--- title: Verifiable Quantum Memory url: https://www.emergentmind.com/topics/verifiable-quantum-memory type: topic --- # Verifiable Quantum Memory Searching arXiv for the core papers and recent work on verifiable quantum memory. Search query: "verifiable quantum memory entanglement-breaking certification arXiv" Verifiable quantum memory denotes a family of certification tasks aimed at establishing that a storage device, channel, or dynamical process preserves genuinely quantum information rather than admitting a purely classical simulation. In the channel-theoretic formulation, the minimal requirement is that the memory be non-entanglement-breaking, so that it cannot be replaced by measurement followed by classical storage and state preparation; in non-Markovian open-system theory, verification asks whether observed memory effects require a genuinely quantum environmental memory rather than a stored classical record; and in interactive cryptographic settings, verification asks whether a remote prover actually maintained a prescribed quantum state or number of qubits for a specified time [1710.04710][2310.01205][2510.04159]. Across these settings, verification methods range from unconditional state tomography and channel witnesses to measurement-device-independent, device-independent, and proof-based protocols [1907.02521][2601.14191]. ## 1. Channel-theoretic and dynamical definitions A standard formalization models a quantum memory as a completely positive trace-preserving map $\mathcal E:\mathcal B(\mathcal H_A)\to\mathcal B(\mathcal H_B)$. In this language, a classical memory is an entanglement-breaking channel, equivalently a measure-and-prepare map of the form \[ \mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i, \] or, in the continuous-variable setting, \[ E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}. \] A memory that cannot be written in such a form is non-entanglement-breaking and therefore qualifies as a genuine quantum memory [1710.04710][2305.07513]. The same boundary appears in resource-theoretic form. The free set is the set of entanglement-breaking memories, and free transformations are classically correlated pre/post-processing supermaps \[ \Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i, \] which map entanglement-breaking channels to entanglement-breaking channels. This induces a preorder $\mathcal E\succeq \mathcal F$ when $\mathcal F$ can be obtained from $\mathcal E$ by such a free supermap, and the associated monotones become verification targets [1710.04710]. A distinct but related formulation concerns open-system dynamics. For two times $t_11/2$, $F_{\mathrm{nc}}=2/3$, and $T^++T^-\ge 1$, $V^+V^-\le 1$ | | Robustness of quantum memory $R(\mathcal N)$ [1907.02521] | Distance from entanglement-breaking channels | $R(\mathcal N)=\min\{s\ge 0:\exists M\in EB,\ (\mathcal N+sM)/(1+s)\in EB\}$ | | Coherence-based channel quality [1809.03403] | Basis-independent coherence preservation | $Q_-\le Q_0\le Q_+\le 1$ | | CV adversarial-metrology witness [2305.07513] | Non-entanglement-breaking CV memory | $\langle\mathcal W\rangle_{EB}\ge \frac{\sigma_\alpha^2}{1+\sigma_\alpha^2}+\frac{\sigma_\beta^2}{1+\sigma_\beta^2}$ | | Device-independent causal witness [2601.14191] | Black-box non-entanglement-breaking memory | $\Gamma\ge 1$ classically, while quantum mechanics can reach $2-\sqrt 2$ | For weak coherent-state memories, fidelity is often complemented by state-independent quadrature benchmarks. In the warm-vapor gradient-echo memory, the verification used \[ F=\langle\psi|\rho_{\mathrm{out}}|\psi\rangle, \] together with conditional variances \[ V^+=\mathrm{Var}(X_{\mathrm{out}}\mid X_{\mathrm{in}}),\qquad V^-=\mathrm{Var}(P_{\mathrm{out}}\mid P_{\mathrm{in}}) \] and transfer coefficients \[ T^+=\mathrm{SNR}(X_{\mathrm{out}})/\mathrm{SNR}(X_{\mathrm{in}}),\qquad T^-=\mathrm{SNR}(P_{\mathrm{out}})/\mathrm{SNR}(P_{\mathrm{in}}). \] The classical limits are $T^\pm\le 1/2$ and $V^\pm\ge 1$, while beating the no-cloning bound requires $T^++T^-\ge 1$ and $V^+V^-\le 1$ [1412.8235]. A more universal quantifier is the robustness of quantum memory. The quantity $R(\mathcal N)$ has four operational meanings: robustness to entanglement-breaking noise, the number of noiseless qubits needed for synthesis, classical-simulation overhead, and maximal advantage in non-local games. In low dimensions, whenever the PPT relaxation is tight, one has \[ R(\mathcal N)=\max\{0,d_A\lambda_{\max}(\Phi_{\mathcal N})-1\}, \] with closed forms for dephasing, erasure, and stochastic damping channels [1907.02521]. For qubit memories, geometric quantification is also possible. A single-qubit channel $\Lambda$ induces an ellipsoid $E^\Lambda$ in the Bloch sphere via $r_{\mathrm{out}}=Tr_{\mathrm{in}}+c$, with shape matrix $Q=TT^\top$ and volume \[ V^\Lambda=\frac{4\pi}{3}\sqrt{\det Q}. \] The same work states the relation \[ QM(\Lambda)\le \left[\frac{3V^\Lambda}{4\pi}\right]^{1/4}, \] and reconstructs the Choi state from the ellipsoid geometry in a semi-device-independent manner [2312.06939]. ## 3. Direct experimental demonstrations of verified storage A prominent unconditional demonstration is the warm-$^{87}$Rb $\Lambda$-GEM device operated in a 10 cm vapor cell of isotopically enriched $^{87}$Rb with 0.5 Torr Kr buffer gas at approximately $80^\circ$C. The two long-lived hyperfine ground states $F=1$ and $F=2$ of the $5^2S_{1/2}$ manifold serve as $|g_1\rangle$ and $|g_2\rangle$, with an excited state $|e\rangle$ in a $\Lambda$ scheme. A magnetic field gradient along the optical axis produces a controllable Raman bandwidth of about $0.5$ MHz. The signal and control fields are co-circularly polarized and derived from the same laser, while a second Rb cell suppresses the control by more than $60$ dB with about $1.5$ dB signal loss. The storage protocol combines Raman absorption, a $\sim1\ \mu$s control-off interval following a $\sim2\ \mu$s input pulse, and gradient inversion for time-reversed recall. More than $99\%$ of the incident signal can be absorbed, and the storage efficiency was $\eta\approx78\%\pm5\%$ for a $3\ \mu$s total delay. For weak coherent states, the measured recall fidelity reached $98\%$ at mean photon number $\langle n\rangle=0.67$, about $93\%$ at $\langle n\rangle=3.4$, and about $82\%$ at $\langle n\rangle=16$. Balanced homodyne detection with a scanned local-oscillator phase, about $100{,}000$ shots per input amplitude, a detuned phase-reference pulse delayed by $9\ \mu$s, and iterative maximum-likelihood reconstruction yielded density matrices up to $20$–$30$ photons and corresponding Wigner functions. The noise spectra were at or below the shot-noise level, and for $\langle n\rangle=3.4$ one representative point was $T^++T^-\approx1.27>1$ and $V^+V^-\approx0.19<1$, placing the recalled states inside the no-cloning region [1412.8235]. Another foundational demonstration stored entangled two-mode squeezed states in two room-temperature cesium vapor cells, one for each mode, with memory time about $1$ msec. The optical input consisted of two-mode squeezed states by $6.0$ dB with variable squeezing orientation and coherent displacements by a few vacuum units. The memory used two paraffin-coated cesium cells, off-resonant light-atom interaction, homodyne detection, and feedback-enhanced mapping to collective spin variables. For an alphabet of displaced squeezed states with $d_{\max}\approx3.8$, the maximal fidelity achievable by an entanglement-breaking channel was $0.45$, while the measured average fidelity was \[ F_{\mathrm{exp}}=0.52\pm0.02. \] This was taken as rigorous proof that the memory preserved quantum coherence for the tested ensemble [1002.1920]. These demonstrations exemplify two experimentally important verification styles. The warm-vapor GEM experiment used unconditional tomography together with state-independent $T$-$V$ criteria to show that the recalled field was the best quantum copy of the input, while the cesium-EPR experiment used Gaussian-state fidelity against an entanglement-breaking benchmark for a family of displaced entangled inputs [1412.8235][1002.1920]. ## 4. Minimal-assumption, MDI, and device-independent certification A general minimal-assumption framework is provided by semiquantum signaling games. Here only the input states are trusted, while the memory and all measurements can be untrusted or adversarial. Given trusted inputs $\{\xi_x\}$ and $\{\psi_y\}$ and a payoff function $\wp$, one defines the maximal average payoff \[ \langle\wp\rangle_{\mathcal E} = \max_{p(b|x,y)\in\mathcal S(\mathcal E,S)} \sum_{x,y,b}\wp(b,x,y)\,p(b|x,y). \] The family of such game payoffs is complete for the resource preorder induced by classically correlated supermaps, and a channel is non-entanglement-breaking if and only if there exists some semiquantum game for which its payoff exceeds the entanglement-breaking value, which can be shifted to zero. The same framework is explicitly loss-tolerant: if losses are modeled by an erasure channel $\mathcal E_\eta(\rho)=\eta\rho+(1-\eta)|e\rangle\langle e|$, then $\mathcal E_\eta\circ\mathcal E$ remains non-entanglement-breaking for every $\eta>0$ exactly when $\mathcal E$ is non-entanglement-breaking [1710.04710]. A discrete-variable measurement-device-independent implementation was realized with two cold atomic ensembles. A single photon generated via Rydberg blockade in one ensemble was stored in another ensemble via electromagnetically induced transparency, retrieved after storage, and interfered with a second photon in a Bell-state measurement. The source efficiency was $P_{ph}=6\%$ with antibunching $g^{(2)}(0)=0.03\pm0.02$. The two EIT channels had zero-delay efficiencies about $27\%$ and $25\%$ and lifetimes $58.2\ \mu$s and $56.6\ \mu$s. The Mach-Zehnder visibility was $V=0.875\pm0.018$, corresponding to $\lambda\approx0.24$ in the effective Bell operators. Using randomly chosen input polarizations from $\{H,V,D,R\}$, the entanglement-witness payoff $\langle W\rangle$ remained positive out to about $50\ \mu$s, with typical short-time statistical uncertainty $\pm0.01$ [2104.14723]. For continuous variables, an MDI protocol based on adversarial metrology uses only trusted coherent-state preparation. Alice sends a coherent state $|\alpha\rangle$ at time $t=0$, later sends $|\beta\rangle$ after the delay $\tau$, and the untrusted provider returns estimators $\xi_x,\xi_p$ for quadrature sum and difference. The per-round score is \[ \mathcal W(\alpha,\beta;\xi_x,\xi_p) = (\xi_x-[\alpha_x+\beta_x])^2 + (\xi_p-[\alpha_p-\beta_p])^2. \] For the identity memory, the optimal beam-splitter and homodyne strategy yields $\langle\mathcal W\rangle=1$. Any entanglement-breaking memory satisfies \[ \langle\mathcal W\rangle_{EB} \ge \frac{\sigma_\alpha^2}{1+\sigma_\alpha^2} + \frac{\sigma_\beta^2}{1+\sigma_\beta^2}, \] which tends to $2$ for wide Gaussian priors. The same paper shows that all non-Gaussian-incompatibility-breaking Gaussian memories can be witnessed, and for a pure-loss channel $M_\eta$ one finds $\langle\mathcal W\rangle=1/\eta$, so $\eta>1/2$ is sufficient for certification [2305.07513]. A fully device-independent formulation replaces trusted state preparation by temporal causal inequalities. The relevant variables are a free instrumental choice $X$ at time $t_1$, a first outcome $A$, and a second outcome $B$ at time $t_2$, under assumptions of no retrocausation, free choice of $X$, and forward-in-time order $A\to B$. Classical instrumental models satisfy \[ \Gamma := \sum_{b_0,b_1=0,1} \inf_{x\in\mathcal X} \bigl[ P(A=0,B=b_0|X=x)+P(A=1,B=b_1|X=x) \bigr] \ge 1, \] while quantum mechanics can reach $\Gamma_Q^{\min}=2-\sqrt2\approx0.586$. In a trapped-ion processor with two outer $^{40}\mathrm{Ca}^+$ ions, dynamical decoupling echos every $2.5$ ms, and storage times up to $35$ ms, the observed value was \[ \Gamma_{\mathrm{exp}}=0.642\pm0.052, \] about $7\sigma$ below the classical bound. Auxiliary checks gave $\Delta=0.883\pm0.023\le 1$ and $\mathrm{ACDE}=0.0325\pm0.0339\approx0$, and the device-independent fidelity bound was $F_{\mathrm{memory}}\gtrsim92\%$ [2601.14191]. ## 5. Local witnesses for quantum memory in open-system dynamics In open-system theory, verification targets the quantumness of the environmental memory that mediates non-Markovian behavior. A local certification criterion compares the Choi states $\chi_1=\chi(E_1)$ and $\chi_2=\chi(E_2)$ of the maps at two times. If, for some entanglement monotone $E_m$, \[ E_m^{\mathrm{assi}}[\chi_1] < E_m[\chi_2], \] then no classical-memory decomposition exists, so the process requires genuinely quantum memory. The protocol needs only channel tomography of $E_1$ and $E_2$ by preparing a maximally entangled system-ancilla state and reconstructing the two Choi states. The criterion is sufficient but not necessary. The paper gives examples where partial dephasing followed by its inverse admits a random-unitary, hence classical-memory, realization, and where partial amplitude damping followed by its inverse requires genuine quantum memory. It also exhibits a non-Markovian time-local master equation with a single classical-bit memory, showing that non-Markovianity alone does not imply quantum memory [2310.01205]. A computationally simpler witness uses the von Neumann entropy. With an ancilla $A$, an initial state $\rho_0^{SA}$, and $\rho_t^{SA}=(E_t\otimes id_A)[\rho_0^{SA}]$, the witness states that if \[ S_S(\rho_{t_1})< \max\{-S(S|A)_{\rho_{t_2}},-S(A|S)_{\rho_{t_2}}\}, \] then the two-time dynamics cannot be realized with classical memory. For a maximally entangled initial state, one defines \[ \Delta S(t_1,t_2):=S_S(\rho_{t_1})+S(S|A)_{\rho_{t_2}}, \] and $\Delta S<0$ certifies quantum memory. The method is given both for finite-dimensional qudits and for continuous-variable Gaussian systems; in the damped-harmonic-oscillator example, choosing $\eta_1\approx0.3$ at $t_1=1$, $\eta_2\approx0.2$ at $t_2=1.2$, and squeezing $r=0.1$ gives $\Delta S\approx-0.08<0$. The witness is again sufficient but not necessary, with perturbative stability $\Delta S\to\Delta S+O(\delta\epsilon\ln\delta\epsilon)$ under small estimation errors [2501.17660]. These ideas have been implemented on present-day superconducting hardware. Using a collision model on the IBM Quantum device ibm_sherbrooke, with system, ancilla, and environment qubits and repeated applications of \[ U(g)=\exp[-ig(\sigma_-\otimes\sigma_+ + \sigma_+\otimes\sigma_-)], \] the reconstructed Choi states at different collision numbers gave, for $g=\pi/4$, the measured values $C^\sharp(2)=0.51$ and $C(4)=0.62$, so $0.51<0.62$ certified quantum memory for the single-qubit dynamics. The paper also presented a two-qubit toy model with measured $C^\sharp_>(t_1)=0.72$ and $C_<(t_2)=0.89$, again satisfying the sufficient certification inequality [2510.19522]. ## 6. Interactive verification, memory checking, and proofs of possession A computational precursor to proof-style verification is quantum online memory checking. Here a user stores an $n$-bit string $x$ on an untrusted public memory and later performs online retrieve-bit operations. Classical memory checkers obey the lower bound $s\cdot t\in\Omega(n)$, where $s$ is private storage and $t$ is communication per retrieve. The quantum construction uses an error-correcting code $E(x)$ and $k$ copies of the fingerprint \[ |\psi_x\rangle=\frac{1}{\sqrt m}\sum_{j=1}^m (-1)^{E_j(x)}|j\rangle, \] compared to freshly queried summaries by repeated SWAP tests. This yields $s\in O(\log n)$ private qubits and $t\in O(\mathrm{polylog}\, n)$ communication, an exponential improvement in the space-query product over the classical setting [1002.2970]. A different unconditional direction assumes only a bound on the prover’s memory. In the bounded-storage model, the cheating device may run arbitrarily long and use unbounded ancillas, but may retain at most $M$ bits or qubits across rounds. One protocol achieves a quadratic gap: the honest prover uses $n+2$ qubits, the verifier uses $O(n)$ classical bits, completeness is exactly \[ \cos^2(\pi/8)\approx 0.8536, \] and any classical device with memory less than $n^2/20$ bits has acceptance probability at most \[ \frac34 + O(2^{-\alpha n}). \] A second protocol achieves an exponential gap, with honest parties using polylogarithmic memory while classical cheating devices with only $O(n)$ bits cannot pass [2505.23978]. The notion is formalized directly in proofs of quantum memory (PoQM), defined as interactive protocols between a classical PPT verifier and a quantum PPT prover over a classical channel. The goal is to certify that the prover possessed an $m$-qubit quantum memory for a specified time interval. Two constructions are given. The first is a four-round PoQM based on $1$-of-$2^k$ puzzles and subexponential hardness of LWE, achieving $(1-\mathrm{negl},\mathrm{negl},m_1,m_2)$-PoQM. The second is a polynomial-round PoQM via verifiable remote state preparation of BB84 states, based on polynomial-time hardness of LWE, achieving \[ \bigl(1-\mathrm{negl},\,1/p(n),\,\lceil 9.1\,m_2(n)\rceil,\,m_2(n)\bigr)\text{-PoQM}. \] The same work shows that PoQM imply one-way state puzzles, and that a restricted extractable version implies QCCC key exchange [2510.04159]. ## 7. Scope, heterogeneity, and limitations The cited literature verifies several nonequivalent targets. Some benchmarks certify preservation of unknown quantum states against classical or no-cloning thresholds; some certify non-entanglement-breaking channels; some certify that environmental memory is irreducibly quantum; and some certify that a remote party actually retained qubits over a designated time interval. The resulting statements are therefore not interchangeable. A memory may beat an ensemble-specific fidelity benchmark without being device-independent, and an open-system witness may certify quantum memory in the environment without quantifying channel robustness or stored-qubit count [1412.8235][1907.02521][2510.04159]. Several limitations are explicit. The entanglement-of-assistance and entropic open-system tests are sufficient but not necessary, so failure to violate their inequalities does not imply classical memory [2310.01205][2501.17660]. Standard channel tomography scales as $d^4$ measurement outcomes for a $d$-level system, while tomography overhead and circuit depth can mask small memory effects on current hardware [2310.01205][2510.19522]. The semiquantum-game framework identifies trusted inputs as the minimal assumption for channel verification, whereas the fully device-independent temporal-correlation approach replaces trusted state preparation by causal assumptions such as free choice and no retrocausation [1710.04710][2601.14191]. A distinct usage of the term appears in the Quantum Memory Matrix hypothesis, where IBM Quantum experiments implement a reversible imprint-retrieval cycle within a coherent circuit. The reported figures include a three-qubit imprint-retrieval fidelity $F=0.67$–$0.77$ with correlation coefficient $r\approx0.72$, a five-qubit parallel fidelity about $0.64\pm0.02$ per channel, an evolution baseline with correction at $0.705\pm0.015$, and a controlled-error case at $0.684\pm0.015$. In that work, fidelity is defined by shot-by-shot equality between the field and output bits, \[ F=\frac{N_{\mathrm{match}}}{N_{\mathrm{total}}}, \] rather than by an entanglement-breaking, no-cloning, or causal-inequality benchmark [2502.15766]. This suggests that “verifiable quantum memory” functions as an umbrella label spanning multiple verification paradigms rather than a single standardized criterion. Source: https://www.emergentmind.com/topics/verifiable-quantum-memory