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arXiv:2204.00374

Repeated Extraction of Scrambled Quantum Data: Sustainability of the Hayden-Preskill Type Protocols

Seok Hyung Lie, Yong Siah Teo, Hyunseok Jeong

Published Apr 1, 2022• 1 citation
arXivHTMLPDF

Abstract

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 $64/(9π^2)\approx0.72$ 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.

DOI
10.48550/arxiv.2204.00374

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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 the problem of quantum hacking, which is the procedure of quantum-information extraction from and installation on a quantum network given only partial access. This problem generalizes a central topic in contemporary physics -- information recovery from systems undergoing scrambling dynamics, such as the Hayden--Preskill protocol in black-hole studies. We show that a properly prepared partially entangled probe state can generally outperform a maximally entangled one in quantum hacking. Moreover, we prove that finding an optimal decoder for this stronger task is equivalent to that for Hayden--Preskill-type protocols, and supply analytical formulas for the optimal hacking fidelity of large networks. In the two-user scenario where Bob attempts to hack Alice's data, we find that the optimal fidelity increases with Bob's hacking space relative to Alice's user space. However, if a third neutral party, Charlie, is accessing the computer concurrently, the optimal hacking fidelity against Alice drops with Charlie's user-space dimension, rendering targeted quantum hacking futile in high-dimensional multi-user scenarios without classical attacks. When applied to the black-hole information problem, the limited hacking fidelity implies a reflectivity decay of a black hole as an information mirror.
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.
Quantum information is scrambled via a chaotic time evolution in many-body systems. Clarifying how and to what extent we can recover the quantum information from the scrambled state is a central issue of today's physics, since it commonly appears as an essential factor in various topics including irreversibility of quantum chaotic dynamics, fault-tolerant quantum computation, and black hole information paradox. In realistic settings, symmetry can ubiquitously exist in scrambling dynamics. Here we establish fundamental limitations on the information recovery arising from the Lie group symmetries in the scrambling dynamics. The results are applicable to many situations, including the Hayden-Preskill black hole model with symmetry where the success rate of the information recovery qualitatively changes from the original no-symmetry case. Our findings show a close relationship between information recovery, symmetry, and quantum coherence, and that symmetry hinders the quantum information recovery.
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 is a qubit-toy model of the black hole information paradox. Based on the assumption of scrambling, it was revealed that quantum information is instantly leaked out from the quantum many-body system that models a black hole. In this paper, we extend the protocol to the case where the system has symmetry and investigate how the symmetry affects the leakage of information. We especially focus on the conservation of the number of up-spins. Developing a partial decoupling approach, we first show that the symmetry induces a delay of leakage and an information remnant. We then clarify the physics behind them: the delay is characterized by thermodynamic properties of the system associated with the symmetry, and the information remnant is closely related to the symmetry-breaking of the initial state. These relations bridge the information leakage problem to macroscopic physics of quantum many-body systems and allow us to investigate the information leakage only in terms of physical properties of the system.
We present two particular decoding procedures for reconstructing a quantum state from the Hawking radiation in the Hayden-Preskill thought experiment. We work in an idealized setting and represent the black hole and its entangled partner by nnn EPR pairs. The first procedure teleports the state thrown into the black hole to an outside observer by post-selecting on the condition that a sufficient number of EPR pairs remain undisturbed. The probability of this favorable event scales as 1/dA21/d_{A}^21/dA2​, where dAd_AdA​ is the Hilbert space dimension for the input state. The second procedure is deterministic and combines the previous idea with Grover's search. The decoding complexity is O(dAC)\mathcal{O}(d_{A}\mathcal{C})O(dA​C) where C\mathcal{C}C is the size of the quantum circuit implementing the unitary evolution operator UUU of the black hole. As with the original (non-constructive) decoding scheme, our algorithms utilize scrambling, where the decay of out-of-time-order correlators (OTOCs) guarantees faithful state recovery.