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Register a new userMonogamy equality in $2otimes 2 otimes d$ quantum systems
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There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {bf 61}, 052306 (2000); Phys. Rev. Lett. {bf 96}, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, $mathcal{C}_{A(BC)}^2=mathcal{C}_{AB}^2+(mathcal{C}_{AC}^a)^2$, in the three-qubit system. In this paper, we consider the monogamy equality in $2otimes 2 otimes d$ quantum systems. We show that $mathcal{C}_{A(BC)}=mathcal{C}_{AB}$ if and only if $mathcal{C}_{AC}^a=0$, and also show that if $mathcal{C}_{A(BC)}=mathcal{C}_{AC}^a$ then $mathcal{C}_{AB}=0$, while there exists a state in a $2otimes 2 otimes d$ system such that $mathcal{C}_{AB}=0$ but $mathcal{C}_{A(BC)}>mathcal{C}_{AC}^a$.
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We revisit qubit-qutrit quantum systems under collective dephasing and answer some of the questions which have not been asked and addressed so far in the literature. In particular, we examine the possibilities of non-trivial phenomena of {it time-invariant} entanglement and {it freezing} dynamics of entanglement for this dimension of Hilbert space. Interestingly, we find that for qubit-qutrit systems both of these peculiar features coexist, that is, we observe not only time-invariant entanglement for certain quantum states but we find also find evidence that many quantum states freeze their entanglement after decaying for some time. To our knowledge, the existance of both these phenomena for one dimension of Hilbert space is not found so far. All previous studies suggest that if there is freezing dynamics of entanglement, then there is no time-invariant entanglement and vice versa. In addition, we study local quantum uncertainity and other correlations for certain families of states and discuss the interesting dynamics. Our study is an extension of similar studies for qubit-qubit systems, qubit-qutrit, and multipartite quantum systems.
Absolute separable states is a kind of separable state that remain separable under the action of any global unitary transformation. These states may or may not have quantum correlation and these correlations can be measured by quantum discord. We find that the absolute separable states are useful in quantum computation even if it contains infinitesimal quantum correlation in it. Thus to search for the class of two-qubit absolute separable states with zero discord, we have derived an upper bound for $Tr(varrho^{2})$, where $varrho$ denoting all zero discord states. In general, the upper bound depends on the state under consideration but if the state belong to some particular class of zero discord states then we found that the upper bound is state independent. Later, it is shown that among these particular classes of zero discord states, there exist sub-classes which are absolutely separable. Furthermore, we have derived necessary conditions for the separability of a given qubit-qudit states. Then we used the derived conditions to construct a ball for $2otimes d$ quantum system described by $Tr(rho^{2})leq Tr(X^{2})+2Tr(XZ)+Tr(Z^{2})$, where the $2otimes d$ quantum system is described by the density operator $rho$ which can be expressed by block matrices $X,Y$ and $Z$ with $X,Zgeq 0$. In particular, for qubit-qubit system, we show that the newly constructed ball contain larger class of absolute separable states compared to the ball described by $Tr(rho^{2})leq frac{1}{3}$. Lastly, we have derived the necessary condition in terms of purity for the absolute separability of a qubit-qudit system under investigation.
A local numerical range is analyzed for a family of circulant observables and states of composite $2 otimes d$ systems. It is shown that for any $2otimes d$ circulant operator $cal O$ there exists a basis giving rise to the matrix representation with real non-negative off-diagonal elements. In this basis the problem of finding extremum of $cal O$ on product vectors $ket{x}otimes ket{y} in mathbb{C}^2otimes mathbb{C}^d$ reduces to the corresponding problem in $mathbb{R}^2otimes mathbb{R}^d$. The final analytical result for $d=2$ is presented.
The quantum steering ellipsoid can be used to visualise two-qubit states, and thus provides a generalisation of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters (CKW) inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication (SLOCC). We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states, and generalize our 3-qubit inequality to n qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any two-qubit state is non-increasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multi-qubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.
We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the {alpha}th power of concurrence and entanglement of formation are presented for N-qubit states. The monogamy relation for entanglement of assistance is also established. Based on these general monogamy relations, the residual entanglement of concurrence and entanglement of formation are studied. Some relations among the residual entanglement, entanglement of assistance, and three tangle are also presented.
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