Quantum speed limit for qubit systems: Exact results

Given an initial state, a target state, and a driving Hamiltonian, how fast can the initial state evolve into the target state according to the Schröchinger dynamics? This problem arises in a variety of contexts such as quantum computation, quantum control, and in particular, the problem of maximum information processing rate of quantum systems, and has been studied extensively due to its fundamental importance. In this paper, we purse further the study in the qubit case in which the particular structure admits stronger results. We use the quantum fidelity as well as relative entropy as a figure of merit to characterize the closeness between a fixed initial qubit state and another one undergoing unitary evolution. We work out explicitly maximal and minimal fidelity and relative entropy by determining the closest and the farthest states to the target state and show that these results are unique for qubit systems. We also determine the minimal time for a state to evolve to the extremal states (that is, the farthest one evolved from the initial state in the sense of minimal fidelity or maximal relative entropy), which generalizes the celebrated Mandelstam–Tamm bound and the Margolus–Levitin bound for qubit systems. We further reveal an interesting fact that this minimal time is independent of the initial states.     

Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system

The concept of entanglement entropy appears in multiple contexts, from black hole physics to quantum information theory, where it measures the entanglement of quantum states. We investigate the entanglement entropy in a simple model, the spin-boson model, which describes a qubit (two-level system) interacting with a collection of harmonic oscillators that models the environment responsible for decoherence and dissipation. The entanglement entropy allows to make a precise unification between entanglement of the spin with its environment, decoherence, and quantum phase transitions. We derive exact analytical results which are confirmed by Numerical Renormalization Group arguments both for an ohmic and a subohmic bosonic bath. The entanglement entropy obeys universal scalings. We make comparisons with entanglement properties in the quantum Ising model and in the Dicke model. We also emphasize the possibility of measuring this entropy using charge qubits subject to electromagnetic noise; such measurements would provide an empirical proof of the existence of entanglement entropy.     

Quantum uncertainty relations of two generalized quantum relative entropies of coherence

In this paper, we consider the quantum uncertainty relations of two generalized relative entropies of coherence based on two measurement bases. First, we give quantum uncertainty relations for pure states in a d-dimensional quantum system by making use of the majorization technique; these uncertainty relations are then generalized to mixed states. We find that the lower bounds are always nonnegative for pure states but may be negative for some mixed states. Second, the quantum uncertainty relations for single qubit states are obtained by the analytical method. We show that the lower bounds obtained by this technique are always positive for single qubit states. Third, the lower bounds obtained by the two methods described above are compared for single qubit states.     

Quantum information capsule and information delocalization by entanglement in multiple-qubit systems

Where do entangled multiple-qubit systems store information? For information injected into a qubit, this question is nontrivial and interesting since the entanglement delocalizes the information. So far, a common picture is that of a qubit and its purification partner sharing the information quantum mechanically. Here, we introduce a new picture of a single qubit in the correlation space, referred to as quantum information capsule (QIC), confining the information perfectly. This picture is applicable for the entangled multiple-qubit system in an arbitrary state. Unlike the partner picture, in the QIC picture, by swapping the single-body state, leaving other subsystems untouched, the whole information can be retrieved out of the system. After the swapping process, no information remains in the system.     

Spin based heat engine: demonstration of multiple rounds of algorithmic cooling

We experimentally demonstrate multiple rounds of heat-bath algorithmic cooling in a 3 qubit solid-state nuclear magnetic resonance quantum information processor. By pumping entropy into a heat bath, we are able to surpass the closed system limit of the Shannon bound and purify a single qubit to 1.69 times the heat-bath polarization. The algorithm combines both high fidelity coherent control and a deliberate interaction with the environment. Given this level of quantum control in systems with larger reset polarizations, nearly pure qubits should be achievable.     

Quantum uncertainty relations of Tsallis relative α entropy coherence based on MUBs

In this paper,we discuss quantum uncertainty relations of Tsallis relative α entropy coherence for a single qubit system based on three mutually unbiased bases.For α∈[1/2,1)U(1,2],the upper and lower bounds of sums of coherence are obtained.However,the above results cannot be verified directly for any α∈(0,1/2).Hence,we only consider the special case of α=1/n+1,where n is a positive integer,and we obtain the upper and lower bounds.By comparing the upper and lower bounds,we find that the upper bound is equal to the lower bound for the special α=1/2,and the differences between the upper and the lower bounds will increase as α increases.Furthermore,we discuss the tendency of the sum of coherence,and find that it has the same tendency with respect to the different θ or φ,which is opposite to the uncertainty relations based on the Rényi entropy and Tsallis entropy.     

Quantum information,oscillations and the psyche

In this paper, taking the theory of quantum information as a model, we consider the human unconscious, pre-consciousness and consciousness as sets of quantum bits (qubits). We view how there can be communication between these various qubit sets. In doing this we are inspired by the theory of nuclear magnetic resonance. In this way we build a model of handling a mental qubit with the help of pulses of a mental field. Starting with an elementary interaction between two qubits we build two-qubit quantum logic gates that allow information to be transferred from one qubit to the other. In this manner we build a quantum process that permits consciousness to “read” the unconscious and vice versa. The elementary interaction, e.g. between a pre-consciousness qubit and a consciousness one, allows us to predict the time evolution of the pre-consciousness + consciousness system in which pre-consciousness and consciousness are quantum entangled. This time evolution exhibits Rabi oscillations that we name mental Rabi oscillations. This time evolution shows how for example the unconscious can influence consciousness. In a process like mourning the influence of the unconscious on consciousness, as the influence of consciousness on the unconscious, are in agreement with what is observed in psychiatry.     

Quantum information capsule in multiple-qudit systems and continuous-variable systems

Quantum correlations in an entangled many-body system are capable of storing information. Even when the information is injected by a local unitary operation to the system, the entanglement delocalizes it. In a recent study on multiple-qubit systems, it is shown that a virtual qubit defined in the correlation space plays a role of perfect storage of delocalized information, which is called a quantum information capsule (QIC). To enhance the capacity of quantum information storage, it is crucial to formulate the cases for multiple-qudit systems and continuous-variable (CV) systems. We analytically prove that it is possible to construct a QIC for general write operations of the systems. It turns out that the extension to quantum field theory is achievable. For Gaussian states, we explicitly construct a QIC for shift write operations. We analyze the time-evolution of QIC in a CV system to demonstrate the diffusion of information in entangled pure states.     

原子系统中偶极-偶极相互作用对量子存储支撑熵不确定度的调控研究

本文研究了Markovian过程和non-Markovian过程两种情况下,原子系统中偶极-偶极(D-D)相互作用和失谐量对两原子系统量子存储支撑(QMA)熵不确定度的调控作用.与Markovian过程相比,在non-Markovian过程中,由于库环境的记忆效应使得具有D-D相互作用的两原子系统QMA熵不确定度的演化行为呈现出振荡上升现象,且熵不确定度的上升趋势得到减缓,从而抑制了由系统量子噪声引起的退相干效应.此外,D-D相互作用对两原子系统QMA熵不确定度具有显著的调控能力,且D-D相互作用比失谐量的调控能力更强.     

Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves

Bennett and Schumacher’s postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch’s model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between “Deutschian” CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.     

Quantum information entropy for one-dimensional system undergoing quantum phase transition

Calculations of the quantum information entropy have been extended to a non-analytically solvable situation. Specifically, we have investigated the information entropy for a one-dimensional system with a schematic "Landau" potential in a numerical way. Particularly, it is found that the phase transitional behavior of the system can be well expressed by the evolution of quantum information entropy. The calculated results also indicate that the position entropy S_x and the momentum entropy S_p at the critical point of phase transition may vary with the mass parameter M but their sum remains as a constant independent of M for a given excited state. In addition, the entropy uncertainty relation is proven to be robust during the whole process of the phase transition.     

Universal Approximation of Multi-copy States and Universal Quantum Lossless Data Compression

We have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE Trans. IT 46, 810–819 (2009)), the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning ‘approximation error’ up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression.     

量子BB84协议在联合旋转噪音信道上的安全性分析

多数在理想条件下设计的量子密码协议没有考虑实际通信中噪音的影响,可能造成机密信息不能被准确传输,或可能存在窃听隐藏在噪音中的风险,因此分析噪音条件下量子密码协议的安全性具有重要的意义.为了分析量子BB84协议在联合旋转噪音信道上的安全性,本文采用粒子偏转模型,对量子信道中的联合噪音进行建模,定量地区分量子信道中噪音和窃听干扰;并且采用冯·诺依曼熵理论建立窃听者能窃取的信息量与量子比特误码率、噪音水平三者之间的函数关系,定量地分析噪音条件下量子信道的安全性;最后根据联合噪音模型及窃听者能窃取的信息量与量子比特误码率、噪音水平三者之间的关系,定量地分析了量子BB84协议在联合噪音条件下的安全性并计算噪音临界点.通过分析可知,在已有噪音水平条件下,窃听者最多能够从通信双方窃取25%的密钥,但是Eve的窃听行为会被检测出来,这样Alice和Bob会放弃当前协商的密钥,重新进行密钥协商,直至确认没有Eve的窃听为止.这个结果说明量子BB84协议在联合旋转噪音信道下的通信是安全的.     

与量子光场相互作用的运动原子的熵压缩

运用量子信息熵理论,研究了与量子光场相互作用的二能级运动原子的熵压缩,讨论了原子运动和场模结构对原子熵压缩的影响,并且比较了分别从基于信息熵测不准关系和海森伯测不准关系出发得出的结果,表明原子的运动导致了原子熵压缩的周期性演化;随着场模结构参量的增大,熵压缩的演化周期缩短,压缩时间延长;选择适当的系统参量,运动原子能够呈现长时间的持续熵压缩效应。当原子反转为零时,基于海伯堡测不准关系的方差压缩定义不再有效,而熵压缩实现了对原子压缩效应的高灵敏量度。     

Quantum information transmission between two qubits through an intermediary photon gas

The theory of the quantum information transmission between two semiconductor two-level quantum dots as two qubits through an intermediary photon gas in a cavity is presented. The reduced density matrix of each two-level quantum dot is the quantum information encoded into this qubit. The quantum information exchange between two distant qubits imbedded in the photon gas is performed in the form of the mutual dependence of their reduced density matrices due to the interaction between the electrons in the qubits and the photon gas. The system of rate equations for the reduced density matrix of the two-qubit system is derived. From the solution of this system of equations it follows the mutual dependence of the reduced density matrices of two distant qubits.     

Information entropy and entanglement of a superconducting qubit coupled to a cavity field with its spontaneous decay

The quantum entanglement between superconducting qubit and cavity field is described quantitatively in the presence of spontaneous decay. Depending on how how a system is quantum correlated with its environment, the entanglement dynamics between the qubit and cavity is evaluated and investigated during the dissipative process. The motivation based on recent experiments wherein the Cooper box can be used to probe the decay of the resonator superposition state due to environmental decoherence, we theoretically investigate the dynamics of entanglement measured by the negativity. Wehrl entropy and Wehrl phase distribution of a superconducting qubit coupled to a cavity field induced by a superconducting qubit-damping reservoir governed by a master equation.     

The fully entangled fraction as an inclusive measure of entanglement applications

Characterizing entanglement in all but the simplest case of a two qubit pure state is a hard problem, even understanding the relevant experimental quantities that are related to entanglement is difficult. It may not be necessary, however, to quantify the entanglement of a state in order to quantify the quantum information processing significance of a state. It is known that the fully entangled fraction has a direct relationship to the fidelity of teleportation maximized under the actions of local unitary operations. In the case of two qubits we point out that the fully entangled fraction can also be related to the fidelities, maximized under the actions of local unitary operations, of other important quantum information tasks such as dense coding, entanglement swapping and quantum cryptography in such a way as to provide an inclusive measure of these entanglement applications. For two qubit systems the fully entangled fraction has a simple known closed-form expression and we establish lower and upper bounds of this quantity with the concurrence. This approach is readily extendable to more complicated systems.     

Entropy Dynamics in the System of Interacting Qubits

The classical second law of thermodynamics demands that an isolated system evolves with a nondiminishing entropy. This holds as well in quantum mechanics if the evolution of the energy-isolated system can be described by a unital quantum channel. At the same time, the entropy of a system evolving via a nonunital channel can, in principle, decrease. Here, we analyze the behavior of entropy in the context of the H-theorem. As exemplary phenomena, we discuss the action of a Maxwell demon (MD) operating a qubit and the processes of heating and cooling in a two-qubit system. Further we discuss how small initial correlations between a quantum system and a reservoir affect the entropy increase during the quantum-system evolution.