Abstract
The generation and quantification of quantum entanglement is crucial for quantum information processing. Here we study the transition of Gaussian correlation under the effect of linear optical beamsplitters. We find the singlemode Gaussian coherence acts as the resource in generating Gaussian entanglement for two squeezed states as the input states. With the help of consecutive beamsplitters, singlemode coherence and quantum entanglement can be converted to each other. Our results reveal that by using finite number of beamsplitters, it is possible to extract all the entanglement from the singlemode coherence even if the entanglement is wiped out before each beamsplitter.
Introduction
Quantum correlation, especially quantum entanglement, is a key ingredient in quantum information science. Since the EinsteinPodolskyRosen (EPR) paradox has been put forward^{1}, quantum correlation has been investigated both in theory and in experiments^{2,3,4,5,6,7,8}. During the past decades, the characterization and definition of quantum correlation have attracted much research interest, such as Bell nonlocality, quantum steering, quantum entanglement, quantum discord, and quantum coherence^{9,10,11,12,13,14,15,16,17}. It has been pointed out that these definitions satisfy hierarchy relations, i.e., quantum coherence > quantum discord > quantum entanglement > quantum steering > Bell nonlocality^{18,19,20,21}, where A > B represents B is a subset of A, meaning all steerable states are entangled, but not all entangled states have steering, etc. On the other hand, quantum entanglement characterizes the nonclassical property of multipartite quantum systems. An entangled state can not be decomposed as a convex sum of product states^{14}, if two subsystems are entangled, the quantum state of each subsystem can not be described independently. Quantum entanglement only represents the nonclassical property between the subsystems which can not characterize the nonclassical property of a singleparty system. Nonclassicality can be used to characterize its nonclassical property of the quantum states, such as squeezed states, antibunched states, and subPoissonian states^{22}. The nonclassicality characterizes the quantumness of a singleparty quantum system. The concept of quantum steering comes from the EPR paradox. For a quantum system consisting of subsystems marked with A and B, A can steer B if A can affect B’s subsystem by using a local operation on A’s subsystem. Quantum steering has many useful applications, such as subchannel discrimination, oneside deviceindependent quantum key distribution, etc^{23,24,25,26}.
In addition to quantum correlation, quantum coherence can also be used to characterize the nonclassical property of both multipartite and singleparty quantum systems. Quantum coherence is one of the key features of quantum mechanics, several experiments have revealed that quantum coherence exists in photosynthetic complexes and indicating that it plays a key role in high efficiency achievements in photosynthesis^{27,28}. Quantum coherence can be quantified by the offdiagonal elements in the density matrix of the system under the reference basis, which is called the l_{1} norm coherence. Another way to measure coherence is the relative entropy coherence defined as the entropy difference between the density matrix of the quantum state and the diagonal density matrix that eliminating all the offdiagonal elements of the original density matrix.
Quantum systems can be roughly partitioned into discrete and continuous variable ones. As the most important family of continuous quantum states, Gaussian states are widely used because of its availability and controllability in experiments^{29}. Recently, some representative applications have been investigated, such as quantum communication, ultrasensitive sensing, detection and imaging, and so on ref. 30. Different from discrete variable quantum states^{13,14,15}, the approaches of quantifying the quantum correlation of continuous variable quantum states are more complicated^{12,31,32}. Here in this study, we will give a detailed mathematical expression of Gaussian correlation and investigate the Gaussian correlation generated by beamsplitters in experiments. Beamsplitters are key components in optics, and have been used in many experiments including Bell test experiment, Wheeler’s delayed choice experiment, etc^{33,34}. Especially in quantum information processing, Knill et al. pointed that efficient quantum computation is possible to be realized by using only beamsplitters, phase shifters, single photon sources and photodetectors^{35}. Although beamsplitters are linear optical devices, they can generate quantum entanglement^{36,37,38,39}.
In this work, we study the behavior of the quantum correlation of Gaussian states under the effect of beamsplitters. We find that the nonclassicality of singlemode Gaussian state generates entanglement by beamsplitters^{40} is not valid for all singlemode Gaussian states. Instead, singlemode coherence plays a vital role in such process. By using consecutive beamsplitters, we find the singlemode coherence and quantum entanglement could be converted to each other for two squeezed states as the input states. Moreover, if we further wipe out the entanglement before each beamsplitter, all the quantum entanglement can be extracted from the singlemode coherence by a finite number of beamsplitters.
Results
Entanglement and steering generation using single beamsplitter
Here we consider two singlemode Gaussian optical beams send into the two input ports of one beamsplitter. The covariance matrix of the output Gaussian optical beam can be characterized by
where σ_{out(in)} is the covariance matrix of the output (input) optical beam, U(θ, ϕ) could be described by^{40}
where denotes the transmittance and ϕ represents the phase difference between the reflected and transmitted fields. Here we consider the case where two singlemode input Gaussian optical beams are squeezed lights, one (say mode A) is squeezed in position quadrature and the other one (say mode B) is squeezed in momentum quadrature. The covariance matrices of modes A and B are given by
where r_{A(B)} is the squeezing parameter for mode A (B). The covariance matrix of the output optical beam becomes σ_{out} = U^{†}(θ, ϕ) (σ_{A}⊕σ_{B})U (θ, ϕ).
The quantum steering of the output optical beams could be calculated by
where . It is obvious that quantum steering can be created by the beamsplitter except the case where r_{A} = −r_{B} (meaning two optical beams are identical) or θ = nπ/2 (n is an integer). By considering the case , the maximal value of quantum steering is found at the point θ = (2n + 1)π/4. And the quantum entanglement of the output state could be obtained by
where .
In the following, we will study the change of the nonclassicality of the two singlemode Gaussian states after the action of a beamsplitter. By using
one can find N_{Δ} exhibits the similar behavior as quantum entanglement. In a previous work^{40}, the authors have found that the nonclassicality creates entanglement in the case where the input states are nonclassical Gaussian states and vacuum states. Here we do not use these states as the input states because mixing these two states by a beams splitter can not generate quantum steering. However, when the two input singlemode Gaussian states are two squeezed lights, their finding is not valid since N_{Δ} ≥ 0 does not always satisfied. Here Fig. 1(a) describes the region where N_{Δ} < 0 (r_{A} = r_{B} = r), and Fig. 1(b) displays the transition of quantum correlation when we regard θ as an independent variable and assume that r_{A} = r_{B} = 0.2. One may find that when 0 < θ < 0.33 and , both quantum steering and quantum entanglement increase and the nonclassicality also increase. Therefore, quantum entanglement and quantum steering are not generated by singlemode nonclassicality.
Then we consider singlemode coherence as
we find the relation C_{Δ} ≥ 0, and C_{Δ} equals to zero only when r_{A} = −r_{B} or θ = nπ/2. Meanwhile, C_{Δ} shows similar behavior as quantum entanglement and quantum steering. Here we numerically study the change of singlemode coherence in Fig. 1(b), where we choose θ as an independent variable and r_{A} = r_{B} = 0.2. Since C_{in} keeps as a constant with a fixed θ, when C_{out} = C_{(A)out} + C_{(B)out} decreases, both quantum entanglement and quantum steering increase, meaning that they can be generated by singlemode coherence. Here we can conclude that quantum entanglement between twomode Gaussian states can be regarded as an intrinsic coherence, i.e., coherence between the two modes. The intrinsic coherence and singlemode coherence are complementary to each other. In other words, intrinsic coherence increases along with the decrement of singlemode coherence^{41}. According to the definition^{42}, quantum correlations originates from the superposition of quantum states. The singlemode coherence can characterize the superposition of the subsystem (the local superposition), the intrinsic coherence (or quantum entanglement) represents the superposition between the subsystems (the unlocal superposition). Our observation indicates that the beamsplitters can be used to convert local superposition to nonlocal superposition, and vice versa.
Conversion between entanglement and quantum steering using Gaussian optical beams
We study in the following the case when more than one beamsplitters are used to mix the squeezed lights. From Eq. (2), we find that U(θ, ϕ) is a periodic function by considering θ as an independent variable. By choosing the input optical beams at the beamsplitters with (see Fig. 2(a)), we find both quantum entanglement and quantum steering increase after the first and second beamsplitters, and decrease when undergoing the third and fourth ones while quantum coherence shows the opposite behavior. Here in Fig. 3(a,b,c), we plot the evolution of quantum steering, quantum entanglement and quantum coherence for different number of beamsplitters as n = 1, 2, 3, 4, respectively. It is obvious that under the effect of consecutive beamsplitters, quantum entanglement (quantum steering) and quantum coherence convert to each other. Figure 3(d) illustrates the evolution of quantum correlation when θ = π/128 and r_{A} = r_{B} = 0.2. In this case, the evolution of the quantum correlation behaves like a nonMarkov process without loss. Therefore, with the memory effect, consecutive beamsplitters can be used to simulate the nonMarkovian environment.
To obtain the condition under which quantum entanglement is wiped out before the input optical beams and further mixed by the next beamsplitter, as illustrated by Fig. 2(b), we find that the quantum entanglement of the input beams after the last beamsplitter will decrease with the increment of the number of the beamsplitters. And the number of beamsplitters depends on θ, for instance, when , quantum entanglement can only be generated by the first beamsplitter. When θ → 0, quantum entanglement can always be generated. Figure 4 shows the quantum entanglement, quantum steering and singlemode coherence when considering n = 1, 2, 3, 4. The quantity of the generated entanglement depends on singlemode coherence of the input states. Once quantum entanglement has been erased, it can not convert back into singlemode coherence, and the observed entanglement will decrease with respect to n. On the other hand, quantum steering can only be generated by the first beamsplitter regardless of θ (see Fig. 4(b)). Since quantum entanglement has been wiped out before the consecutive beamsplitters, we can conclude that the process could be observed during which quantum entanglement is converted to quantum steering.
Summary
We have studied the progress of quantum entanglement and quantum steering generation using beamsplitters. We have found that instead of the singlemode Gaussian nonclassicality, singlemode Gaussian coherence acts as the fundamental resource for generating twomode Gaussian entanglement and steering for two squeezed states as the input states. The covariance matrix of general onemode Gaussian states can be written as where V_{S} is the covariance matrix of squeezed states, is a constant, R represents the phase rotation operator. By further calculations, we find when the two input squeezed states have the same phase rotation, the conclusion is also correct. Therefore, we suppose the conclusion is valid for general onemode Gaussian states. Based on the former results, it has been confirmed by the observation that singlemode coherence and the entanglement are complementary to each other, namely, quantum entanglement increases while singlemode coherence decreases, and vice versa.
Meanwhile, we have discussed the evolution of quantum entanglement under the action by several beamsplitters on two single squeezed states, and discovered that quantum entanglement and singlemode coherence transfers to each other periodically. If we wipe out the entanglement in the consecutive beamsplitters, only the first beamsplitter generates quantum steering. We hope these results on quantum correlation could further develop the applications in quantum information processing.
Method
Definition of Gaussian states
The continuous variable quantum system (Gaussian state) is always difficult to be characterized by using the density matrix since the dimensionality of the density matrix is infinite. Here we consider the canonical operators in the vector by , where , and () are the creation (annihilation) operators for the nbosonic mode system. The elements in satisfy the commutation relation as where is denoted the symplectic form.
The covariance matrix σ contains the elements of which is defined by^{29}
Here the diagonal elements of σ are the variance of and the offdiagonal elements encode the intermodal correlations among subsystems. And the quantity is defined as the mean value of . The physical meanings of characterizes the center of the probability distribution in phase space and σ describes its shape.
The covariance matrix of a twomode Gaussian state consisting of parties A and B could be expressed as
where and are the covariance matrices for subsystems A and B. And characterizes the correlation between A and B.
Quantum correlation of Gaussian states
The definition of negativity characterizes by the entanglement of twopartite discrete variable systems^{14,15}. Similar to discrete variable condition, we use logarithmic negativity to characterize the entanglement of Gaussian states, which can be calculated by E = max{0, −log_{2}v}, where v is the smallest symplectic eigenvalue of the partial transpose of the covariance matrix. This definition also originates from the positive partial transpose (PPT) criteria^{15,31}. In this work, we consider twomode Gaussian states, the partial transpose is defined as
where R = diag{1, −1, 1, 1}, and σ_{AB} denotes the covariance matrix of the twomode Gaussian state.
Quantum steering is also a type of quantum correlation whose definition is located between quantum entanglement and Bell nonlocality^{12}. It describes how local operations on one subsystem effects another. Assuming that a twomode Gaussian state consisting of subsystems A and B, A can steer B by A’s Gaussian measurements if the condition σ_{AB} + i(0_{A} ⊕ Ω_{B}) ≥ 0 is violated^{43}, where Gaussian measurements consists a measurement set that maps Gaussian states into Gaussian ones. The quantum steering from A to B can be obtained by
The definition above is only suitable for twomode Gaussian states, consisting of subsystems A and B, by implementing Gaussian measurements on A’s subsystem. In this paper, we use Eq. (11) to calculate the quantum steering of twomode Gaussian states.
The nonclassicality represents the nonclassical property of the nonclassical states, such as squeezed states, antibunched states, and subPoissonian states whose correlation function can not be reproduced by any classical field^{22}. The nonclassicality of the Gaussian states can be characterized by the covariance matrix of the system. For singlemode Gaussian states, some quantities (for example, the degree of squeezing) are used to characterize the nonclassicality. In this paper, we use N = −log_{2}λ_{min} to calculate the nonclassicality^{40}, where λ_{min} is the minimum eigenvalue of the covariance matrix (ref. 40 gives the equation N = −log_{2}2λ_{min} since they defined .
Quantum coherence characterizes the superposition property of quantum systems. The relative entropy coherence for discrete variable systems can be written as ref. 13
where S(ρ) = −Tr(ρlog_{2}ρ), and ρ_{diag} denotes the diagonal density matrix that eliminating all the offdiagonal elements of ρ under the reference basis. In this paper, we use the coherence measure as , where δ is the nearest incoherent Gaussian state of ρ. The coherence of onemode Gaussian states is ref. 32:
where and denotes the mean value of the thermal state number, which can be calculated by , where x_{i}(i = 1, 2) are the components of . In this study, we choose x_{i} = 0 and we have .
Additional Information
How to cite this article: Wang, Z.X. et al. Gaussian entanglement generation from coherence using beamsplitters. Sci. Rep. 6, 38002; doi: 10.1038/srep38002 (2016).
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Acknowledgements
The authors gratefully acknowledge the support from the National Natural Science Foundation of China through Grants Nos. 61622103, 61471050 and 11404031, the Fok YingTong Education Foundation for Young Teachers in the Higher Education Institutions of China (Grant No. 151063), the Open Research Fund Program of the State Key Laboratory of LowDimensional Quantum Physics, Tsinghua University Grant No. KF201610.
Author information
Affiliations
State Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, P. R. China
 ZhongXiao Wang
 , TieJun Wang
 & Chuan Wang
State Key Laboratory of LowDimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, P. R. China
 Shuhao Wang
 & Teng Ma
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Contributions
Z.W., S.W. and C.W. wrote the main manuscript text, Z.W. and S.W. prepared figures 1–5. All the authors reviewed the manuscript and discussed the results, drew conclusions and edited the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Shuhao Wang or TieJun Wang or Chuan Wang.
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