Verifiable Quantum Memory

• Verifiable quantum memory is a family of certification tasks that confirm storage devices preserve genuine quantum information rather than an entirely classical simulation.
• Techniques involve state tomography, channel witnesses, and various protocols to benchmark coherence and fidelity against entanglement-breaking limits.
• Experimental demonstrations using warm-vapor and cold-atom systems validate the approach by achieving high storage fidelity and robustness with practical metrology tools.

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 (Rosset et al., 2017, Bäcker et al., 2023, Hhan et al., 5 Oct 2025). Across these settings, verification methods range from unconditional state tomography and channel witnesses to measurement-device-independent, device-independent, and proof-based protocols (Yuan et al., 2019, Santos et al., 20 Jan 2026).

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 (Rosset et al., 2017, Abiuso, 2023).

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 (Rosset et al., 2017).

A distinct but related formulation concerns open-system dynamics. For two times $t_1<t_2$, a process has a classical-memory realization if there exist Kraus operators $\{M_i\}$ or $\{K_i\}$ and conditional CPT maps $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$0 such that

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$1

In that case the relevant history can be stored in the classical outcome label $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$2; if no such decomposition exists, the process requires genuinely quantum memory in the environment (Bäcker et al., 2023, Bäcker et al., 29 Jan 2025).

A parallel line of work defines memory quality through preserved coherence. For a channel $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$3, a valid quality measure $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$4 should satisfy the ideal-unitary condition $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$5 for unitary channels and a classical-limit condition $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$6 for measure-and-prepare channels. The coherence-based quantities $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$7, $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$8, and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$9 instantiate this program and sharpen the distinction between unitary storage and classical simulation (Simnacher et al., 2018).

2. Verification quantities and benchmarks

Verification criteria differ in what they certify: state fidelity relative to ensemble-dependent classical bounds, entanglement preservation, coherence preservation, distance from the entanglement-breaking set, or the impossibility of a classical-memory realization. Several of the most used quantities are listed below.

Framework	Certified property	Representative condition
Fidelity and $$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$0-$$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$1 benchmarks (Hosseini et al., 2014)	Beating classical storage and the no-cloning limit for coherent states	$$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$2, $$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$3, and $$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$4, $$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$5
Robustness of quantum memory $$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$6 (Yuan et al., 2019)	Distance from entanglement-breaking channels	$$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$7
Coherence-based channel quality (Simnacher et al., 2018)	Basis-independent coherence preservation	$$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$8
CV adversarial-metrology witness (Abiuso, 2023)	Non-entanglement-breaking CV memory	$$E[\rho]=\int da\ \mathrm{Tr}[N^a\rho]\cdot \rho^{(a)}.$$9
Device-independent causal witness (Santos et al., 20 Jan 2026)	Black-box non-entanglement-breaking memory	$$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$0 classically, while quantum mechanics can reach $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$1

For weak coherent-state memories, fidelity is often complemented by state-independent quadrature benchmarks. In the warm-vapor gradient-echo memory, the verification used

$$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$2

together with conditional variances

$$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$3

and transfer coefficients

$$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$4

The classical limits are $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$5 and $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$6, while beating the no-cloning bound requires $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$7 and $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$8 (Hosseini et al., 2014).

A more universal quantifier is the robustness of quantum memory. The quantity $$\Lambda[\mathcal E]=\sum_i D_i\circ \mathcal E\circ I_i,$$9 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

$\mathcal E\succeq \mathcal F$0

with closed forms for dephasing, erasure, and stochastic damping channels (Yuan et al., 2019).

For qubit memories, geometric quantification is also possible. A single-qubit channel $\mathcal E\succeq \mathcal F$1 induces an ellipsoid $\mathcal E\succeq \mathcal F$2 in the Bloch sphere via $\mathcal E\succeq \mathcal F$3, with shape matrix $\mathcal E\succeq \mathcal F$4 and volume

$\mathcal E\succeq \mathcal F$5

The same work states the relation

$\mathcal E\succeq \mathcal F$6

and reconstructs the Choi state from the ellipsoid geometry in a semi-device-independent manner (Chang et al., 2023).

3. Direct experimental demonstrations of verified storage

A prominent unconditional demonstration is the warm-$\mathcal E\succeq \mathcal F$7Rb $\mathcal E\succeq \mathcal F$8-GEM device operated in a 10 cm vapor cell of isotopically enriched $\mathcal E\succeq \mathcal F$9Rb with 0.5 Torr Kr buffer gas at approximately $\mathcal F$0C. The two long-lived hyperfine ground states $\mathcal F$1 and $\mathcal F$2 of the $\mathcal F$3 manifold serve as $\mathcal F$4 and $\mathcal F$5, with an excited state $\mathcal F$6 in a $\mathcal F$7 scheme. A magnetic field gradient along the optical axis produces a controllable Raman bandwidth of about $\mathcal F$8 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 $\mathcal F$9 dB with about $\mathcal E$0 dB signal loss. The storage protocol combines Raman absorption, a $\mathcal E$1s control-off interval following a $\mathcal E$2s input pulse, and gradient inversion for time-reversed recall. More than $\mathcal E$3 of the incident signal can be absorbed, and the storage efficiency was $\mathcal E$4 for a $\mathcal E$5s total delay. For weak coherent states, the measured recall fidelity reached $\mathcal E$6 at mean photon number $\mathcal E$7, about $\mathcal E$8 at $\mathcal E$9, and about $t_1<t_2$0 at $t_1<t_2$1. Balanced homodyne detection with a scanned local-oscillator phase, about $t_1<t_2$2 shots per input amplitude, a detuned phase-reference pulse delayed by $t_1<t_2$3s, and iterative maximum-likelihood reconstruction yielded density matrices up to $t_1<t_2$4–$t_1<t_2$5 photons and corresponding Wigner functions. The noise spectra were at or below the shot-noise level, and for $t_1<t_2$6 one representative point was $t_1<t_2$7 and $t_1<t_2$8, placing the recalled states inside the no-cloning region (Hosseini et al., 2014).

Another foundational demonstration stored entangled two-mode squeezed states in two room-temperature cesium vapor cells, one for each mode, with memory time about $t_1<t_2$9 msec. The optical input consisted of two-mode squeezed states by $\{M_i\}$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 $\{M_i\}$1, the maximal fidelity achievable by an entanglement-breaking channel was $\{M_i\}$2, while the measured average fidelity was

$\{M_i\}$3

This was taken as rigorous proof that the memory preserved quantum coherence for the tested ensemble (Jensen et al., 2010).

These demonstrations exemplify two experimentally important verification styles. The warm-vapor GEM experiment used unconditional tomography together with state-independent $\{M_i\}$4-$\{M_i\}$5 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 (Hosseini et al., 2014, Jensen et al., 2010).

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 $\{M_i\}$6 and $\{M_i\}$7 and a payoff function $\{M_i\}$8, one defines the maximal average payoff

$\{M_i\}$9

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 $\{K_i\}$0, then $\{K_i\}$1 remains non-entanglement-breaking for every $\{K_i\}$2 exactly when $\{K_i\}$3 is non-entanglement-breaking (Rosset et al., 2017).

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 $\{K_i\}$4 with antibunching $\{K_i\}$5. The two EIT channels had zero-delay efficiencies about $\{K_i\}$6 and $\{K_i\}$7 and lifetimes $\{K_i\}$8s and $\{K_i\}$9s. The Mach-Zehnder visibility was $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$00, corresponding to $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$01 in the effective Bell operators. Using randomly chosen input polarizations from $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$02, the entanglement-witness payoff $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$03 remained positive out to about $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$04s, with typical short-time statistical uncertainty $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$05 (Yu et al., 2021).

For continuous variables, an MDI protocol based on adversarial metrology uses only trusted coherent-state preparation. Alice sends a coherent state $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$06 at time $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$07, later sends $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$08 after the delay $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$09, and the untrusted provider returns estimators $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$10 for quadrature sum and difference. The per-round score is

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$11

For the identity memory, the optimal beam-splitter and homodyne strategy yields $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$12. Any entanglement-breaking memory satisfies

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$13

which tends to $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$14 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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$15 one finds $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$16, so $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$17 is sufficient for certification (Abiuso, 2023).

A fully device-independent formulation replaces trusted state preparation by temporal causal inequalities. The relevant variables are a free instrumental choice $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$18 at time $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$19, a first outcome $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$20, and a second outcome $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$21 at time $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$22, under assumptions of no retrocausation, free choice of $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$23, and forward-in-time order $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$24. Classical instrumental models satisfy

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$25

while quantum mechanics can reach $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$26. In a trapped-ion processor with two outer $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$27 ions, dynamical decoupling echos every $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$28 ms, and storage times up to $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$29 ms, the observed value was

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$30

about $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$31 below the classical bound. Auxiliary checks gave $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$32 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$33, and the device-independent fidelity bound was $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$34 (Santos et al., 20 Jan 2026).

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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$35 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$36 of the maps at two times. If, for some entanglement monotone $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$37,

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$38

then no classical-memory decomposition exists, so the process requires genuinely quantum memory. The protocol needs only channel tomography of $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$39 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$40 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 (Bäcker et al., 2023).

A computationally simpler witness uses the von Neumann entropy. With an ancilla $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$41, an initial state $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$42, and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$43, the witness states that if

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$44

then the two-time dynamics cannot be realized with classical memory. For a maximally entangled initial state, one defines

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$45

and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$46 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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$47 at $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$48, $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$49 at $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$50, and squeezing $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$51 gives $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$52. The witness is again sufficient but not necessary, with perturbative stability $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$53 under small estimation errors (Bäcker et al., 29 Jan 2025).

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

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$54

the reconstructed Choi states at different collision numbers gave, for $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$55, the measured values $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$56 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$57, so $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$58 certified quantum memory for the single-qubit dynamics. The paper also presented a two-qubit toy model with measured $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$59 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$60, again satisfying the sufficient certification inequality (Bäcker et al., 22 Oct 2025).

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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$61-bit string $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$62 on an untrusted public memory and later performs online retrieve-bit operations. Classical memory checkers obey the lower bound $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$63, where $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$64 is private storage and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$65 is communication per retrieve. The quantum construction uses an error-correcting code $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$66 and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$67 copies of the fingerprint

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$68

compared to freshly queried summaries by repeated SWAP tests. This yields $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$69 private qubits and $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$70 communication, an exponential improvement in the space-query product over the classical setting (Dam et al., 2010).

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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$71 bits or qubits across rounds. One protocol achieves a quadratic gap: the honest prover uses $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$72 qubits, the verifier uses $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$73 classical bits, completeness is exactly

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$74

and any classical device with memory less than $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$75 bits has acceptance probability at most

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$76

A second protocol achieves an exponential gap, with honest parties using polylogarithmic memory while classical cheating devices with only $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$77 bits cannot pass (Malavolta et al., 29 May 2025).

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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$78-qubit quantum memory for a specified time interval. Two constructions are given. The first is a four-round PoQM based on $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$79-of-$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$80 puzzles and subexponential hardness of LWE, achieving $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$81-PoQM. The second is a polynomial-round PoQM via verifiable remote state preparation of BB84 states, based on polynomial-time hardness of LWE, achieving

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$82

The same work shows that PoQM imply one-way state puzzles, and that a restricted extractable version implies QCCC key exchange (Hhan et al., 5 Oct 2025).

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 (Hosseini et al., 2014, Yuan et al., 2019, Hhan et al., 5 Oct 2025).

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 (Bäcker et al., 2023, Bäcker et al., 29 Jan 2025). Standard channel tomography scales as $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$83 measurement outcomes for a $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$84-level system, while tomography overhead and circuit depth can mask small memory effects on current hardware (Bäcker et al., 2023, Bäcker et al., 22 Oct 2025). 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 (Rosset et al., 2017, Santos et al., 20 Jan 2026).

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 $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$85–$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$86 with correlation coefficient $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$87, a five-qubit parallel fidelity about $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$88 per channel, an evolution baseline with correction at $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$89, and a controlled-error case at $$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$90. In that work, fidelity is defined by shot-by-shot equality between the field and output bits,

$$\mathcal E(\rho)=\sum_i \mathrm{Tr}[M_i\rho]\sigma_i,$$91

rather than by an entanglement-breaking, no-cloning, or causal-inequality benchmark (Neukart et al., 15 Feb 2025). This suggests that “verifiable quantum memory” functions as an umbrella label spanning multiple verification paradigms rather than a single standardized criterion.