1. IntroductionIt is well known that the Hermiticity of any physical observables is one of basic demands in standard quantum mechanics. For example, the Hamiltonian operator of a closed system must be a Hermitian operator, which means that the eigenvalues (or called energy spectra) of the Hamiltonian is real and the time evolution of the system is unitary. However, in Ref. [1] Bender and his co-workers showed that a non-Hermitian operator, i.e., parity–time (
) symmetric Hamiltonian, may still have real energy spectra under some conditions, which is a novel result differed from the conventional viewpoint. At present, non-Hermitian systems attract a great deal of interest, which is no longer only of academic interest since it is becoming experimentally accessible in the optical system.[2–7] The study of non-hermitian systems was generalized to various fields, i.e., quantum thermodynamics and quantum information science. It has been shown that for a special class of non-hermitian systems, namely in the
-symmetric systems, the quantum Jarzynski equality and some fluctuation relations still hold.[8–11] Moreover, in Ref. [12] the authors have shown that the evolution time between two quantum states under the
-symmetric operation can be arbitrarily small. In Ref. [13] the authors found that the no-signalling principle can be violated when applying the local
-symmetric operation on one of the entangled particles. More recently, it has also been shown that
-symmetric operation can be used to suppress the decoherence of amplitude damping model,[14–20] and slow down the decoherence of pure phase damping model.[21]
On the other hand, the entanglement of assistance (EOA)[22,23] as a special case of the localizable entanglement[24] has recently attracted intense attention,[25–31] which quantifies the entanglement that can be generated between two parties, Alice and Bob, given assistance from a third party, Charlie, when three of them share a tripartite state and where the assistance consists of Charlie initially performing a measurement on his share and communicating the result to Alice and Bob through a one-way classical channel. Inspired by the
symmetric operation with non-trace preserving feature, in the present paper we study the influence of a
symmetric operation on the EOA of a type of W-like state without and with decoherence. By investigation, we derive some interesting results. First we obtain the explicit expressions of the EOA of the
symmetric operation for both the cases without and with decoherence. Further, for the unconsidered decoherence case we find that the EOA shared by two parties can be obviously enhanced with the assistance of the third party by a local
symmetric operation. For the considered decoherence case we find that in the larger amplitude damping case the scheme of
symmetric operation performed on one of two parties with the influence of noise is superior to that of
symmetric operation performed on the third party without the influence of noise in suppressing amplitude damping noise. However, for the smaller amplitude damping case the opposite result is obtained. Hence local
symmetric operation may provide an alternative method to realize effective tripartite-to-bipartite entanglement localization.
The paper is organized as follows. In Section 2, we give a brief reviews of
symmetric operation and EOA. In Sections 3 and 4, we investigate the influences of
symmetric operation on the EOA without and with decoherence. Finally, we give the conclusion of our results in Section 5.
2. A brief reviews of
symmetric operation and EOATo begin with, we would like to give a brief review of the so-called
-symmetric operation adopted in the present paper. From the result of Ref. [32], we learn that for a qubit system a famous
-symmetric Hamiltonian can be given by
where the real number
s is a scaling constant, and the real number
θ is a parameter characterizing the non-Hermiticity of
, i.e., the condition |
θ| <
π/2 ensures the eigenvalues of
real as
symmetry. Obviously, the condition
θ = 0 means that
is Hermitian. When
,
is strongly non-Hermitian. Note that such a non-Hermitian Hamiltonian can be simulated in a nuclear magnetic resonance quantum system.
[4]Equivalently, the
-symmetric Hamiltonian can also be rewritten as
The corresponding time evolution operator of Hamiltonian
can be obtained as follows:[12,13]
where
t′ =
tΔ
E, with Δ
E ≡ (
E+ −
E-)/2 =
scos
θ. Here,
E± = ±
scos
θ are the eigenvalues of the
.
Generally speaking, the time evolution of a system with a non-Hermitian Hamiltonian is not trace preserving, namely
To this end, the density operator of the system has to be renormalized by
It is worth noting that the non-trace preserving feature is a characterization of a system with non-Hermitian Hamiltonian, which may correspond to a nonlocal effect, rather than a genuine local operation as explained by Ref. [
14]. Naturally, from the point of view of the non-trace preserving feature of non-Hermitian Hamiltonian, we may define the success probability of realizing various tasks via non-Hermitian operation as
As for the entanglement of assistance, we take a
W-like tripartite state as an example. In what follows, we assume a
W-like tripartite state as
shared by three parties referred to as Alice (subscript 1), Bob (subscript 2), and Charlie (subscript 3), where |
α|
2 + |
β|
2 + |
γ|
2 = 1. Then Charlie implements a measurement on his qubit to yield a known bipartite entangled state shared by Alice and Bob, which leads to the concept of the entanglement of assistance (EOA). For a pure 2 × 2 ×
n state |
ψ⟩
123 a popular measure of EOA
[33] is defined as
where
ρ12 = Tr
3 (|
ψ⟩
123 ⟨
ψ|), and
C(|
φi⟩
12) is the famous concurrence defined by Ref. [
34]. The maximization is taken over all decompositions,
. Equation (
8) can be adopted to quantity the maximal average entanglement between Alice and Bob with the assistance of Charlie by LOCC. Especially, for a pure 2 × 2 ×
n state Thomas Laustsen
et al.[35] have obtained an analytical expression for the EOA as
where
with
σy Pauli matrix, and
λi are the square roots of the eigenvalues of
. More importantly, it has been proved that equation (
1) is an entanglement monotone for 2 × 2 ×
n-dimensional pure states. Clearly, for here considered state as Eq. (
7) the EOA with the assistance of Charlie by local measurements can be obtained as
Ea (|
ψ⟩
123) = 2|
α ||
β|. Moreover, if Charlie makes a projective measurement in the computational basis |0⟩,|1⟩ an general EPR pair between Alice and Bob can be obtained by the projective measurement with a certain success probability as |
α|
2 + |
β|
2. It should be noted that authors in Ref. [
36] found that weak measurements and their reversal can be useful for probabilistically increasing the EOA between Alice and Bob with the assistance of Charlie.
3. Generation of entanglement of assistance without decoherence via local
symmetric operationIn this section, we discuss the influence of
symmetric operation on the EOA without decoherence. To more conveniently compute the effect of
symmetric operation as shown in Eq. (3), we first give some interesting relationships for
symmetric operation on |i⟩ (i = 0, 1) as
which is used to simplify some calculations. In the following, we give the concrete scheme to show the implement of
symmetric operation.
Scheme Before or after qubits 1 and 2 are sent to Alice and Bob, Charlie carries out a local
symmetric operation on his qubit by Eq. (10). The density operator of the total system consisting of Alice and Bob in the basis {|00⟩12,|01⟩12,|10⟩12,|11⟩12} can be expressed as
where
According to Eq. (9), it is easy to obtain the EOA of the state Eq. (7) under
symmetric operation as
 | |
In what follows, using Eq. (12) we plot Ea as functions of time t and parameter θ for famous W state, i.e.,
in Eq. (7) as shown in Fig. 1. First, Ea of the W state without local
symmetric operation can be easily obtained as Ea = 2/3. From Fig. 1(a) we can clearly see that if only θ ≠ 0 (because θ = 0 corresponds to a local Hermitian operation, which does not alter the EOA) the Ea of using local
symmetric operation in the certain time intervals will exceed the original Ea = 2/3 without local
symmetric operation, which can be further revealed in Fig. 1(b) (see the blue line→green line→red line, i.e., the local
symmetric operation case and the black line-local Hermitian symmetric operation case, but the EOA cannot be altered). By simple calculation, we also find that the maximal Ea can reach Ea ≈ 0.91995, which is greater than Ea = 2/3 without the local
symmetric operation case. This may seem counterintuitive, since in the known entanglement theory one of important axioms is that local operation and classical communication can not increase quantum entanglement. However, this is not the case here. the non-trace preserving feature is a characterization of a system with non-Hermitian Hamiltonian, i.e.,
symmetric Hamiltonian as Eq. (4). The renormalization procedure as Eq. (5) induces a nonlocal effect, and the entanglement monotonicity is violated as explained by Ref. [14]. Further, it should be noted that the enhancement of the EOA comes at the expense of the success rate. Similar to the weak measurement method in suppressing amplitude-damping decoherence,[37] there is a trade-off between success probability as Eq. (6) and the amount of entanglement as Eq. (12). Moreover, from Figs. 1(a) and 1(b) we can find that Ea shows a periodic oscillation feature with time t for different parameters θ, which means that the EOA between two parties can be restored when we apply the
-symmetric operation on the third party. Especially, at the θ → ± π/2 (corresponding to the strongly non-Hermitian case), i.e., θ = 5π/11 the Ea exhibits a strangely periodic evolution with time t, namely a change will suddenly happen (similar to quantum phase transition) at some special time. Moreover, from Fig. 1(b) we can also find that the maximal Ea between two parties is also improved with the increase of
-symmetric operation strength (i.e., θ = 0 → π / 8 → π/4 → 5π/11). From the above analysis, we can learn that entanglement of assistance shared by two parties can be obviously enhanced with the assistance of the third party by local
symmetric operation in some proper time.
4. Protection of entanglement of assistance with decoherence via local
symmetric operationIn this section, we further discuss the influence of
symmetric operation on the EOA with decoherence. In the following, we provide two schemes to show the implementation of
symmetric operation.
Scheme I First, qubits 1 and 2 are sent to Alice and Bob. Then two qubits of Alice and Bob individually undergoe a zero-temperature amplitude damping channel, the density operator evolution is governed by
Obviously, the Kraus operator of total system
Eij can be expressed as a tensor product of the Kraus operator of each qubit system, i.e.,
Eij =
Ei ⊗
Ej. For a single qubit system, there are
and
d denotes the amplitude of the upper level of a qubit. At certain time, Charlie performs a local
symmetric operation on his qubit by Eq. (
10). After two qubits of Alice and Bob going through two independent zero-temperature amplitude damping channels the combined state of Alice, Bob, and Charlie at time
tc can be given by
where |
W′⟩
123 =
α d1|100⟩
123 +
β d2 |010⟩
123 +
γ |001⟩
123,
μ = |
α|
2 (1 − |
d1|
2) + |
β|
2 (1 − |
d2|
2),
d1 and
d2 are the amplitudes of the upper two qubits for Alice and Bob, respectively. Then at time
tc, Charlie performs a local
symmetric operation and the corresponding state of Alice and Bob can represented as
where
Here
G is the same as
g of Eq. (
11). In view of the same form of Eqs. (
11) and (
15), the corresponding EOA has a similar solution to
Equivalently, equation (
16) can be rewritten as
Interestingly, equation (
17) shows that after the sequence of two amplitude damping channels and a local
symmetric operation,
Ea of the
W-like state can be expressed as the products of the EOA under a local
symmetric operation, i.e., Eq. (
12) and the channels’ action on the maximally entangled state, i.e., |
d1|, |
d2|. Moreover, when we only consider two decoherence channels but no local
symmetric operation (i.e.,
θ = 0 or
t′ = 0),
Ea has a simple form as
Ea = 2|
α||
β||
d1||
d2|. Recall that for the case without considering amplitude damping channel there are
Ea = 2|
α||
β| and Eq. (
12). Therefore, in this case the improvement of a local
symmetric operation with two amplitude damping channels is same as that of having no amplitude damping channel. Here it is worth noting that Ref. [
38] has derived the similar factorization law of entanglement evolution of two qubits under noisy environments as Eq. (
17). Instead, in the present paper we consider the EOA involved in a three-qubit system and the implementation of local
symmetric operation.
Scheme II In scheme I, after qubits 1 and 2 of Alice and Bob undergoing two amplitude damping channels a local
symmetric operation is performed on the qubit of Charlie without the influence of noise. An interesting question naturally arises: What would happen if the local
symmetric operation is applied to the qubit of Alice or Bob with the influence of noise? For two qubits of Alice and Bob individually undergoing a zero-temperture amplitude damping channel as given by Eq. (14), at time tc Alice makes a local
symmetric operation on her qubit. After the sequence of two amplitude damping channels and a local
symmetric operation, by some simple calculations the reduced density operator of two qubits of Alice and Bob in the basis {|00⟩12,|01⟩12,|10⟩12,|11⟩12} can be written as
where
![]()
Using Eq. (
9), it can be found that the corresponding EOA of two qubits of Alice and Bob can be expressed as
As is expected, if we choose
θ = 0 and
t′ = 0 in the local
symmetric operation as Eq. (
3),
Ea will reduce to 2|
α||
β||
d1||
d2|, which is just the cases of Hermitian operation and identity matrix, respectively. Now having Eqs. (
17) and (
19) in mind, we can discuss the local
symmetric operation on the improvement of EOA for both schemes I and II.
To show the effects of schemes I and II in protecting the EOA against decoherence, in Fig. 2 we plot the comparisons of their Ea as a function of the time t for different d1 and d2.
First figure 2 shows that using the local
symmetric operation (see red, green and blue lines) in certain time can effectively suppress the amplitude damping noise (see the black line-Hermitian operation case) for the non-Hermitian operation applied to both qubit of Charlie without the influence of noise and qubit of Alice with the influence of noise, which can reach the maximum value by choosing a proper time. Further, from Figs. 2(a) and 2(c) we can find that for the larger amplitude damping case (i.e., d1 = d2 = 0.1) Ea of scheme II (see Fig. 2(c)) is better than that of scheme I (see Fig. 2(a)). The reason is that for the larger amplitude damping case, the schemes I and II correspond to the local
symmetric operation performed on qubit of Charlie without the influence of noise and qubit of Alice with the influence of noise, respectively. However, for smaller amplitude damping case (i.e., d1 = d2 = 0.9) the opposite is true, which means that the effect of scheme I (see Fig. 2(b)) is obviously superior to that of scheme II (see Fig. 2(d)), which is an interesting phenomenon. Therefore, the different effects of schemes I and II on the protection of EOA of two parties may be distinguished by different amplitude damping cases. In addition, it can be seen from Fig. 2 that Ea shows a periodic evolution with period π with the change of time t. To explicitly show the merits of a
symmetric operation including pre-
and post-
symmetric operations on suppression of the amplitude damping noise, we also numerically calculate the maximal Ea with and without the
symmetric operation under concrete parameter conditions (i.e., θ = 0 → π/8 → π/4 → 5π/11), which are listed in Table 1. Before ending the section, it should be noted that although we have focused on the famous W state case, actually for other W-like states with unbalanced coefficients, i.e., α = β = 1/2,
in Eq. (7), similar effects of the local
symmetric operation can also be obtained, which is not given in the text to avoid repetition.
5. ConclusionsBefore ending this paper, it is necessary to discuss the experimental realizations of the preparation of W state and the
symmetric operation in current experimental techniques. First a three-photon polarization-entangled W state can be created by a second-order emission process of type-II spontaneous parametric down-conversion (SPDC).[39] The
symmetric operation as shown in Eq. (1) can be implemented in a nuclear magnetic resonance quantum system[4] using the idea of duality quantum computing, in which an ancillary qubit is used as a conventional quantum computer. The concrete procedure of implementation for such a Hamiltonian is as follows:[4] The system we use contains two qubits: an ancillary qubit a and a work qubit e, which are initially in state |0⟩a |0⟩e. First we perform an unitary operation V on the ancillary qubit a. Then we apply two controlled unitary operations. Finally, a Hadamard operation is used on the work qubit e.
In summary, we have studied the effects of the
symmetric operation on the EOA shared by two parties with the assistance of the third party for a general W-like state without and with decoherence. Through an analytical approach, we obtain the explicit expressions of the EOA. For a famous W state without decoherence, we find that the EOA shared by two parties can be obviously enhanced with the assistance of the third party by a local
symmetric operation. Further, for the decoherence case, effective protection of the EOA against decoherence via local
symmetric operation can be also obtained. Especially, we find that for the larger amplitude damping case the scheme of the
symmetric operation performed on one of two parties with the influence of noise is superior to that of the
symmetric operation performed on the third party without the influence of noise in suppressing amplitude damping noise. However, for the smaller amplitude damping case the opposite result is obtained. Therefore, using
symmetric operation method to protect entanglement of assistance against amplitude damping noise could be distinguished by different amplitude damping cases. Moreover, although we have focused on the famous W state case we believe that the similar effect of
symmetric operation also exists in other W-like states with unbalanced coefficients, i.e., α = β = 1/2,
. The result that a local
symmetric operation of the third party can lead to the enhancement of entanglement of other two parties, which may have potential applications in the quantum information processing. Finally, it is worth noting that in Refs. [40] and [41] authors studied the enhancement of intensity-difference squeezing via energy-level modulations in atomic and atomic-like ensemble. Although the effect of the
symmetric operation on quantum state as Eq. (10) has some similarities to the energy-level modulations in the Refs. [40] and [41], the methods for energy-level modulations and subjects for study are completely different.
