How an autonomous quantum Maxwell demon can harness correlated information

We study an autonomous quantum system which exhibits refrigeration under an information-work trade-off like a Maxwell demon. The system becomes correlated as a single “demon” qubit interacts sequentially with memory qubits while in contact with two heat reservoirs of different temperatures. Using strong subadditivity of the von Neumann entropy, we derive a global Clausius inequality to show thermodynamic advantages from access to correlated information. It is demonstrated, in a matrix product density operator formalism, that our demon can simultaneously realize refrigeration against a thermal gradient and erasure of information from its memory, which is impossible without correlations. The phenomenon can be even enhanced by the presence of quantum coherence.

Figure 1

(a) A snapshot of our quantum Maxwell demon. The demon qubit $D$ interacts sequentially with each qubit in a memory $\mathbb{M}$ via an open-system process in contact with two thermal reservoirs of different temperatures. For a given interaction, $M$ is the currently interacting qubit, $\overline{M}$ the subsystem of previously interacting qubits, and $\tilde{M}$ that of qubits yet to interact. The joint state of $D\mathbb{M}$ is described by a matrix product density operator with periodic boundary conditions, though we take sufficiently many interactions that the system has reached a periodic steady state. Interaction with the shared system $D$ allows the possibility to build further correlations within $\overline{M}$, hence the double line. (b) The transition diagram of an individual interaction between the demon qubit and a memory qubit. Solid arrows represent possible stochastic transitions, allowed by exchanging a unit $\mathrm{\Delta }$ of energy with the associated reservoir, with transition rates satisfying the detailed balance condition. For example, intrinsic transitions coupled with the thermal reservoir of temperature $T_h$ change only the state of the demon, whereas cooperative transitions coupled with that of $T_c(<T_h)$ change the joint state of the demon and the memory qubit, leaving a record on the memory. The dashed arrows indicate the cyclic processes which refrigerate (pump energy against the gradient), with the horizontal dot-dashed transitions provided by the memory shift. Note both refrigerative processes result in a net flip of the memory qubit from “0” to “1.”

Figure 2

A pictorial representation of our time evolution method, which relies on a matrix product density operator (MPDO) description, for $n=3$. (a) On the left is a schematic depiction of our interaction sequence, with quantum operations vectorized to let us represent the sequence as a quantum circuit. The shared degree of freedom is the demon qubit $D$, and the remaining degrees of freedom are the memory $\mathbb{M}$. We indicate that a particular qubit is not acted upon by a given quantum operation by drawing the qubit's line over the quantum operation's box. We parametrize the sequence as a matrix product operator (MPO), shown on the right, with physical indices labeled according to the corresponding qubit degrees of freedom. (b) We multiply our initial MPDO, for which the state of $\mathbb{M}$ is translationally invariant and uncorrelated with that of $D$, by the MPO describing our interaction sequence to obtain an approximation to the steady state. This is done by performing a sequence of index contractions, shown on the left as dotted boxes. The approximation to the steady state, which is valid for sufficiently large $n$, is shown as the resulting MPDO on the right. All boundary conditions are taken to be periodic.

Figure 3

(a) The nonequilibrium phase diagram for input states from the GHZ-correlated family described in Observation t4, analogous to Fig. 2 in Ref. [9], for $\tau =0.3$ (units of $1/\mathrm{\Delta }$ for $\hslash =1$). For $\zeta =\pm 1$, the input is a product, and it is entangled for $\zeta \in \left(-1,1\right)$, with the strongest correlations at $\zeta =0$, for which it is the GHZ state. Note that for every state in this family, the reduced state of any qubit is diagonal in the $z$ basis. We see that correlations shift the phase boundaries to induce a region of simultaneous refrigeration and erasure near the uncorrelated phase transition triple point ($\varepsilon =\zeta =0$). (b) Dependence of the Clausius terms, $Q_{h\to c}({\beta }_c-{\beta }_h)$ and $\mathrm{\Delta }{S}_M$, on the interaction time, $\tau$ when $\varepsilon =0.01$ in comparing the GHZ state for $\zeta =0$—solid black and orange (gray) curves, respectively—to the product of its single-qubit reduced states, which are all maximally mixed—dotted black and orange (gray) curves, respectively. The reduced evolution of the interacting memory qubit is monotonic when there is a strict trade-off of refrigeration and erasure; both cannot be simultaneously negative. This is obeyed by the uncorrelated input, but the correlated input shows a departure from monotonicity for $\tau \lesssim 2.2$. For long-enough interaction times, the effect of correlations is “washed out” as the terms for correlated input approach the values for their uncorrelated-input counterparts.

Figure 4

Generation of correlations $\mathrm{\Delta }{I}_{M:\tilde{M}}$ between the interacting qubit $M$ and the qubits $\tilde{M}$ yet to interact, as a function of the interaction time $\tau$, for the GHZ input of Fig. 3. $\tau$ is in units of $1/\mathrm{\Delta }$ for $\hslash =1$, and $\varepsilon =0.01$. Correlations are always being consumed, and so this term offsets the others in the Clausius inequality such that simultaneous refrigeration and erasure does not violate the second law for this input family. The dotted line is at $\mathrm{\Delta }{I}_{M:\tilde{M}}=-ln2$ and is meant to provide a visual aid, as it represents the maximal possible consumption of correlations for this state.

Figure 5

(a) The three-dimensional surface whose enclosure is the set of local Bloch vectors for which the locally coherent GHZ-correlated family attains a quantum erasure advantage over its corresponding classically correlated family, whose off-diagonal elements in the $z$ basis have been projected out. Here $\tau =0.3$ and $\varepsilon =0.01$. We see clearly the predicted azimuthal symmetry about the $z$ axis, and the advantage vanishes along this axis as we expect, since the two inputs are the same in this case. (b) The difference between the erasure performance for the quantum correlated locally coherent input state and that of its decohered counterpart for $\varepsilon =0.01$, $\varphi =\pi /2$, $\theta =0.01$, and ${\zeta }_{\vec{n}}=-0.02$ as a function of the interaction time $\tau$. A quantum advantage is demonstrated when this quantity is negative. We see that the advantage is several orders of magnitude smaller and lost much faster than the advantage afforded by classical correlations of Fig. 3.