Properties of the max- relative entropy of entanglement are investigated, and its significance as an upper bound to the one shot rate for perfect entanglement dilution, under a particular class of quantum operations, is discussed. It is shown that it is in fact equal to another known entanglement monotone, namely the log robustness. It is known that the latter is not asymptotically continuous and it is not known whether it is weakly additive. However, by suitably modifying the max- relative entropy of entanglement we obtain a quantity which is seen to satisfy both these properties. In fact, the modified quantity is shown to be equal to the regularised relative entropy of entanglement.
It is shown that Smolin four-qubit bound entangled states [Phys. Rev. A, 63 032306 (2001)] can maximally violate two-setting Bell inequality similar to standard CHSH inequality. Surprisingly this entanglement does not allow for secure key distillation, so neither entanglement nor violation of Bell inequalities implies quantum security. It is also pointed out how that kind of bound entanglement can be useful in reducing communication complexity.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of…
It shown that when one of the components of a product channel is entanglement breaking, the output state with maximal p-norm is always a product state. This result complements Shor's theorem that both minimal entropy and Holevo capacity are additive for entanglement breaking channels. It is also shown how Shor's results can be recovered from the p-norm results by considering their behavior for p close to one.
Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies, and allow us to define the analogues o…
When a two-qubit system is initially maximally-entangled, two independent decoherence channels, one per qubit, would greatly reduce the entanglement of the two-qubit system when it reaches its stationary state. We propose a method on how to minimize such a loss of entanglement in open quantum systems. We find that the quantum entanglement of general two-qubit systems with controllable parameters can be protected by tuning both the single-qubit parameters and the two-qubit coupling strengths. Indeed, the maximum fidelity $F_{\rm max}$ between the stationary entangled state, $ρ_{\infty}$, and th…
Very recently, it was shown by Ghosh, Kar, Roy and Sen (\emph{Entanglement vs. Noncommutativity in Teleportation}, quant-ph/0010012) that if it is \emph{a priori} known that the state to be teleported is from a commuting set of qubits, a separable channel is sufficient. We show that 1 ebit of entanglement is a necessary resource to teleport a qubit even when it is known to be one of two noncommuting states.
We show that the state with the highest known average two-particle von Neumann entanglement entropy proposed by Sudbery and one of the authors gives a local maximum of this entropy. We also show that this is not the case for an alternative highly entangled state proposed by Brown et al.
This paper demonstrates that how well a state performs as an input to Grover's search algorithm depends critically upon the entanglement present in that state; the more entanglement, the less well the algorithm performs. More precisely, suppose we take a pure state input, and prior to running the algorithm apply local unitary operations to each qubit in order to maximize the probability P_max that the search algorithm succeeds. We prove that, for pure states, P_max is an entanglement monotone, in the sense that P_max can never be decreased by local operations and classical communication.
The inference of entangled quantum states by recourse to the maximum entropy principle is considered in connection with the recently pointed out problem of fake inferred entanglement [R. Horodecki, {\it et al.}, Phys. Rev. A {\it 59} (1999) 1799]. We show that there are operators $\hat A$, both diagonal and non diagonal in the Bell basis, such that when the expectation value $<\hat A>$ is taken as prior information the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of $\hat A^2$ (unlike what happens when the partial information is given by …
The three-tangle-dependence of $S_{max} = \max <S>$, where $S$ is Svetlichny operator, are explicitly derived when one party moves with an uniform acceleration with respect to other parties in the generalized Greenberger-Horne-Zeilinger and maximally slice states. The $π$-tangle-dependence of $S_{max}$ are also derived implicitly. From the dependence we conjecture that the multipartite entanglement is not the only physical resource for quantum mechanical multipartite non-locality.
A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pu…
There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numeric…
The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the maximum possible Schmidt number of an arbitrary mixed state. We derive a framework for lower bounding the appr…
Correlation beyond entanglement is a subject of interest in quantum information. Here we have shown the existence of quantum correlation without entanglement in Heisenberg $XYZ$ spin model with external magnetic field, using different versions of measurement-induced nonlocality. However maximally entangled states are shown to possess maximum correlation.
The discrepancy between maximally entangled states and maximally non-classical quantum correlations is well-known but still not well understood. We aim to investigate the relation between quantum correlations and entanglement in a family Bell inequalities with $N$-settings and $d$ outcomes. Using analytical as well as numerical techniques, we derive both maximal quantum violations and violations obtained from maximally entangled states. Furthermore, we study the most non-classical quantum states in terms of their entanglement entropy for large values of $d$ and many measurement settings. Inter…
A global quantum quench can be modeled by a quantum circuit with local unitary gates. In general, entanglement grows linearly at a rate given by entanglement velocity, which is upper bounded by the growth of the light cone. We show that the unitary interactions achieving the maximal rate must remain unitary if we exchange the space and time directions -- a property known as dual unitarity. Our results are robust: approximate maximal entanglement velocity also implies approximate dual unitarity. We further show that maximal entanglement velocity is always accompanied by a specific dynamical pat…
We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is accomplished by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a "thermodynamic-like" description of the entanglement arising out of the maximum entropy principle.
We study maximal bipartite entanglement in valence-bond states and show that the average bipartite entanglement $E_v^2$, between a sub-system of two spins and the rest of the system, can be maximized through a homogenized superposition of the valence-bond states. Our derived maximal $E_v^2$ rapidly increases with system size and saturates at its maximum allowed value. We also demonstrate that our maximal $E^2_v$ states are ground states of an infinite range Heisenberg model (IRHM) and represent a new class of resonating-valence-bond (RVB) states. The entangled RVB states produced from our IRHM…
We study the effect of space anisotropy in the entanglement generated by expanding universe on spin 0 and 1/2 fields. For massive scalar field we find revivals of entanglement entropy vs momentum after decreasing from the maximum at k = 0. In massive Dirac field the effect is a slight distortion of the non-monotonic profile giving rise to the maximum of entanglement entropy at k > 0. More interestingly, massless field of both type can only get entangled through anisotropy, with a maximum of entanglement entropy occurring at k > 0.
We study the entanglement dynamics of quantum many-body systems and prove the following: (I) For any geometrically local Hamiltonian on a lattice, starting from a random product state the entanglement entropy is bounded away from the maximum entropy at all times with high probability. (II) In a spin-glass model with random all-to-all interactions, starting from any product state the average entanglement entropy is bounded away from the maximum entropy at all times. We also extend these results to any unitary evolution with charge conservation and to the Sachdev-Ye-Kitaev model. Our results hig…
In 4 dimensional Maxwell gauge theory, we study the changes of (Renyi) entangle-ment entropy which are defined by subtracting the entropy for the ground state from the one for the locally excited states generated by acting with the gauge invariant local operators on the state. The changes for the operators which we consider in this paper reflect the electric-magnetic duality. The late-time value of changes can be interpreted in terms of electromagnetic quasi-particles. When the operator constructed of both electric and magnetic fields acts on the ground state, it shows that the operator acts o…
The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein-Podolosky-Rosen pair of two spin-1/2 particles. Given the data on the Bell-Clauser-Horne-Shimony-Holt observable, the analytic expression is given for the inferred quantum entangled state. It is shown that for q greater than unity, indicating the sub-additive feature …
We optimize a quantum walk of multiple fermions following a quench in a spin chain to generate near ideal resources for quantum networking. We first prove an useful theorem mapping the correlations evolved from specific quenches to the apparently unrelated problem of quantum state transfer between distinct spins. This mapping is then exploited to optimize the dynamics and produce large amounts of entanglement distributed in very special ways. Two applications are considered: the simultaneous generation of many Bell states between pairs of distant spins (maximal block entropy), or high entangle…
The conservation of information of evaporating black holes is a very natural consequence of unitarity which is the fundamental symmetry of quantum mechanics. In order to study the conservation of information, we need to understand the nature of the entanglement entropy. The entropy of Hawking radiation is approximately equal to the maximum of entanglement entropy if a black hole is in a state before the Page time, i.e., when the entropy of Hawking radiation is smaller than the entropy of the black hole. However, if there exists a process generating smaller entanglements rather than maximal ent…
