اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe generalized Gell-Mann representation and violation of the CHSH inequality by a general two-qudit state
108
0
0.0
(
0
)
تأليف
Elena R. Loubenets
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $dgeq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH result of Horodeckis, and also, for the Greenberger-Horne-Zeilinger (GHZ) state with an odd $dgeq2,$ where the new upper bound is less than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension $dgeq2$ and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2sqrt{2}$ if $dgeq2$ is even and to $frac{2(d-1)}{d}sqrt{2}$ if $d>2$ is odd.
قيم البحث
اقرأ أيضاً
Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a $d$-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formali
sm describing the unitary evolution of a qudit $(dgeq2)$ in terms of the Bloch-like vector space and specify how in a general case this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case $(d=2)$, we specify the unitary evolution of a qubit via the evolution of a unit vector in $mathbb{R}^{4}$ and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the
maximal violation of the original Bell inequality is upper bounded by 3/2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3/2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.
By virtue of the integration method within P-ordered product of operators and the property of entangled state representation, we reveal new physical interpretation of the generalized two-mode squeezing operator (GTSO), and find it be decomposed as th
e product of free-space propagation operator, single-mode and two-mode squeezing operators, as well as thin lens transformation operator. This docomposition is useful to design of opticl devices for generating various squeezed states of light. Transformation of entangled state representation induced by GTSO is emphasized.
The Gell-Mann grading, one of the four gradings of sl(3,C) that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations i
s reduced and solved. Each non-trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra sl(3,C). The resulting 53 contracted algebras are divided into two classes - the first is represented by the algebras which are also continuous Inonu-Wigner contractions, the second is formed by the discrete graded contractions.
For an even qudit dimension $dgeq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violati
on of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $dgeq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $frac{3}{2}$ specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
