Quantum discord with weak measurement operators of quasi-Werner states based on bipartite entangled coherent states a
E. Castro*a, R. Gómeza, C. L. Laderaa and A. Zambranoa Departamento de Física, Universidad Simón Bolívar, Apdo. 89000, Caracas 1086, Venezuela ABSTRACT
Among many applications quantum weak measurements have been shown to be important in exploring fundamental physics issues, such as the experimental violation of the Heisenberg uncertainty relation and the Hardy paradox, and have also technological implications in quantum optics, quantum metrology and quantum communications, where the precision of the measurement is as important as the precision of quantum state preparation. The theory of weak measurement can be formulated using the pre-and post-selected quantum systems, as well as using the weak measurement operator formalism. In this work, we study the quantum discord (QD) of quasi-Werner mixed states based on bipartite entangled coherent states using the weak measurements operator, instead of the projective measurement operators. We then compare the quantum discord for both kinds of measurement operators, in terms of the entanglement quality, the latter being measured using the concept of concurrence . It’s found greater quantum correlations using the weak measurement operators. Keywords: Quantum discord, weak measurement, bipartite entangled non-orthogonal states.
1. INTRODUCTION The definition of a quantum measurement procedure is fundamental for the concept of the value of a variable in quantum mechanics. If we define a measurement procedure of a variable, the possible outcomes of this measurement is associated with a set of eigenvalues. Every state of a quantum system is associated with a probability distribution for outcomes of measurement of every variable. The standard model of quantum measurement is the von Neumann procedure [1]. In a typical von Neumann measurement an observable of a system is coupled to a measurement apparatus or pointer via its momentum. This coupling leads to an average shift in the pointer position that is proportional to the expectation value of the system observable. In a strong measurement, this shift is large relative to the initial uncertainty in pointer position, so that significant information is acquired in a single shot. In 1998, Aharonov, Albert and Vaidman proposed an alternative quantum measurement scheme called weak measurement [2]. In a weak measurement, the pointer shift is small and little information can be generated on a single shot, introducing a little disturbance in the system. A weak measurement involves three stages generally; (1) the system to be measured and a measurement probe are prepared in initial states (pre-selection), (2) the system and the probe are coupled by such a weak interaction that the state of the system remains almost undisturbed after the interaction, (3) the ⟩, the pointer shift of the probe is then recorded, system is observed, and it is found to be in a specific final state otherwise, discarded (post-selection). The weak value obtained from such a measurement can lie outside the range of the eigenvalues of A, in sharp contrast to the expectation value. For this reason, unlike the results of strong measurement, weak values are not constrained to lie within the eigenvalues spectrum of the observable being measured. On the other hand, it has been shown that weak measurements are universal in the sense that any generalized measurement can be realized as a sequence of weak measurements which results in small changes to the quantum state for all outcomes [3]. This kind of measurement approach may be useful in quantum optics and quantum communication, where the precision is as important as the precision of quantum state preparation. The technique of weak measurement have proven very useful tool for investigating fundamental questions in quantum mechanics and technological applications [4-7]. Quantum correlations have been comprehensively accepted as the main resource for different quantum optics and quantum information task. The traditional belief was that entanglement, between the subsystems that the parties share in a system was the most important correlation. However, the quantum discord initially introduced by Olliver and Zurek [8], *
[email protected] 8th Iberoamerican Optics Meeting and 11th Latin American Meeting on Optics, Lasers, and Applications, edited by Manuel Filipe P. C. Martins Costa, Proc. of SPIE Vol. 8785, 87858U © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2021625 Proc. of SPIE Vol. 8785 87858U-1
and Henderson and Vedral [9], quantifies nonclassical correlations beyond the standard classification of quantum states into entangled and unentangled. In general, the quantum discord is different from entanglement and may be nonzero even for certain separable states. ) ( Quantum discord, for bipartite systems, is defined as a difference ( of two classical equivalent expressions for the classical mutual information [8, 9] (
)
( )
( )
(
)
( )
(
( )
)
),
(
) between generalizations
(1)
( )
(
),
(2)
where ( ), ( ) and ( ) are Shannon entropies of the random variables X, Y and the pair ( ) respectively, while ( ) and ( ) are conditional entropies. In the quantum word the situation is fundamentally different, since measurement can disturb quantum systems and the mutual information ( ) and ( ), are different. The Shannon entropy of the random variables X, Y and ( ) are replaced by the von Neumann entropies of , and , respectively. Then, is the density matrix of the whole system AB, ( ) and ( ) are the reduced matrix density matrix of subsystem A and subsystem B. The quantum mutual information are given by (
)
(
)
(
(
)
(
)
(
)
(
),
(3)
),
(4)
where ( ) ( ) is the von Neumann entropy of a quantum state and ( ) is the quantum conditional entropy. In QD it is assumed that we perform a set of local projective measurement (von Neumann measurements) ( ) ⟩⟨ on subsystem B. Then the quantum condition entropy is given by { } (
)
{
( )
(
}
( )
{
}
)
{
( )
}
∑
(
( )
with the minimization being over all projection-valued measurements quantum correlation called quantum discord QD, is given by (
⃖
)
(
)
( )
∑
(
{
( )
}
),
(5)
. Consequently, the amount of genuinely
),
(6)
This process usually induces strong perturbations on the subsystem B, and the whole system AB simultaneously. Due to this measurement procedure we lose coherence, and possibly constrain one’s ability to extract as much quantum correlations as possible. However, if instead of projective measurement, we performed a measurement which couples the subsystem b to the measurement device weakly, then the subsystem will be perturbed weakly and it will not lose its coherence completely. This process of extracting quantum correlations using weak measurement is known in the literature as super quantum discord (SQD). The weak measurement operator formalism was introduced by Oreshkov and Brun [3]. The weak measurement operators are given by ( )
(
√
)
(
)
(
√
√ )
(
√
)
(
, )
(7)
,
Where x is a parameter that denotes the strength of the measurement orthogonal positive-operator valued measure (POVM) projectors, which satisfies ( ) ( ) ( ) ( ) operators satisfy: , ( ) , ( ) ( ) ( √
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(8) process, and and are two . These weak measurement ), [ ( ) ( )] , and in the
( )
strong measurement limit we recover the projective measurement operator; .
and
(
)
Here, if x=ε, where , the measurement is weak. When we perform weak measurements on B, ( ) , these measurements will disturb weakly both the subsystem B and the whole system AB. After these weak measurements, the post-selection for the subsystem A is given by (
where
)
(
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(
(
))
(
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(
(
),
(9)
is the 2x2 identity matrix. ( ) (
[(
(
))
(
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(
(
(
)] ,
(10)
) ),
(11)
Here, (
) are the probabilities of weak measurements occurs. The von Neumann conditional entropy is given by ( ) ( ( ) ( ( ) ) ), this is a function of the measurement strength parameter x and ( ) ( ) ( ) [10] [10,11] using the concavity of entropy is easy to prove that ( ) ( . The amount of QD for bipartite ( ) ) ( ) quantum state
, using the weak measurement operators ⃖
(
)
(
(
)
),
(12)
Note that the SQD on the subsystem B, is a function of the weak conditional entropy and it is straightforward to show [10] that . Thus ⃖ ⃖ ⃖ is a measure of the pair-wise mutual information that is locally inaccessible, and it is a function of the measurement strength parameter x. This is non-negative, and in general it is not symmetric, i.e. . On the other hand, SQD remain unchanged by performing local unitary transformations, and in the ⃖ [10,11]. strong limit when , we obtain the normal quantum discord; ⃖ ⃖ The importance of introducing weak measurements in the analysis of QD is that when we disturb the subsystem A or B, of the whole system AB, we disturb this subsystem weakly introducing weak quantum decoherence in the measurement. As a consequence we capture more quantum correlations using than ⃖ . On the other hand, the weak ⃖ measurement with different strengths can impose different orders of quantum states, and can change the monogamous character for certain classes of states, inducing counterintuitive effects in the quantum discord [12-16]. In this work, we consider bipartite entangled non-orthogonal states. We attempt to compare QD with the SQD of Werner states formed with four bipartite entangled non-orthogonal states. We use the ⟩ to encode qubits when they are ⟩. The ⟩ states of difference and the overlaps ⟨ ⟩ are different from zero. Since ⟩, superposed with are not orthogonal, we choose an orthogonal basis by considering odd and even states, given by ⟩
( ⟩
⟩)
⟨
(
,
⟩)
.
(13)
⟩]
(14)
Now we consider the four bosonic bipartite entangled f-deformed states ⟩ With
(
⟨
[
⟩
⟩]
⟩) . Under the orthogonal basis ⟩
[
⟩
[
⟩ (
)
(
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⟩ (
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] ]
,
⟩
⟩ we can write, ⟩
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[
⟩
,
⟩
√
√
[
⟩
⟩]
(15)
[
⟩
⟩]
(16)
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⟩ and ⟩ are maximally entangled states, while the states ⟩ and ⟩ are non-maximally The two states entangled states and are not-orthogonal mutually. They form a complete orthogonal basis just like standard Bell states. The Werner states of a bipartite system, formed by 2 dimensional subsystems, is a mixture of the fully mixed states with probability (1-p) and a singlet state ⟩ with probability p, given by (
⟩ )
(
)
⟩⟨
(17)
⟩ ⟩ ⟩ ⟩ . The Werner states are rotationally invariant giving the same results There we assume that ⟩ for the conditional entropy in any measurements basis, therefore it is not necessary to minimize it over all measurement ( ) ( ) and , and they are independent of the ⟩ states basis. The eigenvalues of are ⟩ ) are known as quasi-Werner states. parameters. The Werner states ( ⟩ ) and ( To study the relation between quantum discord and entanglement, we choose concurrence as a measure of entanglement [17] . The concurrence is defined as the nonnegative real number , where are the square roots of eigenvalues positive matrix ̃, in descending order. Where ̃ is the result of applying the spin-flip operation ⟩ and to , i.e., ̃ ( ) ( ) and the complex conjugation is taken in the standard basis. The two states ⟩ are maximally entangles states with unit concurrence. The states ⟩ and ⟩ are non-maximally entangled with concurrence ⟨
(
⟩ )(
⟨
⟩ )
(18)
Below, we study the SQD of several bipartite entangled non-orthogonal states. The first one is the bipartite entangled coherent state. It is well know that these states are an important type of robust states which are extensively applied to various continuous processing and transmission task. Those states are chosen since they are easy to generate experimentally and convenient to use. The second one is the set of displaced, squeezed and squeezed displaced Fock states. These states are very important both theoretically and experimentally in non-linear quantum optics. In the more general form the inner product for the squeezed displaced Fock states is given by [18] ⟨
⟩ (
√
)
( ⁄
⁄
(
) (
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] ∑ ∑
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∑
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(19)
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́
( ⁄ ) ) (
[
]
The bipartite entangled non-orthogonal states for displaced, squeezed and squeezed displaced state can be obtained ⟩ and ⟩ by ⟩. replacing ⟩ by
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2. RESULTS AND DISCUSSION We have a special interest to study the QD of the Werner and quasi-Werner mixed states based on bipartite entangled coherent states, using the weak measurements operator, and for this reason we considered in this work the quasi-Werner and Werner states ( ) and ( ). In order to simplify our analysis in this work we considered that the squeezing parameter is real, fixing the squeezing angle θ to zero. Then the inner product only depends on the real squeezing parameter r and upon the Fock displacement number m. In figure 1 we plot the SQD as a function of the measurement strength parameter x, for different quantum states. We consider the following states: coherent states (dotted line), squeezed states (small dashed line), and two squeezed displaced states (large-dashed line m = 1, and dot-dashed line m = 2). ⟩ ) state and in figure 1.b we consider the SQD of the In figure 1.a we consider the SQD of quasi-Werner ( ⟩ ) state. Both figures are plotted for Werner ( , the mixing parameter p = 0.5 and the measurement angles θ =0.5 rad and ϕ = 1 rad. The continuous line represent the QD obtained using the von Neumann measurement projection operators and , obtained in the limit when . Figure 1, demonstrates that SQD of all the quantum states considered, have the same asymptotic behavior for large values of x. It was also found that SQD for , is greater for the squeezed displaced states than for the squeezed and coherent states. SQD
0.8
SQD
0.9
a b c d e
0.6
0.4
a b c d e
0.8 0.7 0.6 0.5
0.2 0.4 0.3 0.0 0
1
2
3
(1.a)
4
5
x
6
0
1
2
3
(1.b)
4
5
x
6
⟩) and Werner Figure 1. Plot of the SQD as a function of the strength measurement parameter x,for the quasi-Werner ( ⟩). For the (a) line we have the coherent state ( , and , for the (b) line we have the coherent , for the (c) line we have the squeezed state , for the (d) line we have the coherent displaced state and for the (e) .
In Figure 2 we plotted the SQD of Werner ( ) (Fig. 2.a) and quasi-Werner (Fig. 2.b) states as a function p, for , x = 0.5 (weak measurement), θ = 0.5 rad and ϕ = 1 rad in order to illustrate the difference of SQD between the coherent, the squeezed and squeezed displaced state. In fig. 2.a and 2.b we plot the SQD for coherent states (dotted line), squeezed states (continuous line) and two squeezed displaced states (large-dashed line m = 1, and dot-dashed line m = 2). In the figure it can be easily seen, that the SQD of the quasi-Werner states is similar for all states for small p. However, for large values of p the difference between all states is notable. The SQD of squeezed displaced states is higher than for squeezed and coherent states. In the case of Werner States we can see that all states have the same SQD for all values of p. ⟩) states is studied The dependence of the SQD as a function of the mixture parameter p for the quasi-Werner ( in figure 3. In this figure we took , rad, rad and x = 0.5. We considered the coherent state ⟩ (Fig. 3.a), squeezed state ⟩ (Fig. 3.b) and two squeezed displaced states ⟩ (Fig. 3.c) and ⟩ (Fig. 3.d). with In figure 3, we find that apart from entanglement SQD show more non-local quantum correlations in all cases. We also find that the concavity of the SQD is similar for the coherent and squeezed quasi-Werner like states. But the squeezed
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displaced states have similar concavities between them and they are different to the concavities of coherent and squeezed states. SQD
SQD
1.5
2.0
a b c d e
1.0
a b c d e
1.5
1.0 0.5 0.5
0.0
0.0 0.0
0.2
0.4
0.6
0.8
(2.a)
p
1.0
0.0
0.2
0.4
0.6
0.8
(2.b)
p
1.0
⟩) and Werner ( ⟩). For the Figure 2. Plot of the SQD as a function of the mixing parameter p,for the quasi-Werner ( (a) line we have the coherent state , and , for the (b) line we have the coherent , for the (c) line we have the squeezed state , for the (d) line we have the coherent displaced state and for the (e) . SQD
SQD0.5
0.5
0.4
0.4
a b c
0.3
a b c
0.3
0.2
0.2
0.1
0.1
0.0
0.0 0.0
0.2
0.4
0.6
0.8
(3.a)
p
1.0
0.0
0.2
0.4
0.6
0.8
(3.b)
p
1.0
SQD
SQD
1.5
1.0 a b c
0.8
0.6
a b c
1.0
0.4
0.5
0.2
0.0
0.0 0.0
0.2
0.4
0.6
(3.c)
0.8
p
1.0
0.0
0.2
0.4
0.6
(3.d)
0.8
p
1.0
⟩). For 3.a we have the coherent Figure 3. Plot of the SQD as a function of the mixing parameter p, for the quasi-Werner ( state , for 3.b we have the squeezed state , for 3.c we have the squeezed displaced state and for 3.d we have the squeezed displaced state . In each plot the (a) line is for , (b) line for and (c) line is the concurrence for each state considered.
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3. CONCLUSION In this work, we have analyzed the advantage of using weak measurement operators instead of projective measurements POVM in the study of QD. We considered the Werner and quasi-Werner entangled coherent states in order to compare the effects of implementing weak measurements in the study of quantum discord, and we extended the analysis to nonlinear quantum states of quantum optics the squeezed and squeezed displaced states. We find that the SQD is greater than the normal discord for small values of the strength measurement parameter x, and approaches asymptotically to the normal discord for larger values of x. The Coherent, squeezed and squeezed displaced states behave in a similar way in the weak measure scheme in terms of discord variation under mixing and measure angle parameters, but with the previously discussed enhance in the value of discord over their counterparts under the Von Neumann measurement scheme. For pure states our analysis could be resumed that they are robust in the sense that nor squeezing, nor displacement affect their nature, only the measure strength parameter affect the value of quantum discord, growing considerably as for the mixed states. The advantage of using weak measurement compared with standard projective measurement is that we may obtain extra quantity of quantum correlations, and this advantage is gradually diminishing as the measurement strength parameter grows. It implies that the quantum correlation, in a quantum state not only depends upon the measurement scheme but it also strongly depend on the interaction with the related measure instrument. We have a significant improvement when considering quantum correlations, which could be very useful performing tasks of communication or quantum computation where performance is dependent of the quantity of quantum correlations available and squeezed and displaced states present a fine tuning for state preparation and also benefit from the measure scheme, they may be suitable for experiments where few photons are used or required, such as in very sensitive cryptographic protocol or even a computation designed for few photons. So the weak measurement scheme, forbids us a possible powerful tool in quantum optics, were precision in state preparation and greater quantum correlation available is required to perform communication or quantum computation protocols or tasks.
4. ACKNOWLEDGMENTS This work partially was supported by the “Programa de Estímulo a la Investigación (PEI) del Ministerio del Poder Popular para Ciencia, Tecnología e Industrias Intermedias, proyecto No 1.120: Desarrollo de un sistema de transmisión segura de información, basado en información cuántica”.
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[11]Singh, U., and Pati, A. K., “Weak measurement induced super discord can resurrect lost quantumness,”arXiv 1305.4393v1 (19 May 2013) [12] Hu, M-L., Fan, H., and Tian, D-P., “Dual role of weak measurements for quantum correlation,”arXiv: 1304.5074v2 (15 May 2013) [13] Hu, M-L., Fan, H., and Tian, D-P., “Dual role of weak measurements for quantum correlation,” arXiv: 1304.5074v2 ( 15 May 2013) [14] Wang, Y-K., Ma, T., Fan H., Fei,S-H., Wang, Z-X., “Super quantum correlation and geometry for Bell-diagonal states with weak measurements,” arXiv:1302.4039v3 (24 Feb 2013) [15] Huai, L-P., Li, B., Qu S-X., Fan H., “Quantum advantage by weak measurements,” arXiv:1305.6366v3 (30 May 2013) [16] Mishra M. K., Maurya, A. K., and Prakash, H., “Quantum discord and entanglement of quasi-Werner states based on bipartite entangled coherent states.” arXiv:1209.3706v1 (17 Sep 2012) [17] Wootters, W.K., “Entanglement of Formation of an Arbitrary State of Two Qubits,” Phys. Rev. Lett. 80 (10), 2245 (1998) [18] Abdel-Aty, M., Al-Kader, G. M. A., and Obada, A-S. F., “Entropy and entanglement of an effective two-level atom interacting with two quantized field modes in squeezed displaced Fock states, ” Chaos, Solitons and Fractas 12, 24552470 (2001)
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