The mean king's problem: Spin 1

Solution to the Mean King's problem with mutually unbiased bases for arbitrary levels

The Mean King's problem with mutually unbiased bases is reconsidered for arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of d-1 mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when e.g., d=6 or d=10. In contrast to their result, we show that the King's problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing t…

Geometrical view of the Mean King Problem

Finite geometry is used to underpin finite, $d^2$, dimensional Hilbert space accommodating two particles, d dimensional each. d=prime $\ne2$. Central role is allotted to states with mutual unbiased bases (MUB) labelling underpinned with points of finite dual affine plane geometry (DAPG). The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space and provide a novel, geometrical view to a new solution of the Mean King Problem (MKP). Brief expositions to the topics considered: MUB, DAPG and the MKP are included rendering the paper self cont…

A Meaner King uses Biased Bases

The mean king problem is a quantum mechanical retrodiction problem, in which Alice has to name the outcome of an ideal measurement on a d-dimensional quantum system, made in one of (d+1) orthonormal bases, unknown to Alice at the time of the measurement. Alice has to make this retrodiction on the basis of the classical outcomes of a suitable control measurement including an entangled copy. We show that the existence of a strategy for Alice is equivalent to the existence of an overall joint probability distribution for (d+1) random variables, whose marginal pair distributions are fixed as the t…

Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem

We analyze and further develop a new method to represent the quantum state of a system of $n$ qubits in a phase space grid of $N\times N$ points (where $N=2^n$). The method, which was recently proposed by Wootters and co--workers (Gibbons {\it et al.}, quant-ph/0401155), is based on the use of the elements of the finite field $GF(2^n)$ to label the phase space axes. We present a self--contained overview of the method, we give new insights on some of its features and we apply it to investigate problems which are of interest for quantum information theory: We analyze the phase space representati…

Comments on "Best conventional solutions to the King's problem"

Conventional solutions to the (Mean) King's problem without using entanglement have been investigated by Aravind [P. K. Aravind, ``Best conventional solutions to the King's problem'', Z. Naturforsch. 58a, 682 (2003)]. We report that the upper bound for the success probability claimed there is not valid in general, and give a condition for the claim to be justified.

Solution to the King's problem with observables being not mutually complementary

We investigate the King's problem of the measurement of operators $\vec{n}_k \nobreak \cdot \nobreak \vecσ (k=1,2,3)$ instead of the three Cartesian components $σ_x$, $σ_y$ and $σ_z$ of the spin operator $\vecσ$. Here, $\vec{n}_k$ are three-dimensional real unit vectors. We show the condition over three vectors $\vec{n}_k$ to ascertain the result for measurement of any one of these operators.

A 2-Categorical Analysis of Complementary Families, Quantum Key Distribution and the Mean King Problem

This paper explores the use of 2-categorical technology for describing and reasoning about complex quantum procedures. We give syntactic definitions of a family of complementary measurements, and of quantum key distribution, and show that they are equivalent. We then show abstractly that either structure gives a solution to the Mean King problem, which we also formulate 2-categorically.

Bell states, mutually unbiased bases and the Mean King's problem

When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations.These Bell states are in one to one correspondence with the N^2 elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function.Thanks to the properties of general…

刻薄国王的新故事

New Tales of the Mean King

本文回顾刻薄国王问题并提出组合解法，证明仿射可分解设计可定义并求解类似的态重构问题，且无需辅助量子比特即可确定三量子比特系统中九次测量的结果。

高维平均国王问题的实验解决方案

Experimental Solutions to the High-Dimensional Mean King's Problem

本文提出一种基于PyTheus框架的通用实验方案，首次在素数维度（三维、五维、七维）解决平均国王问题，成功概率分别达72.8%、45.8%和34.8%，超过经典概率两倍，展示了量子纠缠实验实施的潜力。

Experimental Test of Tracking the King Problem

In quantum theory, the retrodiction problem is not as clear as its classical counterpart because of the uncertainty principle of quantum mechanics. In classical physics, the measurement outcomes of the present state can be used directly for predicting the future events and inferring the past events which is known as retrodiction. However, as a probabilistic theory, quantum-mechanical retrodiction is a nontrivial problem that has been investigated for a long time, of which the Mean King Problem is one of the most extensively studied issues. Here, we present the first experimental test of a vari…

Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values

The structural connections between the Kochen-Specker (KS) theorem, pre- and post-selection (PPS) paradoxes, and anomalous weak values are explored in detail. All PPS paradoxes, such as the 3-box paradox, the Quantum Cheshire Cat, and the Quantum Pigeonhole principle, construct a particular type of ontological model that assigns an eigenvalue to each observable (independent of context) of a system such that these assignments are consistent with the PPS. It is shown that such an ontological model must be explicitly contextual in the sense of the KS theorem, or otherwise implies either a restric…

平行与反平行自旋中的量子不相容性

Quantum Incompatibility in Parallel vs Antiparallel Spins

该研究探讨了平行与反平行自旋配置中量子不相容性的联合测量问题，揭示了反平行配置在预测自旋分量方面的独特优势，并探讨了其在加密协议和实验实现中的应用。

国际象棋中的量子国王-环支配：一种QAOA方法

Quantum King-Ring Domination in Chess: A QAOA Approach

本文提出量子国王-环支配基准，通过结构化实例评估QAOA性能，发现约束保持混合器收敛更快，且优于经典启发式方法。

Mean field master equation for self-interacting baths: comparison with exact spin--spin-bath dynamics

A mean field approximation is employed to derive a master equation suitable for self-interacting baths and strong system-bath coupling. Solutions of the master equation are compared with exact solutions for a central spin interacting with a spin-bath.

Mean spin entanglement of two massive Dirac particles under Lorentz transformations

We have studied the relativistic effects on the mean spin entanglement of two massive Dirac particles using the simultaneous eigen-spinors of the Foldy-Woutheysen mean spin operator and the Dirac Hamiltonian. We have obtained the transformation matrix from the spinor with specific momentum to the spinor with a transformed momentum under an arbitrary Lorentz transformation. Using the transformation matrix we have shown the consistent monotonic behavior between the concurrence and the maximum value of Bell parameter in Bell inequality of transformed spin states.

Quantifying geometric measure of entanglement by mean value of spin and spin correlations for pure and mixed states

We quantify the geometric measure of entanglement in terms of mean values of observables of entangled system. For pure states we find the relation of geometric measure of entanglement with the mean value of spin one-half for the system composed of spin and arbitrary quantum system. The geometric measure of entanglement for mixed states of rank-2 is studied as well. We find the explicit expression for geometric entanglement and the relation of entanglement in this case with the values of spin correlations. These results allow to find experimentally the value of entanglement by measuring a value…

Superposition principle for qubit states in spin-projection mean representation

The superposition principle of two qubit states is formulated as nonlinear addition rule of mean spinprojection onto three perpendicular directions. The explicit expression for the mean value determining the superposition state in term of the two initial pure states of the qubit is obtained. Possibility to check the superposition expression in experiments is discussed.

在量子计算机上探测高自旋系统的平均值与关联

Probing mean values and correlations of high-spin systems on a quantum computer

本文提出在量子计算机上模拟高自旋并测量自旋平均值与关联的协议，在 ibmq-santiago 和 ibmq-lima 上验证了 spin-1 及双自旋系统的时间演化，并将协议推广至任意自旋。

Spin Squeezing of Superposition of Biaxial State and two qubit Bell State

In this study we investigate the spin squeezing in superposition of a Biaxial state [1,2] and Bell state. Numerical and analytical solutions for the length of mean spin, mean spin direction and spin squeezing are given. It is shown that both the mean spin direction and spin squeezing parameter are determined by the coefficients of superposition and the relative phase.

Von Neumann Entropy from Mean Spin Vector

We show for a general pure entangled state of two two-level atoms, the von Neumann entropy of the partial traces can be directly measured from the magnitude of the mean spin vector of a single atom of the pair. We emphasize the fact that the von Neumann entropy of the partial traces for such a system can be obtained without knowing the exact form of the quantum state of the two atoms, if we have the value of the magnitude of the mean spin vector of a single atom of the pair. Mean spin vector, used in the context of spin squeezing and spectroscopic squeezing in population spectroscopy, is exper…

Pseudo-PT symmetric Dirac equation : effect of a new mean spin angular momentum operator on Gilbert damping

The pseudo-PT symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PT symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization - which is the quantity usually measured experimentally - is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phen…

基于自旋的动力学平均场理论对核磁共振信号的微观理解

Microscopic understanding of NMR signals by dynamic mean-field theory for spins

本文提出非局域自旋动力学平均场理论（nl-spinDMFT），通过计算自旋自相关和配对相关来准确捕捉核自旋动力学，揭示了自由感应衰减的微观来源，并在CaF$_2$和金刚烷实验数据中验证了其有效性。

Monogamy of entanglement and improved mean-field ansatz for spin lattices

We consider rather general spin-$1/2$ lattices with large coordination numbers $Z$. Based on the monogamy of entanglement and other properties of the concurrence $C$, we derive rigorous bounds for the entanglement between neighboring spins, such as $C\leq 1/\sqrt{Z}$, which show that $C$ decreases for large $Z$. In addition, the concurrence $C$ measures the deviation from mean-field behavior and can only vanish if the mean-field ansatz yields an exact ground state of the Hamiltonian. Motivated by these findings, we propose an improved mean-field ansatz by adding entanglement.

Rapid filling of the spin gap with temperature in the Schwinger-boson mean-field theory of the antiferromagnetic Heisenberg kagome model

Using Schwinger-boson mean-field theory, we calculate the dynamic spin structure factor at low temperatures $0<T\ll J$ for the spin-$1/2$ antiferromagnetic Heisenberg kagome model, within the gapped $\mathbb{Z}_2$ spin liquid phase Ansatz. We find that the spectral gap rapidly fills with temperature, with robust low-energy spectral weight developing by a temperature of $Δ/3$, where the spin gap is $2Δ$ (i.e., $Δ$ is the spinon gap), before any appreciable rise in spinon density or change in zero-temperature mean-field parameters. This is due to deconfinement of spinons which leads to terms sup…