Observation of genuine multi-time quantum memory

The paper in detail

A probe qubit evolves from time 0 under a dynamical map \(\mathcal E_t\); tomography gives the channels \(\mathcal E_1,\mathcal E_2,\mathcal E_3\) at times \(t_1<t_2<t_3\). The pair \((\mathcal E_1,\mathcal E_2)\) has a classical-memory explanation if \[\mathcal E_1[\rho]=\sum_iK_i\rho K_i^{\dagger},\qquad \mathcal E_2[\rho]=\sum_i\Phi_i\bigl[K_i\rho K_i^{\dagger}\bigr]\] for Kraus operators \(K_i\) and channels \(\Phi_i\): the environment keeps only the classical label \(i\) at \(t_1\) and steers the later evolution with it. The three-time version lets the label be updated by a second instrument \(G_{j|i}\) and the final channel \(\Phi_{ij}\) depend on both labels.

The question is whether checking all pairs is enough. For the weaker question of whether the dynamics needs memory at all, pairwise checks do compose; for quantum memory, the paper shows they do not. The witness is \[W_3=\min_{\hat n,\hat m}\Bigl[F_2(\hat n)+F_3(\hat m)-\lvert\vec s_1\rvert\,\lvert\hat n\times\hat m\rvert\Bigr],\] where \(\vec s_1\) is the Bloch vector of \(\mathcal E_1[\mathbb 1/2]\) (the channel's non-unitality) and \(F_k(\hat n)\) is the fidelity between the images under \(\mathcal E_k\) of the two antipodal pure states along \(\hat n\). Every three-time classical-memory model has \(W_3\ge0\).

In the discrete example \(\mathcal E_1\) is amplitude damping with strength \(\gamma\) and \(\mathcal E_2,\mathcal E_3\) are dephasing of strength \(p\) about \(z\) and \(x\). Dephasing keeps antipodal states on its axis orthogonal, so \(F_2(\hat z)=F_3(\hat x)=0\), while \(\lvert\vec s_1\rvert=\gamma\); hence \(W_3=-\gamma\). Explicit classical decompositions exist for each pair whenever \(p\ge(1-\sqrt{1-\gamma})/2\). The intuition: a non-unital first channel singles out a direction, a classical record can keep perfect distinguishability along only one direction afterwards, and the later channels demand it along two orthogonal ones.

The experiment drives a probe transmon coupled to a memory transmon by a parametrically activated exchange interaction (\(g/2\pi=0.994\) MHz) plus a local drive whose sign is averaged over, which reproduces a model with two environment qubits. At \(\omega/g=0.32\) the most violating times are 232, 464 and 832 ns. For small drive both two- and three-time witnesses detect quantum memory, for large drive neither does, and in between only the three-time witness does. Open questions are witnesses for higher dimensions and more times, where semidefinite programs may replace the analytic bound.

Transferable methods

1. Flagged fidelities and a two-direction trade-off

   To rule out every classical-memory model at once, the proof extracts from any such model a single set of Kraus operators \(K_i\) that must serve both later channels, and shows that these operators cannot keep antipodal states distinguishable along two different axes while also making \(\mathcal E_1\) non-unital.

   Summing over the second label, a three-time model gives \(\mathcal E_2=\sum_i\Lambda_i[K_i\cdot K_i^{\dagger}]\) and \(\mathcal E_3=\sum_i\Upsilon_i[K_i\cdot K_i^{\dagger}]\) with the same \(K_i\). Write \(K_i^{\dagger}K_i=a_i\mathbb 1+\mathbf b_i\cdot\vec\sigma\). Since \(K_iK_i^{\dagger}\) has the same spectrum, its Bloch part has the same length, and \(\mathcal E_1[\mathbb 1]=\sum_iK_iK_i^{\dagger}\) gives \(\lvert\vec s_1\rvert\le\sum_i\lvert\mathbf b_i\rvert\).

   Flag the branch: \(\Omega_\pm=\sum_i\ketbra ii\otimes K_i\rho_\pm K_i^{\dagger}\). The fidelity of block-diagonal states is the sum of blockwise fidelities, and for pure inputs each block gives \(\lvert\langle +_{\hat n}\rvert K_i^{\dagger}K_i\lvert -_{\hat n}\rangle\rvert=\lvert\mathbf b_i\times\hat n\rvert\). Applying the channel \(\Omega\mapsto\sum_i\Lambda_i[\langle i\rvert\Omega\lvert i\rangle]\) can only raise fidelity, so \[\sum_i\lvert\mathbf b_i\times\hat n\rvert\le F_2(\hat n),\qquad \sum_i\lvert\mathbf b_i\times\hat m\rvert\le F_3(\hat m).\]

   The geometric inequality \(\lvert\mathbf b\rvert\,\lvert\hat n\times\hat m\rvert\le\lvert\mathbf b\times\hat n\rvert+\lvert\mathbf b\times\hat m\rvert\) (a vector cannot be nearly parallel to two non-parallel axes) chains the bounds into \(\lvert\vec s_1\rvert\lvert\hat n\times\hat m\rvert\le F_2(\hat n)+F_3(\hat m)\), which is \(W_3\ge0\).

   Editorial checks: for \(\gamma\in\{0.3,0.7\}\) the example gives \(\lvert\vec s_1\rvert=\gamma\) and \(F_2(\hat z)=F_3(\hat x)=0\). The decomposition of Appendix C.1 (record the damping jump; on a jump undo it, otherwise dephase with strength \(f=(\sqrt{1-\gamma}+2p-1)/(2\sqrt{1-\gamma})\)) reproduces \(\mathcal E_2\) to machine precision when \(p\ge p_{\mathrm{crit}}\), and below it \(f<0\), so it is no longer a channel. The geometric inequality held in \(2\times10^5\) random trials.

   How to reuse it

   When each pair of observations has a classical explanation but the whole set may not, flag the hidden classical record, use data processing branch by branch, and look for a resource (here distinguishability) that one branch cannot supply along two incompatible directions. The same template applies to compatibility of measurements or channels beyond qubits, though the geometric step must be replaced.

2. Emulating a coherent environment by averaging a conserved sign

   The continuous-time model couples the probe to an exchange partner \(A\) and, through \(\sigma_x^S\sigma_x^B\), to a second qubit \(B\). Because the coupling to \(B\) commutes with everything, \(B\) acts as a coherent control of the sign of a drive, and the probe's reduced dynamics is an equal mixture of two driven two-qubit evolutions. This is what lets two transmons stand in for three.

   With \(H=H_{SA}+\hbar\omega\,\sigma_x^S\otimes\sigma_x^B\) and \(\sigma_x^B\) conserved, \[U_t=\sum_{\pm}\exp\!\bigl[-it(H_{SA}\pm\hbar\omega\sigma_x^S)/\hbar\bigr]\otimes\ketbra{\pm}{\pm}_B.\] Starting \(B\) in \(\ket0=(\ket++\ket-)/\sqrt2\) and tracing it out removes the cross terms, since \(\braket{+}{-}=0\), leaving \(\mathcal E_t=\tfrac12\sum_\pm\mathcal E_t^{(\pm)}\). Editorial derivation of the paper's Eq. (11).

   Experimentally, each evolution time is run with both drive signs and the reconstructed channels are averaged. Tomography uses six input states and three Pauli measurements, with readout error corrected by constrained maximum likelihood (readout fidelity 94%), and the witness uncertainty is propagated from the fit covariance.

   How to reuse it

   A qubit environment coupled through a conserved operator can be replaced by classical averaging over that operator's eigenvalues whenever only the system's reduced dynamics is measured. The trick fails for any test that probes correlations with the emulated qubit itself.

Source: Eqs. (1)–(7); Appendices A–C; Eqs. (8)–(11). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Quantum memory has a strict hierarchy, like the step from pairwise to genuinely multipartite entanglement: classical explanations of each pair can be mutually incompatible. The witness needs only channel tomography at a few times, with no interventions during the evolution, so the effort grows linearly in the number of times probed rather than exponentially as in process-tensor tomography.

Limitations. The analytic witness is for qubits. Pairwise classicality in the experiment is established by non-convex numerical optimisation (a pair counts as classical when the fit reaches infidelity below \(10^{-10}\)), not by an exact criterion. Whether the memory is genuinely three-time depends on the observation window: it holds over a wide range of drive strengths for 850 ns and only in a narrow one for 1200 ns. The second environment qubit is emulated by averaging runs with opposite drive signs. The paper has no AI-use statement.

Description based on the full arXiv HTML of v1, read 2026-11-13: the main text, Appendices A (witness proof), B (discrete example) and C (two-time classical decompositions), and Supplements I–III; Supplements IV–VII were not read. The value \(W_3=-\gamma\), the threshold \(p_{\mathrm{crit}}\), the explicit decomposition of Appendix C.1 and the geometric inequality used in the proof were checked numerically.