help us answer some open questions about their procedure.
Information scrambling refers to the unitary dynamics that quickly spreads and encodes localized quantum information over an entire many-body system and makes the information accessible from any small subsystem. While information scrambling is the key to understanding complex quantum many-body dynamics and is well-understood in random unitary models, it has been hardly explored in Hamiltonian systems. In this Letter, we investigate the information recovery in various time-independent Hamiltonian systems, including chaotic spin chains and Sachdev-Ye-Kitaev models. We show that information recovery is possible in certain, but not all, chaotic models, which highlights the difference between information recovery and quantum chaos based on the energy spectrum or the out-of-time-ordered correlators. We also show that information recovery probes transitions caused by the change of information-theoretic features of the dynamics. Published by the American Physical Society 2024