# Quantum memories: What are they?

Searching the literature for the term "quantum memory" seems to bring up results from two different communities.

On the one hand there are quantum opticians, who see a quantum memory as something used to absorb a photon and store its quantum state before re-emitting it again. The memory in this case could be something as small as a single atom. More details on these are exponded in a review here.

On the other hand, there are those who see a quantum memory as a many-body system in which quantum information is stored. Possibilities for this include error correcting codes, such as the surface code, or the non-Abelian anyons that can arise as collective excitations in certain condensed matter systems. In this case it is not necessarily assumed that the quantum information existed anywhere before it was in the memory, or that it will be transferred elsewhere after. Instead, the state can be initialized, processed and measured all in the same system. An example of such an approach can be found here.

These two concepts of quantum memories seem, to me, to be quite disjoint. The two communities use the same term for different things that store a coherent state, but what they refer to is otherwise pretty unrelated. Is this the case, or are they really the same thing? If so, what exactly are the connexions?

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If you actually discuss with people working on quantum memories, you will notice (at least I did) that they share a vague definition : "a quantum memory is something which stores a quantum state" better than a classical memory could do. Beyond that, they have vastly different ideas on

• how to implement a quantum memory (single qubits, collective degrees of freedom, array of qubits impelmenting a topological error correction code ...)
• what to do with a quantum memory (RAM for a quantum computer, store states to reconstruct them later, store states to measure them later )
• how to evaluate the quality of a quantum memory (fidelity, quantum capacity, cheating probability in a noisy-storage model based quantum cryptography protocol...)

Note that the same kind of differences also apply on classical memories, between a sheet of paper, a magnetic tape, an ECC RAM or a group of neurons in my brain.

I'm convinced however that it is possible to give a generic definition of a quantum memory. In a paper (shameless plug) on a specivic kind of continuous variable quantum memory, I wrote

A quantum memory, by definition, stores informations about a quantum state for a given time interval, and it should do it better than any classical memory (i.e. classical-states based memory). Since an a priori known quantum state has a complete classical description (its density matrix), it can be reconstructed with an arbitrarily high fidelity by a setup only storing this description in a classical memory.

More specifically, following the noisy storage model literature, a quantum memory can be defined by a quantum channel, which itself can be described by a time-dependent completely positive (CP) map $\mathcal T_t$. If the quantum memory has a classical-output (e.g. if it is used for a delayed measurement), it can be modelled by a CP map followed by a measurement. It gives us a straightforward criterion to distinguish a classical memory from a quantum memory, since a classical memory supplemented by measurement and preparation can only implement entanglement breaking channel.

If the memory output is quantum, it can be said to be quantum memory iff $\mathcal T_t$ is not entanglement breaking. If the output is classical, one has to show that the outputs cannot be obtained by direct measurements of the input state.

The question whether the memory uses a (cloud of) atom(s) to store a photonic qubit or topologic error correcting codes to store the state of the nuclear spin of 5 NV-centres is irrelevant for the definition. In the same way that the RAM of my computer differs vastly from a poetry book. Both are classical memories.

Then, by definition, the classical capacity cannot be a figure of merit to characterize a quantum memory. But many figures of merits are possible, depending on the application, as with quantum channels. The quantum capacity seems a natural figure of merit, but a memory storing bound entangled state would be excluded by this figure of merit.

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Yet already Grover introduced an alternative idea of "memory", then used term "database". In a more recent modifications it is some quantum circuit with property $D: |k,0\rangle \to |k,b_k\rangle$ It was called "database" and may be compared with read-only memory, because may be considered as access to $k$-th cell of some storage to retrieve a bit $b_k$. Initially it was suggested a model with $b_k=0,1$ and with single unit element in whole "database". It was used to analyze speed up for an oracle. Later it was generalized to many elements and it is not too difficult to show, that it may be constructed a circuit with property $D : |k\rangle|0\rangle \to |k\rangle(\alpha_k|0\rangle + \beta_k|1\rangle)$ and it may be also generalized to more than one qubit for output. The problem that it is yet the read-only memory for qubits.
Yet, if to compare the model with CD-ROM, for varying, but known $\alpha_k$ and $\beta_k$ we may construct some analogue of CD-RW. It resembles already mentioned in previous answer idea of engineering of quantum state for given density matrix. I only do not agree with term "classical" for such a memory because (1) the output of such device is quantum state and so classical device would not do that, (2) for output with qubits (or other discrete quantum variables) it is possible to use encoding of $\alpha_k$ and $\beta_k$ with continuous quantum variables, e.g., using stereographic projection of Bloch sphere to pairs of continuous real-valued quantum variables $x_k,y_k$.