Thermal devices powered by generalized measurements with indefinite causal order

A quantum-controlled device may produce a scenario in which two general quantum operations can be performed in such a way that it is not possible to associate a definite order for the operation's application. Such an indefinite causal order can be explored to produce nontrivial effects in quantum thermal devices. We investigate a measurement-powered thermal device that consists of generalized measurement channels with adjustable intensity parameters, where energy is exchanged with the apparatus in the form of work or heat. The measurement-based device can operate as a heat engine, a thermal accelerator, or a refrigerator, according to a measurement intensity setting. By employing a quantum switch of two measurement channels, we explore a thermal device fueled by an indefinite causal order. We also discuss how a coherent control over an indefinite causal order structure can change the operating regimes of the measurement-powered thermal device to produce an advantage when compared to a scenario with an incoherent control of the order switch.

Figure 1

Generalized measurements powered thermal device. (a) Schematic representation of the quantum thermal cycle powered by generalized measurements. In the first stroke, the system is initialized in the cold thermal equilibrium state ${\rho }^{\left(1\right)}$ at inverse temperature $\beta$. The second stroke is a generalized measurement channel ${\mathcal{M}}^a$ with a varying measurement (strength) parameter $a$. The third stroke is another generalized measurement ${\mathcal{M}}^b$ with a varying measurement parameter $b$. Both measurement channels represent nonselective measurements. (b) Description of the measurement parameter setting to perform the measurement powered cycle in three operation modes: accelerator, engine, or refrigerator.

Figure 2

Thermal devices powered by non-selective generalized measurements. (a) Engine cycle. Straining from a thermalization with the cold environment two generalized measurement channels are sequentially applied. The ${\mathcal{M}}^a$ channel increases the energy and von Neumann entropy of the working substance (heat absorption from the meter $\mathcal{A}$), while the second measurement ${\mathcal{M}}^b$ is dedicated to work extraction (to the meter $\mathcal{B}$) by changing the internal energy in an isentropic way. (b) Thermal accelerator cycle, with the first measurement parameter adjusted (in the meter $\mathcal{A}$) to increase energy and entropy of the working substance, while the second measurement channel adds energy in an isentropic way (work invested by the meter $\mathcal{B}$). (c) Refrigerator cycle, with a first isentropic measurement channel (meter $\mathcal{A}$), which performs work on the system, while the second measurement parameter is now adjusted to produce a heat flux from the cold environment to the meter $\mathcal{B}$.

Figure 3

Indefinite causal order between measurement channels ${\mathcal{M}}^a$ and ${\mathcal{M}}^b$. (a) Coherent control of generalized measurement channels causal order which is realized by projecting the controller in an orthogonal basis which produces in each postselected results an indefinite causal order. (b) Schematic description of the incoherent mixtures of causal orders which are implemented by tracing (nonobserving) the control qubit. This is equivalent to a stochastic control of the two causal orders of the measurement maps, i.e., ${\mathcal{M}}^b\left[{\mathcal{M}}^a\left({\rho }^{\left(1\right)}\right)\right]$ with probability $p^{a\to b}$ and ${\mathcal{M}}^a\left[{\mathcal{M}}^b\left({\rho }^{\left(1\right)}\right)\right]$ with probability $p^{b\to a}$.

Figure 4

Probability for the projection of the causal order controller in the state $|x_{-}\rangle$. We note that $p_{-}$ varies from $38\%$ to $50\%$. The probability for the projection of the causal order controller in the state $|x_{+}\rangle$ is complementary to the figure since $p_{+}=1-p_{-}$.

Figure 5

Quantum thermal device with indefinite causal order between measurement channels ${\mathcal{M}}^a$ and ${\mathcal{M}}^b$. (a) Accelerator and engine cycles. Starting from the cold thermal equilibrium state at inverse temperature $\beta$ (stroke 1), measurement channels (${\mathcal{M}}^a$ and ${\mathcal{M}}^b$) with indefinite causal order are applied (stroke 2) transferring heat from the meters to the working substance. Next, the isentropic measurement channel ${\mathcal{M}}^c$ is applied to extract or invest work (stroke 3). (b) Refrigerator cycle. Starting from the cold thermal equilibrium state, the isentropic measurement channel ${\mathcal{M}}^d$ is applied to invest work (stroke 2). Next measurement channels (${\mathcal{M}}^a$ and ${\mathcal{M}}^b$) with indefinite causal order are applied (stroke 3) to establish a heat flux from the cold environment to the meters.

Figure 6

Heat and invested work for the quantum engine and accelerator with indefinite causal order of measurement channels. (a) Heat absorbed from the meters $\mathcal{A}$ and $\mathcal{B}$ as a function of the measurement parameter $a$ ($b=a$) and the causal-order-control parameter $\theta$, considering the projection of the causal order controller in $|x_{+}\rangle$. (b) Absorbed heat considering the projection of the causal order controller in $|x_{-}\rangle$. (c) Work by the channel ${\mathcal{M}}^c$ considering the projection of the causal order controller in $|x_{+}\rangle$. (d) Work considering the projection of the causal order controller in $|x_{-}\rangle$. The thermal device operates as an engine when ${\mathcal{W}}^{(\pm )}\le 0$ (blue color gradient) and it is a thermal accelerator when ${\mathcal{W}}^{(\pm )}>0$ (red color gradient). The gray regions are excluded from the respective operation modes. All quantities were computed for $\beta \varepsilon =1.39$.

Figure 7

Engine efficiency and accelerator COP with indefinite causal order of measurement channels. (a) Efficiency in the engine operation mode, considering the projection of the causal order controller in $|x_{+}\rangle$. (b) Efficiency in the engine operation mode, considering the projection of the causal order controller in $|x_{-}\rangle$. (c) Coefficient of performance in the thermal accelerator mode, considering the projection of the causal order controller in $|x_{+}\rangle$. (d) Coefficient of performance in the thermal accelerator mode, considering the projection of the causal order controller in $|x_{-}\rangle$. The gray regions are excluded from the respective operation modes. All quantities were computed for $\beta \varepsilon =1.39$.

Figure 8

Refrigerator COP with indefinite causal order of measurement channels. (a) Coefficient of performance in the refrigerator mode, considering the projection of the causal order controller in $|x_{+}\rangle$. (b) Coefficient of performance in the refrigerator mode, considering the projection of the causal order controller in $|x_{-}\rangle$. The gray regions are excluded from the respective operation modes. Both quantities were computed for $\beta \varepsilon =0.45$.

Figure 9

Thermal device with incoherent causal order control. (a) Heat absorbed from the meters $\mathcal{A}$ and $\mathcal{B}$ as function of parameters $a$ and $\theta$. (b) Work invested by the channel ${\mathcal{M}}^c$. (c) Efficiency in the engine mode. (d) Coefficient of performance in the accelerator mode. The gray regions are excluded from the respective operation modes. All quantities were computed considering the scenario where the causal order controller is not observed (incoherent control) and for $\beta \varepsilon =1.39$.