Persistent Entanglement in Arrays of Interacting Particles

The persistency of the states |φ_N〉 is equal to their Schmidt measure.

“the persistency of the states | φ N 〉 is equal to their Schmidt measure [15]”

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The state |φ_4〉 is not equivalent to a 4-qubit GHZ state.

“| φ 4 〉 is not equivalent to a 4-qubit GHZ state [4].”

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The persistency of |W_N〉 is N-1.

“The persistency of | W N 〉 , on the other hand, is given by N -1 [16]”

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The Schmidt measure of |φ_N〉 is ⌊N/2⌋.

“the Schmidt measure P S ( | φ N 〉 ) [15] of | φ N 〉 is equal to log 2 ( r ) = ⌊ N/ 2 ⌋ .”

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The minimum number of product terms in an expansion of |φ_N〉 is 2^{⌊N/2⌋}.

“it follows by induction that r = 2 ⌊ N/ 2 ⌋ .”

The states |φ_N〉 and the N-qubit GHZ state |GHZ_N〉 are not equivalent for N > 3, i.e. cannot be transformed into each other by LOCC.

“More generally, the states | φ N 〉 and the N -qubit GHZ state | GHZ N 〉 ≡ 2 -1 / 2 ( | 0 〉 1 . . . | 0 〉 N ) + | 1 〉 1 . . . | 1 〉 N ) are not equivalent for N > 3 i.e. cannot be transformed into each other by LOCC (local transformations and classical communication) as we shall see below.”

The persistency of |GHZ_N〉 equals its Schmidt measure.

“This result also holds for the state | GHZ N 〉 .”

The Schmidt measure of |W_N〉 is log2(N).

“The Schmidt measure of this state is equal to log 2 ( N )”

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The N-qubit states generated in arrays of qubits with an Ising-type interaction contain a large amount of entanglement as given by their Schmidt measure.

“These states contain a large amount of entanglement as given by their Schmidt measure.”

For the state |φ_N〉, it is maximally connected.

“for the state | φ N 〉 we show that (i) it is maximally connected”

The persistency of |φ_N〉 is P_e(|φ_N〉)=⌊N/2⌋.

“its persistency is P e ( | φ N 〉 ) = ⌊ N/ 2 ⌋ .”

For N=2, the state |φ_2〉 is a maximally entangled state.

“which is a maximally entangled state.”

The state |φ_3〉 corresponds to a GHZ state of three qubits.

“While | φ 3 〉 corresponds to a GHZ state of three qubits [7]”

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These states have a high persistency of entanglement, meaning that approximately N/2 qubits have to be measured to disentangle the state.

“They have also a high persistency of entanglement which means that ∼ N/ 2 qubits have to be measured to disentangle the state.”

By measuring all neighboring qubits surrounding a path P in the σ_z basis, the qubits on path P are projected into a state that is, up to local unitary transformations, identical to the state |Φ_N〉 of the linear chain.

“By this procedure, we project the qubits on path P into a state that is, up to local unitary transformations, identical to the state | Φ N 〉 of the linear chain.”

Any two of the four qubits in either |φ_4〉 or |GHZ4〉 can be projected into a Bell state by measuring the other two qubits in an appropriate basis.

“The states share the property that any two of the four qubits can be projected into a Bell state by measuring the other two qubits in an appropriate basis.”

Equations (6) can be used to calculate the persistency, as they imply strict correlations among 1-particle measurements.

“Equations (6) can also be used to calculate the persistency, as they imply strict correlations among 1-particle measurements.”

The state |W_N〉 is not maximally connected.

“Note however that, different from | φ N 〉 and | GHZ N 〉 , the state | W N 〉 is not maximally connected.”

For φ=π,3π,5π,..., the chain is in some sense maximally entangled.

“For the values ϕ = π, 3 π, 5 π, . . . the chain is in some sense maximally entangled and we will concentrate on this situation in the following.”

All cluster states are maximally connected.

“We find that all cluster states are maximally connected.”

The Schmidt measure gives a lower bound to the persistency.

“On the other hand, the Schmidt measure gives a lower bound to the persistency.”

From a block C of L^d qubits, one can obtain any state of the form α|00...0〉_{C'} + β|11...1〉_{C'} of any subset C'⊂ C∩{2Z}^d.

“We note without proof [8] that from a block C of L d qubits, one can obtain any state of the form α | 00 . . . 0 〉 C ′ + β | 11 . . . 1 〉 C ′ of any subset of qubits C ′ ⊂ C ∩ { 2 Z } d .”

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In general, the exact value of the persistency depends on the shape of a cluster.

“In general, the exact value of the persistency depends on the shape of a cluster.”

For large convex clusters, the asymptotic persistency per qubit is P_e/N = 1/2 as N→∞.

“For large convex clusters, we can give the asymptotic result P e /N = 1 / 2 where N →∞ is the number of qubits.”

For the chain of qubits in the state |φ_N〉, the value of the persistency agrees with the Schmidt measure of |φ_N〉.

“For the chain of qubits in the state | φ N 〉 , we have shown that the value of the persistency agrees with the Schmidt measure of | φ N 〉 .”

The cluster states (5) satisfy the eigenvalue equations (6).

“The cluster states (5) satisfy the following set of eigenvalue equations:”

It is impossible to destroy all entanglement of |φ_4〉 by a single local operation, such as a von Neumann measurement or complete depolarization of a qubit.

“In fact, it is impossible to destroy all entanglement of | φ 4 〉 by a single local operation, such as a von Neumann measurement or complete depolarization of a qubit.”

These states can be regarded as an entanglement resource because one can generate a family of other multi-particle entangled states, such as generalized GHZ states of fewer than N/2 qubits, by simple measurements and classical communication (LOCC).

“These states can be regarded as an entanglement resource since one can generate a family of other multi-particle entangled states such as the generalized GHZ states of < N/ 2 qubits by simple measurements and classical communication (LOCC).”

The amount of entanglement contained in |W_N〉 is smaller, in fact exponentially smaller, than in |φ_N〉.

“which means that the amount of entanglement contained in | W N 〉 is smaller, in fact exponentially smaller, than in the state | φ N 〉 .”

The state |φ_N〉 of the qubit chain is much more entangled than most of the known N-qubit states.

“In that sense, the state | φ N 〉 of the qubit chain is indeed much more entangled than most of the known N qubit states.”

The entanglement of |W_N〉 is harder to destroy by local measurements than that of |φ_N〉.

“which means that the entanglement of | W N 〉 is harder to destroy by local measurements than that of | φ N 〉 .”

These correlations can be used to minimize the number of measurements required to project |Φ〉_C into a product state.

“These correlations can be used to minimize the number of measurements required to project | Φ 〉 C into a product state.”

The state |φ_N〉 is indeed much more entangled than most known N-qubit states.

“In that sense, the state | φ N 〉 is indeed much more entangled than most known N qubit states.”

Optical lattices can implement the interaction (1) by cold atomic collisions or other techniques.

“one has identified systems such as 'optical lattices' [17,18] in which the interaction (1) can be implemented by cold atomic collisions [18] or other techniques.”

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The unitary transformation U(φ) with φ=π can be realized by a single global operation.

“The unitary transformation U ( ϕ ) [before eq.(2)] with ϕ = π can thereby be realized by a single global operation.”

The parties may use either |φ_4〉 or |GHZ4〉 to teleport a qubit between any of the four parties.

“In other words, the parties may use either of the states | φ 4 〉 or | GHZ4 〉 to teleport [12] a qubit between any of the four parties.”

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The interaction Hamiltonian is equivalent to the quantum Ising model with H'_int = -∑_{a,a'} 1/4 ħ g(t) f(a-a') σ^{(a)}_z σ^{(a')}_z.

“Concerning the entanglement properties of the states we are going to investigate, this interaction Hamiltonian is equivalent to the quantum Ising model with H ′ int = -∑ a,a ′ 1 4 ℏ g ( t ) f ( a -a ′ ) σ ( a ) z σ ( a ′ ) z , where the indices a, a ′ run over all occupied lattices sites.”

This enables one, in principle, to create a variety of multi-particle entangled states such as |GHZ_M〉 with M≫3 by the entanglement operation U(φ), followed by 1-qubit measurements and subsequent 1-qubit rotations.

“This enables one, in principle, to create a variety of multi-particle entangled states such as | GHZM 〉 with M ≫ 3 by the entanglement operation U ( ϕ ), followed by 1-qubit measurements and subsequent 1-qubit rotations (compare Fig. 1b).”

The two- and three-dimensional problem is thereby reduced to the one-dimensional problem.

“We have thereby reduced the two- and three-dimensional problem to the one-dimensional problem.”

The interaction (1) realizes simultaneous conditional phase gates between qubits at neighboring sites a and a'.

“In the language of quantum information, the interaction (1) realizes simultaneous conditional phase gates between qubits at neighboring sites a and a ′ .”

The cluster states can be regarded as a scalable resource for other multi-qubit entangled states, such as multiparticle GHZ states.

“The cluster states can be regarded as a (scalable) resource for other multi-qubit entangled states, such as multiparticle GHZ states.”

These systems allow one to switch on and off the coupling g(t) between all qubits simultaneously by manipulating the parameters of the trapping lasers.

“These systems allow one, in particular, to switch on and off the coupling g ( t ) between all qubits simultaneously by a manipulation of the parameters of the trapping lasers.”

For α=β=1/√2, this includes the family of generalized (multiparticle) GHZ states on this subset.

“For α = β = 1 / √ 2, this includes, in particular, the family of generalized (multiparticle) GHZ states on this subset.”

For the state |GHZ4〉, a single local measurement suffices to bring it into a product state.

“For the state | GHZ4 〉 , in contrast, a single local measurement suffices to bring it into a product state [13].”

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