Reinhold A. Bertlmann, Philipp Krammer
We investigate the entanglement properties of a three--parameter family of states that are part of the magic simplex of two qutrits, which is a simplex of states that are mixtures of maximally entangled two--qutrit Bell states. Using entanglement witnesses we reveal large regions of bound entangled and separable states.
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Non-stabilizer eigenstates of Clifford operators are natural candidates for endpoints of magic state distillation routines. We provide an explicit bestiary of all inequivalent non-stabilizer Clifford eigenstates for qutrits and ququints. For qutrits, there are four non-degenerate eigenstates, and two families of degenerate eigenstates. For ququints, there are eight non-degenerate eigenstates, and three families of degenerate eigenstates. Of these states, a simultaneous eigenvector of all Clifford symplectic rotations known as the qutrit strange state is distinguished as both the most magic qut…
The ternary Golay code -- one of the first and most beautiful classical error-correcting codes discovered -- naturally gives rise to an 11-qutrit quantum error correcting code. We apply this code to magic state distillation, a leading approach to fault-tolerant quantum computing. We find that the 11-qutrit Golay code can distill the "most magic" qutrit state -- an eigenstate of the qutrit Fourier transform known as the strange state -- with cubic error-suppression and a remarkably high threshold. It also distills the "second-most magic" qutrit state, the Norell state, with quadratic error-supp…
Magic state distillation (MSD) is a purification protocol that plays a central role in fault tolerant quantum computation. Repeated iteration of the steps of a MSD protocol, generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. Magic state distillation was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher dimensional qudit-based computation. Here, we describe a general approac…
We investigate the state space of bipartite qutrits. For states which are locally maximally mixed we obtain an analog of the ``magic'' tetrahedron for bipartite qubits--a magic simplex W. This is obtained via the Weyl group which is a kind of ``quantization'' of classical phase space. We analyze how this simplex W is embedded in the whole state space of two qutrits and discuss symmetries and equivalences inside the simplex W. Because we are explicitly able to construct optimal entanglement witnesses we obtain the border between separable and entangled states. With our method we find also the t…
Low Overhead Qutrit Magic State Distillation
本研究构建了适用于魔态蒸馏的三正交qutrit纠错码族，证明其可大幅降低容错量子计算的魔态蒸馏开销，小参数下的码性能已优于同规模的已知qubit三正交码。
Magic state distillation is a crucial component in the leading approaches to implementing universal fault tolerant quantum computation, with existing protocols for both qubit and higher dimensional systems. Early work focused on determining the region of distillable states for qubit protocols, yet comparatively little is known about which states can be distilled and with what distillable region for d>2. Here we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for …
We compute the measure of entanglement for some classes of states belonging to the simplex of Bell - diagonal states of two qutrits.
We investigate the state space of bipartite qutrits. We construct an analog to the "magic" tetrahedron for bipartite qubits--a magic simplex W. It is formed by all convex combination of nine Bell states which are constructed using the Weyl operators. Due to the high symmetry it is enough to consider certain typical slices through W. Via optimal entanglement witnesses we find regions of bound entangled states inside W.
Analytic formulae for non-local nonstabiliserness in bipartite systems of qutrits and ququints
本文提出双分量子三态和量子五态系统的非局域魔解析公式，并发现其在高维系统中不适用于纠缠诊断
Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits
本文从群论角度分析双粒子三能级魔法单纯形中PPT和重新排列标准，揭示其与底层群结构的联系，为理解纠缠与群结构的数学和物理性质提供新视角。
Realization of a fast triple-magic all-optical qutrit in strontium-88
该研究在锶-88中实现了双钟跃迁的同步魔法陷阱，通过双光子和三光子耦合实现高保真度的全光量子三重态操控，并发现魔角调制陷阱下的原子间相干时间可达715(30)毫秒。
A Search for High-Threshold Qutrit Magic State Distillation Routines
本文研究了三态奇特态的高阈值魔态提纯规程，通过分析不同稳定化码的性能，发现超过600个CSS码能够实现该态的提纯，尽管未超越现有高莱码的阈值。
Analytical Landscape of Maximal Magic for Two-Qutrit States and Beyond
研究两量子三态及五态在固定纠缠下魔力与纠缠的关系，通过刻画帕累托前沿，给出魔力上下限前沿相关值及解析表达式等，推测两量子dit系统（素数维\(d\)）最大魔力公式，改进相关理论界限和数值估计。
A popular universal gate set for quantum computing with qubits is Clifford+T, as this can be readily implemented on many fault-tolerant architectures. For qutrits, there is an equivalent T gate, that, like its qubit analogue, makes Clifford+T approximately universal, is injectable by a magic state, and supports magic state distillation. However, it was claimed that a better gate set for qutrits might be Clifford+R, where R=diag(1,1,-1) is the metaplectic gate, as certain protocols and gates could more easily be implemented using the R gate than the T gate. In this paper we show that when we ha…
Stabilizer rank bounds for magic-state orbits
本文研究了qutrit系统中不同Clifford轨道魔法态的稳定子秩，给出了Strange态、Hadamard本征态和Norrell轨道的渐近指数上界和下界，并提供了显式稳定子分解和两qutrit Clifford电路。
Coherence makes quantum systems 'magical'
本文探讨量子相干作为创造魔法的基本资源，统一了量子通信与计算的发展基础，并研究了量子相干与魔法在三能级系统中的相互作用。
We construct entanglement witnesses with regard to the geometric structure of the Hilbert--Schmidt space and investigate the geometry of entanglement. In particular, for a two--parameter family of two--qutrit states that are part of the magic simplex we calculate the Hilbert--Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three--parameter family of states. In this way we discover new regions of bound entangled states. Furthermore we outline how to use our method to distinguish entangled from separable states.
Harvesting magic from the vacuum
本研究提出利用三能级Unruh-DeWitt探测器与真空态量子场相互作用的协议，首次实现从量子场论框架中采集并产生量子魔态资源，拓展了量子场论资源提取的应用范围。
We show that the cost of strong simulation of quantum circuits using $t$ $T$ gate magic states exhibits non-trivial reductions on its upper bound for $t=1$, $t=2$, $t=3$, and $t=6$ with odd-prime-qudits. This agrees with previous numerical bounds found for qubits. We define simulation cost by the number of terms that require Gaussian elimination of a $t \times t$ matrix and so capture the cost of simulation methods that proceed by computing stabilizer inner products or evaluating quadratic Gauss sums. Prior numerical searchs for qubits were unable to converge beyond $t=7$. We effectively incre…
Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation [M. Planat and Rukhsan-Ul-Haq, Preprint 1701.06443]. We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such IC-POVMs, investigated in dimensions $2$ to $12$, exhibit simple finite geometries in their projector products and, for dimensions $4$ and $8$ and $9$, relate to two-qubit, three-qubit and two-qutrit contextuality.
We obtain formulas (bot. p. 12)--including $\frac{2}{121}$ and $\frac{4 \left(242 \sqrt{3} π-1311\right)}{9801}$--for the eight atoms (Fig. 11), summing to 1, which span a 256-dimensional three-set (P, S, PPT) entanglement-probability boolean algebra for the two-qutrit Hiesmayr-Loffler states. PPT denotes positive partial transpose, while P and S provide the Li-Qiao necessary and}sufficient conditions for entanglement. The constraints ensuring entanglement are $s> \frac{16}{9} \approx 1.7777$ and $p> \frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}$. Here, $s$ is the s…
Geometric Characterization and Feasible Region Analysis of Bipartite Qutrit Bound Entanglement
本研究通过结合PPT准则与结构映射，精确刻画了二分量子比特束缚纠缠的几何边界与可行域，并证明该族推广了Horodecki态，其束缚纠缠区域占PPT叶面积的14.76%。
We determine the cost of performing Shor's algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to…
Extremal Magic States from Symmetric Lattices
该研究揭示高维对称格与量子魔法态的联系，通过映射E₈等格的向量构造并分类多量子系统的稳定子态与魔法态，明确三量子比特极值态分类及单qutrit极大态的克利福德轨道拆分，指出极值魔法态结构蕴含深刻代数几何对称性。
Quantum Magic and Computational Complexity in the Neutrino Sector
该研究通过qutrit映射分析致密中微子系统的量子魔法演化，发现初始态类型决定魔法随中微子数的增减趋势，揭示了量子模拟复杂性与标准模型参数的联系。
We study entanglement witnesses that can be constructed with regard to the geometrical structure of the Hilbert-Schmidt space, i.e. we present how to use these witnesses in the context of quantifying entanglement and the detection of bound entangled states. We give examples for a particular three-parameter family of states that are part of the \emph{magic simplex} of two-qutrit states.
Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information theory. Examples include: (1) N=2 STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is [SL(2)]^3 and (2) N=8 black holes and the tripartite entanglement of seven qubits where the common symmetry is E_7 which contains [SL(2)]^7. Here we present another example: N=8 black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-s…
Qutrit entanglement and joint multi-parameter estimation in an optical clock platform
本研究在光晶格钟平台中实验实现真实两qutrit纠缠，构建最优计量探针态与抗噪读出电路，联合估计双钟跃迁注入相位，其方差低于理想两能级单独传感阈值，为高维量子信息科学提供关键基础。
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres--Horodecki positive partial transpose (ppt)-criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevic…
We investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure…
We show how to detect entangled, bound entangled, and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert-Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the "magic simplex", i.e. the set of Bell-state mixtures of arbitrary dimension.
Harvesting Contextuality from the Vacuum
本文提出通过Unruh-DeWitt模型从真空获取量子上下文性，并探讨其与纠缠和魔法的关系。
Focus is on two parties with Hilbert spaces of dimension d, i.e. "qudits". In the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them is included in the chosen simplex which fits exactly into conditions needed for teleportati…
The familiar Greenberger-Horne-Zeilinger (GHZ) states can be rewritten by entangling the Bell states for two qubits with a state of the third qubit, which is dubbed entangled entanglement. We show that in this way we obtain all 8 independent GHZ states that form the simplex of entangled entanglement, the magic simplex. The construction procedure allows a generalization to higher dimensions both, in the degrees of freedom (considering qudits) as well as in the number of particles (considering n-partite states). Such bases of GHZ-type states exhibit a certain geometry that is relevant for experi…
In this contribution we relate two different key concepts: mutually unbiased bases (MUBs) and entanglement; in particular we focus on bound entanglement, i.e. highly mixed states which cannot be distilled by local operations and classical communications. For a certain class of states --for which the state-space forms a "magic" simplex-- we analyze the set of bound entangled states detected by the MUB criterion for different dimensions d and number of particles n. We find that the geometry is similar for different d and n, consequently, the MUB criterion opens possibilities to investigate the t…
We report several witness-parameterized families of bound-entangled probabilities. Two pertain to the $d=3$ (two-qutrit) and a third to the $d=4$ (two-ququart) subsets analyzed by Hiesmayr and L{ö}ffler of "magic" simplices of Bell states. The Hilbert-Schmidt probabilities of positive-partial-transpose (PPT) states--within which we search for bound-entangled states--are $\frac{8 π}{27 \sqrt{3}} \approx 0.537422$ ($d=3$) and $\frac{1}{2}+\frac{\log \left(2-\sqrt{3}\right)}{8 \sqrt{3}} \approx 0.404957$ ($d=4$). We obtain bound-entangled probabilities of $-\frac{4}{9}+\frac{4 π}{27 \sqrt{3}}+\fr…
Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a l…
First, we considerably simplify an initially quite complicated formula -- involving dilogarithms. It yields the total bound entanglement probability ($\approx 0.0865542$) for a qubit-ququart ($2 \times 4$) three-parameter model, recently analyzed for its separability properties by Li and Qiao. An "archipelago" of disjoint bound-entangled regions appears in the space of parameters, somewhat similarly to those recently found in our preprint, "Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities". There, two-qutrit and two-ququart Hiesmayr-L{ö}ffler "magic simplices" and g…
Functional analytic insights into irreversibility of quantum resources
本文以范数压缩映射和泛函、凸分析工具统一刻画量子资源变换，给出单拷贝与渐近可蒸馏资源及资源成本新界，证明纠缠渐近不可逆并发现 qutrit magic states 在 stabiliser 协议下不可逆。
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