⇦Chinese Physics Letters, 2018, Vol. 35, No. 4, Article code 040301 Realization of Quantum Maxwell's Demon with Solid-State Spins * W.-B. Wang(王玮彬)1, X.-Y. Chang(常秀英)1, F. Wang(王飞)1, P.-Y. Hou(侯攀宇)1, Y.-Y. Huang(黄园园)1, W.-G. Zhang(张文纲)1, X.-L. Ouyang(欧阳晓龙)1, X.-Z. Huang(黄晛之)1, Z.-Y. Zhang(张正宇)2, H.-Y. Wang(王海艳)1, L. He(何丽)1, L.-M. Duan(段路明)1,2** Affiliations 1Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084 2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA Received 9 March 2018, online 17 March 2018 *Supported by the Ministry of Education of China, and the National Key Research and Development Program of China under Grant No 2016YFA0301902.
**Corresponding author. Email: lmduan@umich.edu
Citation Text: Wang W B, Chang X Y, Wang F, Hou P Y and Huang Y Y et al 2018 Chin. Phys. Lett. 35 040301 Abstract We report experimental realization of a quantum version of Maxwell's demon using solid state spins where the information acquiring and feedback operations by the demon are achieved through conditional quantum gates. A unique feature of this implementation is that the demon can start in a quantum superposition state or in an entangled state with an ancilla observer. Through quantum state tomography, we measure the entropy in the system, demon, and the ancilla, showing the influence of coherence and entanglement on the result. A quantum implementation of Maxwell's demon adds more controllability to this paradoxical thermal machine and may find applications in quantum thermodynamics involving microscopic systems. DOI:10.1088/0256-307X/35/4/040301 PACS:03.67.-a, 03.65.Ud, 03.65.Ta © 2018 Chinese Physics Society Article Text Maxwell's Demon is a gedanken experiment introduced a century ago by Maxwell that has a seeming contradiction with the second law of thermodynamics.[1,2] Suppose a box full of gas particles is separated into left and right sides and connected through a small gate controlled by a microscopic being called the demon. When a gas particle comes towards the gate, the demon only opens the gate when it observes that the particle is from the left. Due to this selective opening of the gate, eventually all the gas particles move to the right side, and the system entropy thus decreases which seemingly contradicts with the second law of thermodynamics. After decades of controversies in the interpretation, this paradox is finally resolved by the observation that the demon has to acquire information about the gas particle for each gate operation.[1-5] Erasing of this information transfers a finite amount of work to heat (Landauer's principle) and thus the demon's own entropy increases which is always no less than the entropy decrease in the system.[1-8] The essence of Maxwell's demon experiment is the information acquired by the demon and the feedback to the system depending on this information. This thought experiment has been implemented in several physical systems, where the entropy decrease and the transferring of heat to work in the system has been observed.[9-18] The demon's operations could also be implemented by quantum gates using microscopic qubits.[6-8] A quantum implementation allows the investigation of the role of quantum coherence and entanglement in the microscopic limit of thermal machines, a topic that raised significant interest in the rising field of quantum thermodynamics.[19-22] In particular, entanglement could be interpreted as a resource of negative entropy for some thermodynamic process.[21] Very recently, Maxwell's demon has been demonstrated using the nuclear magnetic resonance and the superconducting cavity systems based on quantum gate operations.[23,24] In those experiments, the demon still starts in a classical state and the role of entanglement has not been investigated yet. In this Letter, we report an experimental realization of quantum Maxwell's demon based on control of solid-state spins which demonstrates the important role of memory consumption and entanglement in the functionality of a quantum demon. A key observation to the resolution of Maxwell's demon paradox is that the operation by the demon consumes its memory and without a memory reset it will not be able to continuously reduce the system entropy. To demonstrate this key point, we use electronic spin states to represent the demon's memory and experimentally observe that the same demon's operation successively on two nearby nuclear spins reduces the entropy of the first system but fails to do so for the second one. Then, we prepare the demon in a quantum superposition state and demonstrate that the entropy decrease in the system depends on the observation basis. Finally, to demonstrate the important role of entanglement, we entangle the demon at the beginning with an ancilla representing an insider observer. For an outside observer without access to the ancilla's information, the demon's operation fails to reduce the system entropy. However, for an inside observer with help from the ancilla, we show that the entropy for both the system and the demon can decrease while the ancilla's entropy remains the same. This contradiction with the interpretation of the classical demon therefore experimentally confirms that the entanglement can serve as a resource of negative entropy for the quantum Maxwell's demon.[21] To implement quantum Maxwell's demon, we use electronic spin and nuclear spins in a nitrogen vacancy (NV) center of a room-temperature diamond sample to carry the states of the system and the demon. The spins in diamond NV centers provide a promising solid-state system for realization of quantum information processing and nanoscale sensing, which has attracted a lot of interests in recent years.[25-34] The ground state of the NV center has electronic spin $S=1$, which interacts with the dopant ${}^{14}$N with nuclear spin $I=1$ and the nearby ${}^{13}$C isotopes with nuclear spin $I=1/2$ in the host diamond through hyperfine coupling, forming a small quantum spin register.[25,26] As shown in Figs. 1(a) and 1(b), in our experiment we use the $|0\rangle $ and $|-1\rangle $ states of the NV electronic spin to represent the demon's status for gate closing and opening, respectively. In the first experiment, we use the states of $|\uparrow \rangle $ and $|\downarrow \rangle $ of a ${}^{13}$C nuclear spin with the hyperfine splitting of 6.5 MHz to represent the location of the first particle (particle A) on the left or right side of the box. To test successive operation of the demon, we introduce the second particle (particle B), whose location is represented by the $|0\rangle $ and $|1\rangle $ states of the ${}^{14}$N nuclear spin with the energy splitting about 2.8 MHz. The relevant level structure of the electronic and the nuclear spins in the NV center is Fig. 1(c), and the transitions between these levels can be manipulated through application of microwave or radio frequency (RF) pulses at the corresponding frequencies.
cpl-35-4-040301-fig1.png
Fig. 1. Implementation of quantum Maxwell's demon with solid state spins. (a) Illustration of Maxwell's demon paradox. A box with gas particles is separated into two sides. A demon at the middle detects which side the particle comes from (information acquiring) and only opens the gate if the particle is from the left side (feedback). Eventually all the particles move to the right side and the system entropy decreases which seemingly contradicts with the second law of thermodynamics. (b) Implementation of Maxwell's demon with three solid-state quantum spins. The demon's memory is represented by the electronic spin state, which controls opening or closing of the gate. Two particles A and B are represented by nuclear spins ${}^{13}$C and ${}^{14}$N, where the left or right sides of the particles are encoded with two orthogonal spin states. The information acquiring and feedback operations by the demon are implemented through conditional quantum gates. (c) The hyperfine level structure of the electronic and the nuclear spins. The hyperfine interactions and the applied external magnetic field split all the energy levels and differentiate their transition frequencies. Based on the frequency selection, the applied microwave or RF fields selectively drive a certain set of transitions and make the conditional quantum gates.
The operation of Maxwell's demon has two components. The first component is information acquiring which changes the demon's memory state depending on the side of incoming particle and in our case is realized through the quantum controlled-NOT gate C$_{\rm C}$–NOT$_{\rm E}$ or C$_{\rm N}$–NOT$_{\rm E}$, where the electronic spin state representing the demon's memory flips conditional on the control particle (${}^{13}$C or ${}^{14}$N nuclear spin). The second component is a feedback operation where the demon opens the gate conditional on its memory in the $|1\rangle $ state and the incoming particle changes its side in the box. In our case, this feedback operation is realized through the conditional quantum gate C$_{\rm E}$-NOT$_{\rm C}$ or C$_{\rm E}$–NOT$_{\rm N}$ for the first and the second particles, respectively. The quantum circuit that implements Maxwell's demon operations is shown in Fig. 2(a).
cpl-35-4-040301-fig2.png
Fig. 2. Maxwell's demon under successive operations. (a) The quantum circuit represents the demon's operations successively on the first (A) and the second (B) particles. Each demon's operation consists of two controlled-NOT quantum gates for information acquiring and feedback. We detect the state of the demon, particle A, and particle B at the beginning and after the first (step 1) and the second (step 2) demon's operation. (b) Experimental sequence of laser, microwave, and RF pulses for implementation of the successive demon's operations on particles A and B. Each conditional gate on the electronic spin is implemented by two microwave pulses on the corresponding transitions. The conditional gates on the nuclear spins are realized through RF pulses. The detections at the beginning, step 1, and step 2, are carried out by quantum state tomography, which consists of several microwave and RF pulses to rotate the measurement bases followed by the optical readout of the electronic spin state (see Supplementary Material B). (c) The measured state fidelity of the demon and particles A and B at the beginning and after step 1 and step 2 compared with the ideal states at the corresponding stages. (d) The entropy of the system and particles A and B measured through quantum state tomography at the three corresponding stages. The error bars in all the figures denote the confidence interval with probability of $68\%$, which corresponds to one standard deviation if the underlying distribution is Gaussian.
For Maxwell's demon experiment, the system starts in a mixed state where the particles have equal probabilities to be on the left or right side of the box while the demon's memory is in a pure state so that it can track the particle's information. In our experiment, we apply a weak external magnetic field of 38 Gauss to split the energy levels $|1\rangle $ and $|-1\rangle $ of the electronic spin, and the field is small enough so that the nuclear spins remain in the completely mixed states when we polarize the electronic spin to the $|0\rangle $ state through optical pumping. The CNOT gates for the demon's information acquiring operation are implemented through two microwave pulses and for the demon's feedback operation are implemented through an RF pulse. For instance, with the level structure shown in Fig. 1(c), application of $\pi $-pulses with MW3 and MW4 (MW2 and MW4) can realize the C$_{\rm C}$–NOT$_{\rm E}$ (C$_{\rm N}$–NOT$_{\rm E}$) gate, and application of $\pi $-pulses with RF1 (RF2) can realize the C$_{\rm E}$–NOT$_{\rm C}$ (C$_{\rm E}$–NOT$_{\rm N}$) gate. The typical experimental sequence is shown in Fig. 2(b). The essence of Maxwell's demon is to reduce the entropy of the system at the cost of entropy increase of its own memory. To characterize the entropy flow for the system and the demon, we directly measure the entropy of the system and the demon through quantum state tomography before and after the demon's operation on the first and the second particles. Quantum state tomography is implemented with a set of microwave and RF pulses that rotate the detection basis and then optical read-out of the electronic spin state by detection of fluorescence (see Supplementary Material B). For outcome states $i$ measured to occur with probability $p_{i}$, the entropy is defined as $S=-\sum_{i}p_{i}\log _{2}p_{i}$, where the base is taken to be $2$ for convenience. For comparison between the experimental state (constructed as a density matrix $\rho _{\rm e}$) and the outcome state $\rho _{\rm id}$ in the ideal case, we also measure the state fidelity $F$, which is defined as $F=[{\rm tr}( \sqrt{\sqrt{\rho _{\rm id}}\rho _{\rm e}\sqrt{\rho _{\rm id}}})]^{2}$. The measurement results for the state fidelity $F$ and the entropy $S$ are shown in Figs. 2(c) and 2(d). Before the demon's operation, both particles (nuclear spins) are in the maximally mixed states with entropy $S=1$. The demon (electronic spin) is in a pure state with a high fidelity (our calibration of the fluorescence levels for detection has subtracted the initialization imperfection by the green laser, see Supplementary Material B), but its entropy is measured to be $0.10_{-0.09}^{+0.10}$ as the value of $S$ is extremely sensitive to small imperfection near a pure state by its definition. In the supplementary materials, we provide an analysis of noise and plot the entropy as a function of the system noise and the purity of quantum state. From the plot, one can immediately see the sensitivity of entropy under small drop of the purity of quantum state. After the demon's operation on the first particle, the system (particles A and B) entropy decreases as expected while the entropy for the demon's memory increases. Due to the sensitivity of $S$ to small imperfection, the measured value of $S$ for particle A is $0.36_{-0.05}^{+0.04}$ although it has a high fidelity of $95.5_{-0.8}^{+0.8}\%$ to a pure state, where the numbers in the superscript and the subscript denote a confidence interval with probability of $68\%$. We then continue to apply the demon's operation on the second particle. As the demon has run out of its memory (which is only $1$ bit in this experiment), the successive operation of the demon on the second particle does not cause any entropy decrease for the system as one can see from the measurement result in Fig. 2(d). This confirms that the consumption of information storage capacity in the demon's memory is essential for its operation, a key concept for the resolution of Maxwell's demon paradox with the second law of thermodynamics. From the above data, one can see that the entropy decrease in the system is always bounded by the information acquired by the demon during the readout step, a key observation in interpretation of Maxwell's demon experiment. For instance, for the demon's interaction with particle A, the entropy decrease in the system is $0.58$. while the information acquired by the demon is given by $0.81$. For the demon's interaction with particle B, although the operations are the same, as the demon runs out of its memory, the information acquired by the demon is diminishing. So this limits the entropy decrease of the particle B by the demon's operation, measured to be $0$ within the experimental uncertainty from Fig. 2.
cpl-35-4-040301-fig3.png
Fig. 3. Maxwell's demon in a quantum superposition state. (a) The experimental sequence of laser, microwave, and RF pulses. The first microwave pulse $R_{\pi/2}^{X}$ induces a $\pi/2$ rotation along the $X$ spin-axis and prepares the demon in an equal superposition state. The demon's operation between the electronic spin and the ${}^{13}$C nuclear spin is achieved through the same conditional quantum gates, with the only difference of addition of two dynamical decoupling $\pi$-pulses to remove the dephasing noise of the electronic spin during the slow RF pulse on the nuclear spin. We detect the output state in both the diagonal (computational) and off-diagonal bases. For detection in the off-diagonal basis, we fist apply a conditional gate (a microwave pulse $C_{\rm C}$–$R_{\pi}^{Y}$) to disentangle the electronic and the nuclear spins as the demon's operation entangles them when the demon starts in a superposition state. (b) The measured state fidelity and entropy at the beginning and after the demon's operation. The final state is detected in both the diagonal and the off-diagonal bases. The fidelity denotes the comparison between the experimental states and the ideal states at the corresponding stages.
In our experiment, the measurements are focused on detection of the entropy flow for two reasons: First, the entropy plays a critical role for interpretation of Maxwell's demon paradox and its resolution with the second law of thermodynamics. Second, in our system it is more convenient to measure the entropy flow, which is complementary to the demonstration of classical Maxwell's demon in other systems where it is typically easier to measure the energy extraction instead of the entropy flow.[9-13] Although we cannot directly measure the work extraction in this experimental system, the entropy decrease is closely related to the amount of work that can be extracted from the system as discussed in Ref. [6]. In principle, the work could be extracted through stimulated emission by shining a microwave pulse resonant to this spin although in practice it is hard to realize that experimentally on a single spin. The measured entropy decrease provides an upper bound on the amount of work that could be extracted from the system through Landauer's principle.[2-7]
cpl-35-4-040301-fig4.png
Fig. 4. Maxwell's demon in an entangled state with an ancilla. (a) The experimental Pulse sequence for initial entangled state preparation, the demon's operation, and final state detection under different scenarios. The first part prepares a maximally entangled state $(|0,0\rangle-|-1,1\rangle)/\sqrt{2}$ between the demon (the electronic spin) and the ancilla (the ${}^{14}$N nuclear spin). The demon's operation is the same as those in Fig. 3, with the difference that now the microwave pulse becomes slow as one needs to resolve the level splitting of the ${}^{14}$N nuclear spin and the electronic spin coherence during the slow microwave pulse is protected by dynamical decoupling as well. For the final state detection, an outside observer detects directly the demon (the electronic spin) and the system (the ${}^{13}$C nuclear spin), while an inside observer, with access to the ancilla's information, applies two coherence-protected conditional quantum gates between the electronic spin and the ${}^{14}$N nuclear spin before quantum state tomography on the system and the demon, and obtains results inaccessible to the outside observer and incompatible with interpretation of classical Maxwell's demons. The system and the demon are disentangled by a microwave pulse (corresponding to a $C_{C}$–NOT$_E$ gate) in quantum state tomography. (b) Real and imaginary parts of the reconstructed density matrix elements for the initial entangled state between the demon and the ancilla. (c) The measured state fidelity and entropy at the beginning and after the demon's operation. The results for the final state are shown for both the outside and the inside observers with and without access to the ancilla's information, respectively. The notation "Demon + Ancilla" corresponds to a detection of the joint state and the von Neumann entropy for this composite system, while the others correspond to detection of each individual system.
We prepare Maxwell's demon (the electronic spin in our case) initially in a quantum superposition state $(|0\rangle +|1\rangle )/\sqrt{2}$ and test influence of quantum coherence to the demon's operation. For this experiment, we use the ${}^{13}$C nuclear spin as our system particle, which initially needs to be in a mixed state. However, it is better to polarize the ${}^{14}$N nuclear spin otherwise it works as a decoherence source for the electronic spin when the latter is in a superposition state. We therefore tune the external magnetic field to a value of 340 Gauss, and under this value of magnetic filed, the ${}^{14}$N nuclear spin has a strong enough spin-exchange interaction with the electronic spin in the excited state so that it gets polarized by the optical pumping, while the ${}^{13}$C nuclear spin only undergoes a weak spin-exchange interaction and remains to be mixed. The demon's operation is achieved by the same set of quantum gates. The only difference is that we need to apply dynamical decoupling pulses, which induce a fast $\pi $ rotation along the $X$ or $Y$ spin axis, to the electronic spin before and after the long RF pulse for the C$_{\rm E}$–NOT$_{\rm C}$ gate to remove the coupling of the electronic spin to the spin bath and keep its coherence. The experimental pulse sequence is shown in Fig. 3(a). After the demon's operation, if we measure the system in the classical basis (the computational basis $|\uparrow \rangle $ and $|\downarrow \rangle $), the system entropy does not show any decrease. This can be easily understood as a demon in an equal quantum superposition of classical states is just like a maximally mixed state looking from a measurement in the classical basis. However, if we measure the system in the superposition basis, the system entropy does decrease. The state fidelity and the measured entropy under different bases are shown in Figs. 3(b) and 3(c). So, when the demon hides its information storage capacity in the quantum superposition state, maintaining coherence is important for functioning of the quantum demon. Next, we discuss Maxwell's demon in an entangled state with an ancilla. Entanglement represents the key feature of a quantum system, and it provides new possibilities for a quantum Maxwell's demon. When we prepare the demon in a maximally quantum entangled state with an ancilla, to an outside observer who has no access to information in the ancilla, the demon appears to be in a maximally mixed state and thus it has no information storage capacity to reduce the system entropy by the demon's operation. However, an inside observer who has access to the ancilla can make use of this entanglement and find that the entropy in both the system and the demon can decrease with feedback from the ancilla. This provides a nice demonstration that entanglement can serve as a source of negative entropy for a quantum thermal machine.[21] To demonstrate Maxwell's demon starting from an entangled state, we still tune the external magnetic field to 340 Gauss and polarize both the electronic spin (the demon) and the ${}^{14}$N nuclear spin (the ancilla) by the optical pumping, which leaves the ${}^{13}$C nuclear spin (the system) in an almost maximally mixed state. Then, we can apply the microwave and RF pulses as shown in Fig. 4(a) to prepare the demon and the ancilla in a maximally entangled state $|{\it\Phi} _{\rm DA}\rangle =(|0,0\rangle -{|}-{1,1\rangle })/\sqrt{2}$. The RF pulse for the gate C$_{\rm E}$–NOT$_{\rm N}$ is long, so the electronic spin coherence needs to be protected by the dynamical decoupling pulses. Furthermore, this RF pulse also induces a phase shift for the electronic spin by off-resonant coupling, which is calibrated and compensated for by the phase of subsequent microwave pulses applied on the electronic spin. After the state preparation, we have measured the entanglement between the demon and the ancilla, with the result shown in Fig. 4(b). From the reconstructed density matrix $\rho _{\rm DA}$ of the demon and the ancilla from quantum state tomography, we find that the entanglement fidelity, defined as $F_{\rm e}=\left\langle {\it\Phi} _{\rm DA}\right\vert \rho _{\rm DA}|{\it\Phi} _{\rm DA}\rangle $, is $91.0_{-1.5}^{+1.3}\%$ for this initial state, which confirms entanglement. The demon's operation is achieved by the same set of gates as shown in Fig. 4(a). The coherence of the electronic spin during the conditional gate C$_{\rm C}$–NOT$_{\rm E}$ needs to be protected by the dynamical decoupling pulses, whose spin rotation axis is chosen to be the same as that of the microwave pulses for the C$_{\rm C}$–NOT$_{\rm E}$ gate, therefore the dynamical decoupling commutes with and does not affect the gate operation. After the demon's operation, we immediately perform measurements on the system and demon, emulating the measurement capability of an outside observer who does not have access to the ancilla. In this case, with the result shown in Fig. 4(c), we find that the entropy of both the system and the demon are at the maximum level and do not show any decrease by the demon's operation. This is expected as without access to the ancilla, the demon's entanglement information cannot be retrieved and it has no memory capacity to reduce the system entropy. We then emulate the capability of an inside observer and perform the feedback operation between the demon and the ancilla shown in Fig. 4(a). Same as the demon operation, this is carried out by applying the gates C$_{\rm E}$–NOT$_{\rm N}$ and C$_{\rm N}$–NOT$_{\rm E}$ (with coherence protected by the dynamical coupling). After the gates, we perform quantum state tomography on the system and the demon (they are disentangled by a gate C$_{\rm C}$–NOT$_{\rm E}$), and the resultant entropy is shown in Fig. 4(c). For this inside observer, the entropies for the system and for the demon both decrease. In the ideal case, the entropy should decrease to zero, however, in experiments because of sensitivity of the value of entropy to imperfections near a pure state, we find the entropy only decreases a bit although the fidelity to a pure state is given by $85.0_{-1.8}^{+1.6}\%$ and $75.3_{-2.6}^{+2.5}\%$ for the demon and the system, respectively. As the ancilla's entropy remains almost the same before and after the demon's operation, if we add up the total entropy of the system, the demon, and the ancilla, they decrease by the demon's operation. This is impossible for a classical demon, however, in the quantum case, with feedback from the ancilla, we use one bit of entanglement, which can convert to two bits of negative entropy,[21] and therefore it is possible to reduce the entropy for both the system and the demon. When there is entanglement, we need to count entropy by a collective measurement on the demon and the ancilla in an entangled basis instead of adding up their individual entropies, and as shown in Fig. 4, the total entropy counted in this way then increases and there is no inconsistency with the second law of thermodynamics by taking into account of the contribution of entanglement. We have implemented Maxwell's demon experiment using a quantum system with three solid-state spins in a diamond sample. Our experiment demonstrates for the first time the role of quantum entanglement in Maxwell's demon. We have prepared the demon in a quantum entangled state with an ancilla and demonstrated that the entanglement provides a resource of negative entropy for the system and the demon if the observer has access to the ancilla's information.[21] Through successive application of the demon's operation on system particles, we have also demonstrated that the memory capacity of Maxwell's demon is critical for its functioning, an observation important for the resolution of this paradox with the second law of thermodynamics.[2-6] A quantum implementation of Maxwell's demon, in particular with access to the entangled states, deepens our understanding of the interplay between thermodynamics and quantum entanglement and sheds new light on the profound connection between the fundamental principles of thermodynamics and quantum information theory.[6-8,19-22] The quantum control techniques in Maxwell's demon experiment may also find applications in implementation of other quantum thermal machines involving microscopic systems.[19-22] References Colloquium : The physics of Maxwell’s demon and informationIrreversibility and Heat Generation in the Computing ProcessThe thermodynamics of computation—a reviewThermodynamic cost of computation, algorithmic complexity and the information metricQuantum-mechanical Maxwell’s demonLandauer's erasure, error correction and entanglementQuantum Szilard EngineExtracting Work from a Single Heat Bath via Vanishing Quantum CoherenceA molecular information ratchetComprehensive Control of Atomic MotionExperimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equalityExperimental verification of Landauer’s principle linking information and thermodynamicsExperimental Observation of the Role of Mutual Information in the Nonequilibrium Dynamics of a Maxwell DemonExperimental realization of a Szilard engine with a single electronOn-Chip Maxwell’s Demon as an Information-Powered RefrigeratorPhotonic Maxwell’s DemonExtracting Work from Quantum Measurement in Maxwell’s Demon EnginesThe Second Law, Maxwell's Demon, and Work Derivable from Quantum Heat EnginesMaxwell’s Demon Assisted Thermodynamic Cycle in Superconducting Quantum CircuitsThe thermodynamic meaning of negative entropyA general derivation and quantification of the third law of thermodynamicsExperimental Rectification of Entropy Production by Maxwell’s Demon in a Quantum SystemExperimental observation of the role of mutual information in the nonequilibrium dynamics of a Maxwell demonThe nitrogen-vacancy colour centre in diamondAtom-like crystal defects: From quantum computers to biological sensorsObservation of Coherent Oscillations in a Single Electron SpinMultipartite Entanglement Among Single Spins in DiamondDynamic Polarization of Single Nuclear Spins by Optical Pumping of Nitrogen-Vacancy Color Centers in Diamond at Room TemperatureScalable architecture for a room temperature solid-state quantum information processorDecoherence-protected quantum gates for a hybrid solid-state spin registerRoom-Temperature Quantum Bit Memory Exceeding One SecondExperimental realization of universal geometric quantum gates with solid-state spinsUnconditional quantum teleportation between distant solid-state quantum bitsDecoherence and dynamical decoupling control of nitrogen vacancy center electron spins in nuclear spin bathsCoherent Dynamics of Coupled Electron and Nuclear Spin Qubits in DiamondQuantum Register Based on Individual Electronic and Nuclear Spin Qubits in DiamondSpin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond
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