Resource Theory of Quantum Memories and Their Faithful Verification with Minimal Assumptions

We provide a complete set of game-theoretic conditions equivalent to the existence of a transformation from one quantum channel into another one, by means of classically correlated preprocessing and postprocessing maps only. Such conditions naturally induce tests to certify that a quantum memory is capable of storing quantum information, as opposed to memories that can be simulated by measurement and state preparation (corresponding to entanglement-breaking channels). These results are formulated as a resource theory of genuine quantum memories (correlated in time), mirroring the resource theory of entanglement in quantum states (correlated spatially). As the set of conditions is complete, the corresponding tests are faithful, in the sense that any non-entanglement-breaking channel can be certified. Moreover, they only require the assumption of trusted inputs, known to be unavoidable for quantum channel verification. As such, the tests we propose are intrinsically different from the usual process tomography, for which the probes of both the input and the output of the channel must be trusted. An explicit construction is provided and shown to be experimentally realizable, even in the presence of arbitrarily strong losses in the memory or detectors.

Popular Summary

Storing and processing information in quantum systems could unlock new abilities in computation and communication. For example, quantum protocols can distribute secret keys while mitigating all eavesdropping attempts. These applications require quantum resources (sources, memories, transmission channels) that cannot be simulated using traditional computers or communication links; otherwise, the quantum protocol would not offer any advantage. Among other tasks, the distribution of secret keys in networks requires two abilities: preparing distant systems in a quantum superposition (entanglement) and preserving quantum information in memories. Entanglement is well studied—entangled states can be certified with no assumptions on the experimental devices. However, the theoretical work on quantum memories has been relatively neglected. We fill this gap by describing how quantum memories can be compared and transformed. We also propose experimental tests to certify quantum memories with minimal assumptions.

Given a single quantum memory device, our proposed tests rest on the trusted preparation of a prescribed quantum state without leaking additional information. This assumption is always required; otherwise, the memory could simply store the classical description corresponding to the input state and prepare the required state at retrieval time. Our tests discriminate between entanglement-breaking (classical) memories and useful devices. On a finer level, they quantify the usefulness of quantum memories.

Our work paves the way towards a complete resource theory of quantum memories, and it shows how results for spatially correlated systems can be translated to the temporal case.

Figure 1

The quantum device in diagram (i) is not in the quantum domain whenever its functioning can be simulated by a fixed measurement, arbitrary processing of the classical outcome, and a preparation, as shown in diagram (ii). Such devices are called entanglement-breaking channels.

Figure 2

Quantumness verification via Bell tests.

Figure 3

The test configuration. Time flows from left to right. At time $t=t_0$, a first input ${\xi }_x$ (which must be trusted by the referee who prepares it) is handed over to the experimentalist (whose laboratory is represented by the flying cloud). After some time, at $t=t_1>t_0$, the referee hands over a second input ${\psi }_y$. At this point, the experimentalist is required to output a classical outcome, labeled by $b$. By repeating this procedure many times and observing the input/output correlation $p(b|x,y)$ so obtained, it is possible to verify whether the experimentalist is actually able or not to preserve quantum information through a time $\delta =t_1-t_0>0$.

Figure 4

(i) The channel $\mathcal{N}:\mathsf{D}({\mathsf{H}}_{\mathrm{A}})\to \mathsf{D}({\mathsf{H}}_{\mathrm{B}})$ in our resource theory. (ii) A free channel decomposed as a quantum-classical-quantum or measure-and-prepare channel using POVM elements $\{{\mathrm{\Pi }}_{i|\mu }\}$ and a family of states ${\rho }_{i,\mu }^{\prime }$. (iii) The free transformation of a channel $\mathcal{N}$ into ${\mathcal{N}}^{\prime }$.

Figure 5

A diagram relating the uses of the channels $\mathcal{N}$ and ${\mathcal{N}}^{\prime }$ to produce the signature correlation of ${\mathcal{N}}^{\prime }$ in Eq. (b6). The correlations of both boxes enclosed in double lines are identical, and thus their behavior is essentially indistinguishable.

Figure 6

(i) To correct the effect of the Bell measurement and the preparation of a maximally entangled state, it is sufficient to apply a unitary correction selected using the outcome of the Bell measurement. The resulting channel (bigger box) is equivalent to ${\mathcal{N}}^{\prime }$. (ii) As seen in Fig. 5, the same correction can be applied when using $\mathcal{N}$ to recover ${\mathcal{N}}^{\prime }$. In the dotted blob, we show the parts of the protocol that are regrouped to form a free transformation $\mathrm{\Lambda }$ as defined in Sec. 3.