Measures and dynamics of entangled states

Probably not a substantial claim: may not pass S4 (based).

Concurrence is not additive.

“whereas it is evident that concurrence is not additive.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The concurrence c_N defined with prefactors p{si}=4 except p+...+=0 can be expressed in terms of the reduced density matrices as Eq. (88).

“The concurrence c N , defined with these prefactors for systems with an arbitrary number N of subsystems, can be expressed in terms of the reduced density matrices ̺ i [32] as”

• cites [32] Carvalho, 2004, Decoherence and multipartite entanglement, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

For a separable state, the expectation value <Ψ|⊗<Ψ|A|Ψ>⊗|Ψ> factorises into the product of analogous expressions for the single subsystems and vanishes.

“If a state | Ψ 〉 is separable, the expectation value 〈 Ψ | ⊗ 〈 Ψ | A | Ψ 〉 ⊗ | Ψ 〉 factorises into the product of the analogous expressions corresponding to the single subsystems”

• Also stated in: § 2.3 Representation in product spaces · p. 26 · plain

  “and vanishes because both, | ϕ 〉 ⊗ | ϕ 〉 and | φ 〉 ⊗ | φ 〉 , are symmetric.”

Probably not a substantial claim: may not pass S3 (load-bearing).

I-concurrence coincides with the original definition given in Eq. (34) for two-level systems.

“and it is easily verified that I -concurrence coincides with the original definition given in Eq. (34), for two-level systems.”

May combine two claims.

In two-level systems, there is only one anti-symmetric state, so the projectors onto the anti-symmetric subspaces have only one non-vanishing eigenvalue, and A = |χ><χ|.

“In the case of two-level systems, there is only one anti-symmetric state, namely | 01 〉 - | 10 〉 . Therefore, the projectors onto the anti-symmetric subspaces H 1 -and H 2 -have only one non-vanishing eigenvalue. This, in turn, also holds true for A which reads A = | χ 〉〈 χ | , with”

May combine two claims.

For an n1×n2-dimensional system, A has m = n1(n1-1)n2(n2-1)/4 non-vanishing eigenvalues, so A cannot be expressed with a single eigenvector but with a finite sum.

“For an n 1 × n 2 -dimensional system, A has m = n 1 ( n 1 -1) n 2 ( n 2 -1) / 4 non-vanishing eigenvalues”

• Also stated in: § 2.3.3 Higher dimensional systems · p. 28 · plain

  “Thus, A cannot be expressed with the help of a single eigenvector anymore, but with a finite sum”

Probably not a substantial claim: may not pass S3 (load-bearing).

I-concurrence cannot exceed a given maximum value, with d the dimension of the smallest subsystem.

“Note that I -concurrence cannot exceed a given maximum value,”

• Also stated in: § 2.2.2 I-concurrence · p. 24 · plain

  “with d the dimension of the smallest subsystem.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Eq. (87) defines a continuous family of multipartite concurrences because of the continuous parametrisation in terms of the p{si}.

“Note that, because of the continuous parametrisation in terms of the p { s i } , Eq. (87) actually defines a continuous family of multipartite concurrences.”

Probably not a substantial claim: may not pass S3 (load-bearing).

With the prefactor 4 normalization, the present definition is equivalent to Eq. (59).

“With this normalisation, the present definition is indeed equivalent to Eq. (59).”

Classified as probably not a substantial claim.

The eigenvectors of A carry an undetermined phase factor e^{iφ_α}.

“Furthermore, the eigenvectors of A still carry an undetermined phase factor e iϕ α .”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

In the special case of dephasing environment, the density matrix is always a mixture of two pure states, and hence of rank two, and can be treated analytically.

“Indeed, in the special case of dephasing environment, the density matrix is always a mixture of two pure states, and, hence, of rank two, and can be treated analytically.”

Classified as probably not a substantial claim. The quote only partly states this claim.

The final expression for a lower bound of the concurrence is Eq. (93).

“Our final expression for a lower bound of the concurrence therefore reads”

May combine two claims.

Using only one single matrix T, it is not possible to describe accurately all second order elements, such as A^k1_j1; all terms of third and fourth order are dropped in the present approximation.

“Note that, using only one single matrix T , it is not possible to describe accurately all second order elements, such as A k 1 j 1 . All terms of third and fourth order are dropped in the present approximation.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For four-partite systems, the term A^(1234)_4 = 4 P(1)-⊗P(2)-⊗P(3)-⊗P(4)- does not contain any factor P(i)+.

“Another interesting quantity emerges for four-partite systems: Since the number of subsystems is even, there is a term A (1234) 4 = 4 P (1) -⊗ P (2) -⊗ P (3) -⊗ P (4) -that does not contain any factor P ( i ) + .”

May combine two claims.

The first-order expansions of the finite temperature solutions reproduce the leading order terms of the zero and infinite temperature solutions in the appropriate limits and show the influence of temperature on the short time behaviour of concurrence decay.

“These expressions not only reproduce the leading order terms of the zero and infinite temperature solutions in the appropriate limits, but also show the influence of the temperature on the short time behaviour of concurrence decay.”

• cites [84] A.R.R. Carvalho et al., in preparation.Not checked yet

  Not checked in the free run.

May combine two claims.

The evolution of an initially pure into a mixed state occurs on a rather long time scale, and experimentally interesting states are quasi-pure, with one single eigenvalue μ1 much larger than all the others.

“Indeed the evolution of an initially pure into a mixed state occurs on a rather long time scale, and the experimentally interesting states are - though not exactly pure - at least quasi-pure , i.e. , they have one single eigenvalue µ 1 that is much larger than all the other ones.”

• AssumptionThe authors state this as an assumption: Environmental influences can be assumed to be small under the given experimental conditions.
• AssumptionRests on an assumption: The experimentally interesting states are quasi-pure, i.e., they have one single eigenvalue µ1 much larger than all the other ones.

May combine two claims.

For a GHZ state, c^(1234)_4(Ψ_GHZ)=2√∑_{i≠j} λ_i λ_j, while c^(1234)_4 vanishes for W-states.

“In particular, for a GHZ state, c (1234) 4 (Ψ GHZ ) = 2 √∑ i = j λ i λ j while c (1234) 4 vanishes for W -states - similarly like for tri-partite systems where W -states bear only bipartite correlations .”

Probably not a substantial claim: may not pass S3 (load-bearing).

The lower bound of Eq. (92) was obtained by replacing several matrices Tα by a single suitably chosen T = ∑α Zα Tα.

“The lower bound of Eq. (92) was obtained by the replacement of several matrices T α by a single suitably chosen T = ∑ α Z α T α .”

Classified as probably not a substantial claim. The quote only partly states this claim.

Approximating A in terms of the matrix T, Eq. (76) can be approximated as Eq. (98).

“Approximating A in terms of this matrix T , Eq. (76) can be approximated as”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

A priori, it is not clear up to which degree of mixing the quasi-pure approximation provides reliable estimates.

“A priori, it is not clear up to which degree of mixing the quasi-pure approximation provides reliable estimates.”

Classified as probably not a substantial claim.

Since |ψ1> is assumed entangled, T is well defined.

“Since we were assuming | ψ 1 〉 to be entangled, i.e. , A 11 11 finite by virtue of Eq. (62), T is well defined.”

• AssumptionThe authors state this as an assumption: The subnormalised eigenstate |ψ1> associated with the largest eigenvalue µ1 of ρ is entangled, i.e., not separable.

May combine two claims.

For higher dimensional systems A typically is of higher rank, so Eq. (76) exhibits additional non-linearities and different matrices Tα that in general cannot be diagonalised simultaneously.

“For higher dimensional systems A typically is of higher rank, such that Eq. (76) exhibits two complications with respect to Eq. (72): additional non-linearities, and different matrices T α which, in general, cannot be diagonalised simultaneously.”

May combine two claims.

The ratio of initial concurrences of W states to GHZ states has a maximum at N = 5 and approaches unity for large N, with a value 1.07 for N = 7.

“with a maximum at N = 5, approaches unity for large N (with a value 1 . 07 for N = 7, the maximal size considered in Fig. 14).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The constraint ∑α |Zα|^2 = 1 is simpler to implement than left unitarity Eq. (31), since it is easily parametrised.

“Note that the constraint ∑ α | Z α | 2 = 1 is by far simpler to implement than left unitarity, Eq. (31), since it is easily parametrised.”

Classified as probably not a substantial claim. May combine two claims.

One way to get rid of the ambiguity is to diagonalise A instead of A; for a typical state ρ, A will have no degenerate eigenvalues and thus has uniquely defined eigenvectors Tα.

“One way to get rid of this ambiguity is to diagonalise A instead of A . For a typical state ̺ , A will have no degenerate eigenvalues, and thus has uniquely defined eigenvectors T α .”

Classified as probably not a substantial claim.

The Tα are constructed with the help of eigenvectors |χα> of the projector A, which specifies an eigenspace but does not distinguish states within these subspaces.

“They are constructed with the help of the eigenvectors | χ α 〉 of the projector A , what specifies an eigenspace, but does not distinguish states within these subspaces.”

May combine two claims.

The bound in Eq. (93) depends on the choice of the Zα, which allows to tighten the estimate.

“The bound in Eq. (93) still depends on the choice of the Z α , what allows to tighten the estimate.”

May combine two claims.

With the chosen T, Eq. (97) is exact at lowest and first order, and the second order elements A^11_jk are taken into account correctly.

“With this choice of T , Eq. (97) is exact at lowest and first order, and - in addition - the second order elements A 11 jk are taken into account correctly.”

May combine two claims. The quote only partly states this claim.

If terms proportional to μ_j with j>1 are considered perturbations of the dominant term A^11_11 ∼ μ1^2, then A^11_11 is lowest order, elements with one index j>1 are first order, and elements with two indices j,k>1 are second order.

“Hence, if we consider terms proportional to either one of the µ j , with j &gt; 1, as perturbations of the dominant term A 11 11 ∼ µ 2 1 , we obtain the following classification:”

• AssumptionThe authors state this as an assumption: The eigenvalues µ_i of the state are labeled in decreasing order, µ1≫µ2≥...≥µn.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

There is no clear prescription of how to choose the expansion coefficients Zα, partially because the Tα are determined only up to degeneracy.

“So far, there is no clear prescription of how to chose the expansion coefficients Z α , which is partially due to the fact that the T α are determined only up to degeneracy.”

May combine two claims.

The long-time asymptotics quantity for the singular values of matrix τ is non-positive for all values of n̄ and vanishes only in the case of zero temperature.

“This quantity is non-positive for all values of ¯ n and vanishes only in the case of zero temperature.”

The quote only partly states this claim.

For four ions under the Mølmer-Sørensen scheme, concurrence shows appreciable decrease even for small heating rates.

“while it shows appreciable decrease even for small heating rates.”

May combine two claims.

With the addition of motional heating, the growth of maximum concurrence with N is not monotonic; for example, from a certain value of the coupling strength the maximum concurrence for six ions gets smaller than the one for five.

“With the addition of motional heating, this growth with N is not monotonic anymore and we observe, for example, that from a certain value of the coupling strength the maximum concurrence for six ions gets smaller than the one for five.”

May combine two claims.

States encountered in actual experiments are in general mixed and typically involve several particles.

“require such a quantification to be versatile enough to deal with the states encountered in actual experiments, which are in general mixed and typically involve several particles.”

• cites [5] Monroe, 1995, Demonstration of a fundamental quantum logic gate, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [6] Turchette, 1998, Deterministic entanglement of two trapped ions, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [7] Pan, 2000, Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger-entanglement, Nature sourceNot checked yet

  Not checked in the free run.

• cites [8] Mandel, 2003, Controlled collisions for multi-particle entanglement of optically trapped atoms, Nature sourceNot checked yet

  Not checked in the free run.

• cites [9] Sackett, 2000, Experimental entanglement of four particles, Nature sourceNot checked yet

  Not checked in the free run.

• cites [10] Pan, 2001, Experimental demonstration of four-photon entanglement and high-fidelity teleportation, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [11] Eibl, 2004, Experimental realization of a three-qubit entangled W state, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [12] Roos, 2004, Control and measurement of three-qubit entangled states, Science sourceNot checked yet

  Not checked in the free run.

• cites [13] Rauschenbeutel, 2000, Step-by-step engineered multiparticle entanglement, Science sourceNot checked yet

  Not checked in the free run.

• cites [14] Zhao, 2004, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature sourceNot checked yet

  Not checked in the free run.

Classified as probably not a substantial claim. May combine two claims.

Entanglement is an important resource in quantum computation, teleportation, and quantum cryptography.

“Be it in quantum computation [1,2], teleportation [3] or quantum cryptography [4], entanglement is viewed as an important resource and, as such, must be quantified.”

• cites [1] Feynman, 1982, Simulating physics with computers, Int. J. Theoret. Phys. sourceNot checked yet

  Not checked in the free run.

• cites [2] Deutsch, 1985, Quantum theory, the church-turing principle and the universal quantum computer, Proc. R. Soc. Lond. ANot checked yet

  Not checked in the free run.

• cites [3] Bennett, 1993, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [4] C.H. Bennet, G. Brassard, Quantum cryptography: public key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 1984, p. 175.Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

For bipartite pure states, entanglement measures exist and are straightforwardly computable.

“For bipartite pure states such measures exist and are straightforwardly computable.”

Probably not a substantial claim: may not pass S2 (checkable).

Studying the dynamics of entanglement requires efficiently evaluable quantifiers thereof as an indispensable ingredient, ideally for arbitrary system sizes.

“Studying the dynamics of entanglement requires efficiently evaluable quantifiers thereof as an undispensible ingredient, ideally for arbitrary system sizes.”

• AssumptionRests on an assumption: No entanglement measures are known for arbitrary states of higher dimensional systems.

Probably not a substantial claim: may not pass S3 (load-bearing).

Any evolution consistent with the general rules of quantum mechanics can be described by a linear, completely positive map, called a quantum operation.

“Consequently, any evolution consistent with the general rules of quantum mechanics can be described by a linear, completely positive map, called quantum operation .”

Classified as probably not a substantial claim.

The proper, unambiguous generalisation of a pure state monotone to mixed states uses the infimum over all decompositions into pure states (the convex roof).

“The proper, unambiguous generalisation of a pure state monotone, that we will also use in the following, therefore uses the infimum over all decompositions into pure states - the socalled convex roof [49]”

• cites [49] Uhlmann, 2000, Fidelity and concurrence of conjugated states, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

An explicit evaluation of the convex roof for a specific state implies a high dimensional optimisation problem, in general a very hard computational task.

“An explicit evaluation of this quantity for a specific state - one has to find the infimum over all possible decompositions into pure states - implies a high dimensional optimisation problem - in general a very hard computational task.”

• AssumptionRests on an assumption: All proposed measures that unambiguously fulfill the task of discriminating classical from quantum correlations involve some generally high dimensional optimisation procedure, and hardly allow for an explicit evaluation in concrete cases.

May combine two claims.

The degree of entanglement of pure bipartite states is related to the degree of mixing of the corresponding reduced density matrices: a pure reduced density matrix corresponds to a separable state, whereas a maximally entangled state leads to a maximally mixed reduced density matrix.

“It follows that the concept of Schmidt coefficients allows to relate the degree of entanglement of pure bipartite states to the degree of mixing of the corresponding reduced density matrices - a pure reduced density matrix corresponds to a separable state, whereas a maximally entangled state leads to a maximally mixed reduced density matrix.”

Classified as probably not a substantial claim.

The definition of concurrence given in Section 2 only applies to two-level systems.

“To start with, we have to realize that the definition of concurrence given in Section 2 only applies to two-level systems.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Concurrence can be considered as an independent entanglement measure.

“However, concurrence can also be considered as an independent entanglement measure [27].”

• cites [27] Wootters, 1998, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The decomposition of a mixed state into pure states is not unique, and different decompositions in general lead to different values for a given entanglement monotone.

“Unfortunately, the decomposition into pure states is not unique, and different decompositions in general lead to different values for a given entanglement monotone.”

May combine two claims.

Under global operations, all kinds of correlations can increase as well as decrease; entangled states can be created from initially separable states and vice versa.

“Under this type of operations all kinds of correlations can increase, as well as decrease. Therefore, entangled states can be created from initially separable states and vice versa.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Θ-concurrence is not a single uniquely defined quantity but a family of quantities depending on the choice of Θ.

“Thus Θ-concurrence is not a single, uniquely defined quantity, but rather a family of quantities, depending on the choice of Θ.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Any quantum operation can be composed from elementary operations: unitary transformations, addition of an auxiliary system, partial traces, and projective measurements.

“Any such quantum operation can be composed from elementary operations [36,42,43]:”

• cites [36] Kraus, 1983Not checked yet

  Not checked in the free run.

• cites [42] Kraus, 1971, General state changes in quantum theory, Ann. Phys. sourceNot checked yet

  Not checked in the free run.

• cites [43] Nielsen, 2000Not checked yet

  Not checked in the free run.

May combine two claims.

Under local operations, a product state remains a product state and any separable state remains separable; therefore no correlations can be created by local operations alone.

“Therefore, starting from a separable state no correlations - neither classical nor quantum - can be created by local operations alone.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Any quantum operation can be expressed as an operator sum.

“For a formal mathematical treatment it is useful to note that any quantum operation can be expressed as an operator sum [44,45]”

• cites [44] Hellwig, 1969, Pure operations and measurements, Commun. Math. Phys. sourceNot checked yet

  Not checked in the free run.

• cites [45] Hellwig, 1970, Operations and measurements ii, Commun. Math. Phys. sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Apart from Θ = σy ⊗ σy C* in 2×2 systems, no conjugation is known that is positive exactly for entangled states, so cΘ vanishes for some entangled states.

“However, since, apart from Θ = σ y ⊗ σ y C ∗ in 2 × 2 systems, no conjugation is known that is positive exactly for entangled states, it has the disadvantageous property that c Θ vanishes for some entangled states.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Negativity vanishes for a class of entangled states.

“For example, negativity [26] has become a commonly used quantity although it vanishes for a class of entangled states [26].”

• cites [26] Vidal, 2002, Computable measure of entanglement, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

May combine two claims.

Any entanglement monotone or measure is non-increasing under LOCC and invariant under local unitary transformations.

“Thus, any entanglement monotone and measure has to be non-increasing under both the former and the latter. Since non-increasing behaviour under the latter implies non-decreasing behaviour under the former and vice versa, any monotone or measure has to be invariant under local unitaries.”

• cites [55] Horodecki, 2001, Entanglement measures, Quant. Inform. Comp.Not checked yet

  Not checked in the free run.

May combine two claims.

A single entanglement monotone is in general insufficient to completely characterise the quantum correlations contained in a given pure state; knowledge of all d-1 independent Schmidt coefficients is required.

“Finally, it should be kept in mind that a single monotone is in general insufficient to completely characterise the quantum correlations contained in a given pure state. For such a characterisation, knowledge of all -i.e. d -1 (with d = min(dim( H 1 ) , dim( H 2 ))) independent Schmidt coefficients is required.”

May combine two claims.

Under LOCC operations, any separable state remains separable, and entangled states cannot be created.

“However, any separable state will always remain separable under LOCC operations. Accordingly, entangled states cannot be created with LOCC operations.”

May combine two claims.

The function E(c) is monotonically increasing and convex.

“The function E ( c ) is monotonically increasing and convex.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In systems larger than two-level systems, no conjugation Θ is known such that cΘ(ψ) vanishes for all separable states and is strictly larger than zero for all entangled states.

“In systems larger than two-level systems, no conjugation Θ is known such that c Θ ( ψ ) vanishes for all separable states and is strictly larger than zero for all entangled states.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The standard approach to decide on the separability of a given mixed state relies on positive maps.

“The standard approach to decide on the separability of a given mixed state relies on positive maps .”

May combine two claims.

The Schmidt coefficients allow to distinguish separable from entangled states: a separable state has only one non-vanishing Schmidt coefficient, whereas an entangled state has at least two non-vanishing components.

“The Schmidt coefficients also allow to distinguish separable from entangled states - a separable state is characterised by a vector of Schmidt coefficients with only one non-vanishing entry: ⃗ λ = ⃗ λ s = [1 , 0 , . . . , 0], whereas the Schmidt vector of an entangled state has at least two non-vanishing components.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For mixed states, both classical and quantum correlations must be discriminated against each other by an entanglement monotone.

“For mixed states the situation is much more involved, because there are both classical and quantum correlations that have to be discriminated against each other by an entanglement monotone.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Two inequivalent generalisations of concurrence for systems with more than two levels have been formulated, and both coincide with the original one when restricted to two-level systems.

“So far, two inequivalent generalisations for systems comprising more than two levels have been formulated, which both coincide with the original one, if restricted to two-level systems.”

May combine two claims.

A generalisation of concurrence for tri-partite two-level systems is already available, and it characterises all tri-partite correlations of pure states.

“A generalisation of concurrence [71] for tri-partite two-level systems is already available - it characterises all tri-partite correlations of pure states.”

• cites [71] Coffman, 2000, Distributed entanglement, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

May combine two claims.

Two crucial differences hamper finding the exact infimum: the square and square root in Eq. (76) lead to non-linear expressions in Tα, and the distinct symmetric matrices Tα cannot in general be diagonalised simultaneously.

“There are two crucial differences as compared to Eq. (72) that hamper finding the exact infimum: First, the square and the square root in Eq. (76) lead to non-linear expressions in the T α , and, second, the already mentioned fact that the several distinct symmetric matrices T α cannot , in general, be diagonalised simultaneously.”

May combine two claims.

An expectation value with respect to a single copy of a given state cannot discriminate separable from entangled states, but expectation values with respect to two copies can.

“An expectation value 〈 Ψ | a | Ψ 〉 with respect to a single copy of a given state | Ψ 〉 can not discriminate separable from entangled states. However, this is possible, if expectation values with respect to two copies are used.”

May combine two claims.

Any tensor product of projectors onto symmetric and anti-symmetric subspaces is positive semi-definite and invariant under local unitary transformations.

“Any tensor product of projectors onto symmetric and anti-symmetric sub-spaces is positive semi-definite, and invariant under local unitary transformations, see Fig. 3.”

May combine two claims.

The cardinality of the ensemble that realises the infimum in Eq. (29) can exceed the rank r(ρ), though is bounded by r^2(ρ).

“As mentioned in Section 1.4.2, the cardinality of the ensemble that realises the infimum in Eq. (29) can exceed the rank r ( ̺ ), though is bounded by r 2 ( ̺ ) [52].”

• cites [52] Uhlmann, 1998, Optimizing entropy relative to a channel or a subalgebra, Open System Inform. Dyn. sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Starting the iteration with different initial conditions can give a better intuition on the effective landscape, but the uncertainty about local versus global minimum persists, and more so the higher the dimension of the parameter space.

“While one may start the iteration with different initial conditions parametrised by U , in order to get a better intuition on the effective 'landscape' defining the optimisation problem at hand, this uncertainty persists, and the more so the higher the dimension of the parameter space over which the optimisation is carried out.”

• AssumptionRests on an assumption: There is in general no a priori information available on whether the solution reached by a numerical optimisation procedure is a local or a global minimum.

Probably not a substantial claim: may not pass S3 (load-bearing).

Minima of C can be found by repeated application of the transformation dU = exp(-iεG).

“Minima of C can therefore be found by repeated application of the transformation dU = exp( -iǫG ).”

May combine two claims.

Concurrence vector is not a measure and does not provide an adequate tool for quantitative characterisation of temporally evolving entanglement.

“Concurrence vector is not a measure and as such does not provide an adequate tool for our aims, i.e. the quantitative characterisation of temporally evolving entanglement.”

Probably not a substantial claim: may not pass S3 (load-bearing).

With the definition of A, the elements of A contain partial traces over both subsystems in a balanced way.

“With our above definition, Eq. (63), of A , the elements of A”

• Also stated in: § 2.3.1 Symmetries of A · p. 27 · plain

  “contain partial traces over both subsystems, in a balanced way, what would not have been the case for other choices.”

Probably not a substantial claim: may not pass S3 (load-bearing).

I-concurrence is uniquely defined up to a multiplicative constant.

“Thus -in contrast to Θconcurrence I -concurrence is a quantity that is uniquely defined up to a multiplicative constant.”

Probably not a substantial claim: may not pass S3 (load-bearing).

This is the cause of the appearance of a rectangular, left-unitary matrix V instead of a quadratic, unitary matrix U in Eq. (67).

“This is the cause of the appearance of a rectangular, left-unitary matrix V instead of a quadratic, unitary matrix U in Eq. (67).”

Probably not a substantial claim: may not pass S4 (based).

This symmetry is the crucial ingredient required for the approximations to be discussed in Section 3.

“This symmetry will turn out to be the crucial ingredient required for the approximations to be discussed in Section 3.”

May combine two claims.

GHZ and W states contain fundamentally different correlations, such that none can be created from the other by LOCC alone.

“Both states contain fundamentally different correlations, such that none of the two can be created from the other one by LOCC alone [70].”

• cites [70] Dür, 2000, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

May combine two claims.

During the illumination in the Mølmer-Sørensen scheme, correlations between vibrational and internal degrees of freedom exist, and motional decoherence can decrease the success probability of the preparation process.

“However, during the illumination these correlations do exist, and motional decoherence can decrease the success probability of the preparation process.”

May combine two claims.

The heating of the ions is related to fluctuating fields in the trap electrodes, leads to a thermal motion with steadily increasing temperature, and can be well described by the infinite temperature reservoir with measured heating rates Γ/ν ranging from 10^-4 to 10^-3 [89,90].

“The heating of the ions, related to fluctuating fields in the trap electrodes, leads to a thermal motion with steadily increasing temperature and can be well described by the infinite temperature reservoir discussed previously, Eq. (112), with measured heating rates Γ /ν ranging from 10 -4 to 10 -3 [89,90].”

• cites [89] Turchette, 2000, Heating of trapped ions from the quantum ground state, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

• cites [90] Turchette, 2000, Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The Mølmer-Sørensen scheme [83] produces GHZ-like states and was experimentally implemented at NIST [9] to produce controlled entanglement of two and four trapped ions.

“Specifically, we scrutinize a scalable scheme proposed in [83] to produce GHZ-like states (see Eq. (121)),”

• Also stated in: § 4.2.4 Experimental entanglement production: the Mølmer-Sørensen scheme · p. 55 · plain

  “experimentally implemented at NIST [9] to produce controlled entanglement of two and four trapped ions.”

• cites [83] Mølmer, 1999, Multiparticle entanglement of hot trapped ions, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [9] Sackett, 2000, Experimental entanglement of four particles, Nature sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The decay rate of the electronic states is negligible during the time scale of the state preparation, and therefore one of the main sources of errors in the system arises indirectly through the coupling with these decohering motional modes.

“The decay rate of the electronic states is negligible during the time scale of the state preparation, and therefore one of the main sources of errors in the system arises indirectly through the coupling with these decohering motional modes.”

• AssumptionRests on an assumption: The decay rate of the electronic states is negligible during the time scale of the state preparation.

May combine two claims.

The accelerated decay for |Φ±> states under zero temperature can be understood from time scales: for |Φ±>, each term |01> and |10> corresponds to a single particle decay, leading to a time scale e^{-Γt}; for |Ψ±>, the term |11> allows both particles to undergo an environment induced transition to the ground state, introducing a faster e^{-2Γt} decay.

“while for | Φ ± 〉 each term | 01 〉 and | 10 〉 corresponds to a single particle decay, leading to a time scale e -Γ t , we have the term | 11 〉 in | Ψ ± 〉 , such that both particles can undergo an environment induced transition to the ground state, thus introducing a faster, e -2Γ t , decay.”

Classified as probably not a substantial claim. May combine two claims.

The reliability of the lower bounds derived in Section 3 was demonstrated extensively in Section 4.1, and additional tests at randomly chosen instances confirmed the quality of the quasi-pure approximation with respect to the optimal upper and lower bounds.

“Since the reliability of the lower bounds derived in Section 3 was demonstrated extensively in Section 4.1 above, we will actually use the quasi-pure approximation Eq. (99) in the following, minimising the numerical effort very considerably, notably for large system sizes N (notwithstanding, some additional tests were performed at randomly chosen instances, to convince ourselves of the quality of”

May combine two claims.

Perfect screening is impossible, but experimental techniques are sufficiently advanced to preserve entanglement over appreciable periods of time.

“In general, perfect screening is impossible, but experimental techniques are sufficiently advanced [5,8,74] to preserve entanglement over appreciable periods of time.”

• cites [5] Monroe, 1995, Demonstration of a fundamental quantum logic gate, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [8] Mandel, 2003, Controlled collisions for multi-particle entanglement of optically trapped atoms, Nature sourceNot checked yet

  Not checked in the free run.

• cites [74] Eckert, 2002, Quantum computing in optical microtraps based on the motional states of neutral atoms, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

The fragility of GHZ states was also observed in [86] by an analysis based on their separability and distillability properties [87,88] rather than a dynamical approach.

“In addition, this fragility of GHZ states was also observed in [86] by an analysis based on their separability and distillability properties [87,88] rather than a dynamical approach.”

• cites [86] Dür, 2004, Stability of macroscopic entanglement under decoherence, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [87] Dür, 1999, Separability and distillability of multiparticle quantum systems, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

• cites [88] Dür, 2000, Classification of multiqubit mixed states: separability and distillability properties, Phys. Rev. A sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

GHZ and W states were recently produced in the lab for N = 3, 4 [9,12].

“These were recently produced [9,12] in the lab, for N = 3 , 4”

• cites [9] Sackett, 2000, Experimental entanglement of four particles, Nature sourceNot checked yet

  Not checked in the free run.

• cites [12] Roos, 2004, Control and measurement of three-qubit entangled states, Science sourceNot checked yet

  Not checked in the free run.

May combine two claims. The quote only partly states this claim.

In the infinite temperature case, decay and excitation occur at exactly the same rate, and the noise induced by transitions between the two levels brings the system to a stationary, maximally mixed state.

“In this case, decay and excitation occur at exactly the same rate, and the noise induced by the transitions between the two levels brings the system to a stationary, maximally mixed state.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Cluster states also present a kind of robust behaviour in terms of separability and distillability criteria in the case of a depolarising channel [86].

“Cluster states also present a kind of robust behaviour in terms of separability and distillability criteria in the case of a depolarising channel [86].”

• cites [86] Dür, 2004, Stability of macroscopic entanglement under decoherence, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

It is now within experimental reach to monitor the time dependence of the degree of entanglement of GHZ and W states by means of quantum state tomography [17].

“it is now within experimental reach to monitor the time dependence of their degree of entanglement, by means of quantum state tomography [17].”

• cites [17] Roos, 2004, Bell states of atoms with ultralong lifetimes and their tomographic state analysis, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The statement that initial Bell states evolve into separable states within a finite time for any T > 0 is not true for general states; some initial mixed states may reach separability on finite time scales also in the zero temperature case [85].

“Note that the above is not true for general states. In fact, some initial mixed states may reach separability on finite time scales also in the zero temperature case [85].”

• cites [85] Yu, 2004, Finite-time disentanglement via spontaneous emission, Phys. Rev. Lett. sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Very little is known on the precise temporal evolution of entangled states even under weak but finite environment coupling, one principal obstacle being the lack of computable entanglement measures for arbitrary states.

“Yet, very little is known on the precise temporal evolution of such states even under weak but finite environment coupling, one of the principal obstacles being the lack of computable entanglement measures for arbitrary states.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

In Section 3.3 one of these bounds alone may already yield a satisfactory approximation to the optimised lower bound.

“In Section 3.3 we will see that one of these bounds alone may already yield a satisfactory approximation to the optimised lower bound.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Any degree of mixing, i.e., the presence of classical correlations, decreases quantum correlations and can lead to their complete destruction.

“Any degree of mixing i.e. , the presence of classical correlations - decreases quantum correlations, and can lead to their complete destruction.”

May combine two claims.

For a purely dephasing reservoir (d_i = d = σ+σ- in Eq. (109)), there are no changes in the populations of the ground and excited states, and energy is conserved.

“In this case, there are no changes in the populations of the ground and excited states, and energy is conserved.”

Probably not a substantial claim: may not pass S4 (based).

The general applicability of the various lower bounds of concurrence stems from the algebraic structure of a suitable reformulation of its original definition, with direct generalizations for higher dimensional and/or multipartite quantum states.

“The general applicability of the various lower bounds of concurrence which have been derived stems from the algebraic structure of a suitable reformulation of its original definition, with direct generalizations for higher dimensional and/or multipartite quantum states.”