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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Quantiﬁcation of bipartite and multipartite entangled states in Mach–Zehnder interferometer Mohamad Ali Jafarizaeh a, Farzam Nosraty b,n, Mostafa Sahrai b a b

Department of Physics, University of Tabriz, Tabriz, Iran Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 24 July 2014 Received in revised form 11 October 2014 Accepted 31 October 2014

In the present study, the amount of entanglement of the quantum states capable of being used in an optical interferometer such as the Mach–Zehnder one was characterized and determined, and their relationship to the phase sensitivity was also examined. In other words, the multipartite entanglement of single-input, twin-input and N00N states were investigated in terms of the number of photon states entering Mach–Zehnder interferometer. In addition, the spatial modes of entanglement produced by Mach–Zehnder interferometer-as bipartite entangled states for Fock and squeezed states-was also determined. The amount of spatial modes of entanglement showed the interference of the quantum states. It was found that the more amount of the photon entanglement in twin-input and N00N states leads to the improvement of the phase sensitivity of the interferometer. In the squeezed state, by increasing the squeezed parameter of the two modes, their entanglement as well as their phase sensitivity was enhanced. & 2014 Published by Elsevier B.V.

Key words: Quantum entanglement Geometric measure Von Newumann entropy Phase sensitivity

1. Introduction Entanglement is a speciﬁc feature of quantum mechanics and the most relevant resource for quantum information processing [1–3]. Therefore, in the recent years, quantum entanglement has been known as a powerful source to extract information from a physical system. Entanglement measures are the most important part of quantum information theory. By deﬁnition, entanglement measures should quantify its amount in a given state. The numerous entanglement measures have been utilized for the purpose of describing different properties of the quantum state from among which von Newumann entropy, entanglement formation, and geometric measure of entanglement can be mentioned [4–9]. One of these properties is the phase sensitivity in optical interferometer. Interferometry is a fundamental component of sensing (e.g. measurement precision) and imaging techniques. Optical interferometry is usually described in the Mach–Zehnder conﬁguration. The phase sensitivity of the conventional Mach–Zehnder interferometer (MZI) is bounded by the shot noise limit δφ = 1/ n (where n is the total photon number in the interferometer). The non-classical light is used to enhance phase sensitivity because enhancing sensitivity to detect the phase in the optical system plays a highly important role in lithography and imaging. Also the n

Corresponding author. E-mail address: [email protected] (F. Nosraty).

so-called Heisenberg limited δφ = 1/n is achieved by using N00N states [10–24]. This study set out to distinguish and quantify the amounts of entanglement in MZI as a source of phase sensitivity enhancement. Two kinds of entanglement, the entanglement between spatial modes and photon entanglement were studied. Firstly, the entanglement between spatial modes (lower and upper arms) in MZI for various input states including Fock and squeezed states was investigated. In this case, the investigation was based on the von Newumann entropy. Secondly, the photon entanglement of the input Fock states fed into MZI was examined based on the geometric measure of the multipartite entangled states. Furthermore, the phase sensitivity of various input states which was described by optimal detection strategy was investigated. It was found that as the photon entanglement of the twin-input and the N00N states increases, the phase sensitivity of the interferometer improves as well. To the best of researchers' knowledge, the results presented here have not been previously reported. The present study is organized as follows: Section 1 is concerned with the introduction part. In Section 2, the multipartite entanglement of the single- and twin-input states is studied, and the amount of entanglement between spatial modes produced by MZI is determined. The phase sensitivity obtained by the aforementioned states and their relation to the entangled ones (similar procedure being held for Sections 3 and 4) is investigated. In Section 3, the bipartite and multipartite amounts of the Schrodinger cat state entanglement, more commonly called the N00N

http://dx.doi.org/10.1016/j.optcom.2014.10.072 0030-4018/& 2014 Published by Elsevier B.V.

Please cite this article as: M. Ali Jafarizaeh, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.10.072i

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M. Ali Jafarizaeh et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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state, are studied. In Section 4, the entanglement amount between spatial modes of squeezed light fed into MZI was investigated. Finally, the conclusion was presented in Section 5.

2. Single- and twin-input states In this section, the geometric measure of entanglement is used to obtain the amount of single-input and twin-input state entanglement. The geometric measure is a distance-like measure, in other words, geometric measure assesses the entanglement in terms of “remoteness” of given state from the set of separable states. Besides, the deﬁnition of the geometric measure can be viewed as an optimization problem in the sense of ﬁnding closest product state. The aforementioned remoteness is expressed by the maximal overlap of a given pure multipartite state with all pure product states [5–7]. The logarithm geometric measure of entanglement is described as follows:

E = min − 2 log2( <φ ψ > |).

(1)

|φ >∈ Hsep

The formula of |ψ >in = |na , nb> in which na shows the number of photons entering the upper arm of MZI (mode A) and nb indicates the number of photons going into its lower arm (mode B) describes the quantum states in MZI regarding the number of photons. By representing a Fock input state as a symmetrized combination of N constituent Spinors (Majorana representation):

⎛ N ⎞−1/2 |ψ >in = ⎜ ⎟ ⎝ na ⎠

∑

|0 > |0 > …|0 > |1 > |1 > …|1 > , 0 ≤ na ≤ N

Perm

na

nb

(2)

this state is known as a Dicke state [5] where N is the total number of photons. The closest product state of |ψ >in is known [25] to be

⎛ n |φ > = ⎜ a 0 > + ⎝ N

⎞⊗N nb 1>⎟ . N ⎠

na

(3)

nb ⎞

( )( ) N na

N nb

⎛N ⎞ ⎜ ⎟ ⎝ na ⎠

⎟ ⎟. ⎟ ⎟ ⎠

δφ ≥

1 sin φ 2j(j + 1) − 2m2

(4)

max

E = N − log2(N /N /2 ) in which N must be an even number is the maximum amount of the above entanglement and can be achieved when two modes contain the same number of photons. An upper bound of geometric measure of n symmetric qubit entanglement is Emax ≤ log2(N + 1). In this case, the single-input state of |ψ >in = |n, 0> is the product state containing uncorrelated photons. A two-mode N-photon state, in the Schwinger repre sentation, can be described by means of the angular momen tum operator in terms of two-mode operator such as jx = a†b + b†a , jy = − i(a†b − b†a) and jz = a†a − b†b in which a and a† are, respectively, annihilation and creation of photon operators in upper arm of MZI, and where b and b† are, respectively, annihilation and creation of photon operators in the lower arm. ^ ^ ^†^ ^ Also, UMZI = UB UφUB where U^φ = e−ijz φ is phase shift unitary transiπ ^ ^ formation and UB = e− 2 jx is 50–50 beam-splitter unitary transformation represents MZI according to the unitary su(1, 1) transformation. Using the Baker–Hausdorff lemma [26], MZI transforma^ ^ tion can be shown as UMZI = e−ijx φ which is originally studied by Yurke et al. [20].

(5)

n −n

n +n

where m = a 2 b is the photon number difference and j = a 2 b is the total number photons. If uncorrelated photons enter the MZI (single-input state, m = j ) so-called shot noise limit δφ ≥ 2j , for π/2 phase shift, is obtained. Also, the upper bound of phase sensitivity δφ ≥ 2j(j + 1) is obtained by the maximum entangled twine-input state which includes the same number of photons in each mode. Now the entanglement properties produced by MZI can be studied. It must be noted that MZI as unitary local operation does not change the intrinsic amount of entanglement between subsystem (photons) [8], while it can produce quantum correlation (as entanglement) between lower and upper arms because of the superposition of states. The von Newumann entropy is a measure of entanglement for the pure bipartite states. It is equal to zero for separable states and log2 N for maximal entangled N -dimensional states [8,9]. The von Newumann entropy of reduced single- and twin-output states is

E=−

2

(j) (φ ) ∑ (dmm ′ ) m′

Using Eqs. (1) and (2) and the closest product state mentioned above, the researchers arrived at the entanglement of the twineinput state, |na , nb>in

⎛ ⎜ E(|ψ >in ) = log2⎜ ⎜ ⎜ ⎝

Dirac [27] ﬁrst proposed the uncertainty relationship between photon number and phase ΔnΔφ ≥ 1. Consequently, the phase sensitivity or the error in the estimate phase can be deﬁned as δφ ≥ 1/2Δjz in the Schwinger representation. It should be noted that the phase sensitivity equation is equal to the Cramer–Rao bound, i.e. the highest sensitivity allowed by quantum mechanics [13,21,22]. The phase sensitivity can be obtained by MZI acting as a unitary transformation in the Heisenberg picture on the jz angular momentum operator:

(

2

)

(j) log2 dmm (φ ) . ′

(6)

(j) dmm

where (φ) is Winger's coefﬁcient [26]. ′ Fig. 1 shows the von Newumann entropy as a function of phase shifts (s ), and the photon number difference (m) between two modes produced by MZI. According to Fig. 1, it is not necessary to have π/2 phase shift to achieve maximum entanglement. The von Newumann entropy of the single-input state (uncorrelated photons), for each number of photon, as a function of the phase shift (s) is presented in Fig. 2. It can be clearly seen that the maximum amount of entanglement in single-input state happens when the phase shift is π/2 for each number of photon(j) . In Fig. 3, the entanglement amount of twin-input state which has the same number of photon in each mode is numerically investigated. As can be seen, the entanglement is not maximized when π/2 phase shift is applied to MIZ. It is why the odd-number states destructively interfere with each other and, do not exist in

Fig. 1. The measure of entanglement E (r) is plotted. The Fock state input |j, m> has a total number photon N = 40 ( j ¼20) and phase shift is s = sin2 φ /2.

Please cite this article as: M. Ali Jafarizaeh, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.10.072i

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3

as

⎧ α ≤ 1−α ⎪ E|ψ >n = − log2(1 − α), ⎨ . ⎪ E|ψ >n = − log2 α α > 1 − α ⎩

(9)

The maximum amount of entanglement E(|ψ >n ) = 1 is possible when α = 1/2. The geometric measure of entanglement of GHZ state is less than the Dicke state (twin-input state). It should be noted that, the maximal entangled state and its entanglement amount depend on the chosen entanglement measure [8]. Using similar procedure, the error in detecting phase as a function of α is Fig. 2. The measure of entanglement E (j, s) of the single-input state as a function of total number of photon and phase shift is = sin2 φ /2 s plotted.

(

)

δφ = 1/4N2(α(1 − α)) . According to this equation, the Heisenberg limit is achieved when α = 1/2. In bipartite case, the entanglement between modes A and B is

E|ψ > = − α log2 α − (1 − α)log2(1 − α).

(10)

When α = 1/2, the maximum value of the entropy (E|ψ > = 1) is achieved. Thus, the N00N state is the maximal entangled state.

4. Two modes squeezed state

Fig. 3. The measure of entanglement E (j, s) of twin-input state as a function of total photon number and phase shift s = sin2 φ /2 is plotted.

the output states. In addition, for each number of photon, the maximum amount of entanglement is changing.

The main property of the squeezed state in quantum optics is that the quantum ﬂuctuation in phase is reduced as long as the Heisenberg relation between quadratures ΔφΔn ≥ 1 is not violated. Therefore, the squeezed state will have less “noise” in one of their quadratures [29–32]. As already stated, the entanglement of the modes ‘A’ and ‘B’ – when each port of MZI fed by two single squee zed vacuum states – is to be studied. The output state obtained ^ ^ ^ is out > = U^MZI (φ)Sa(ξ)Sb(ξ) in>. Sa = exp (ξa2/2) − (ξ†(a†)2/2) and

(

)

^ Sb = exp (ξb2 /2) − (ξ†(b†)2/2) are single squeezed operators where

(

)

iθ

3. N00N state In the previous section, it has been shown that twin-input state with the same number of photon in each mode can be utilized to exhibit super sensitivity [28] in MZI. The simple beam splitter transformation, as one of the components of MZI, exerts inﬂuence on these states and converts them to low-N00N states (maximally for two photons in 50–50 beam splitter or π/2 phase shift in the MZI). Thus, high-N00N state previously was proposed to satisfy the Heisenberg limit. [10–15,24]. As previously stated, this study aimed to investigate the entanglement properties of the N00N states by means of the following equations which include the geometrical measures of photon entanglement and von Newumann entropy as the measure of entanglement between modes ‘A’ and ‘B’. The N00N state is described in the following way:

|ψ > =

α |N >A |0 >B + e

iNφ

1 − α |0 >A |N >B

(7)

where α is the probability amplitude which satisﬁes normalization conditions. The relative phase eiNφ is accumulated when each particle in mode B acquires a phase shift of eiφ . The Majorana representation of the N00N state is

|ψ >n = ( α |0 >⊗n +

1 − α |1>⊗n)

(8)

where it is known as GHZ state [5,6]. Disregarding the number of qubits, the geometric measure of the entanglement can be written

(ξ = re ) r is known as squeezed parameter. If initial states include the same number of photon in each mode, phase sensitivity can be found as

1

δφ = 2

cosh (2r)

(

n(n + 2) 2

) + sinh (2r)((n + 2) 2

2

. + n2)

(11)

As the squeezed parameter r increases, the phase sensitivity rises. The two-mode entangled state depends on the phase difference between the initial beams entering MZI. If the phase difference is Δ(φn − φn ) = π /2, as the following equation demona

a

strates, MZI can produce a two-mode squeezed state as ^ Sab(ξ) = exp(ξab − ξ†a†b†),

^ ^ ^ ^ ^ ^ UMZI (φ)Sa(ξ)Sb(ξ) = Sa(ξ1)Sb(ξ1)Sab(ξ2) ξ1 = ξ cos φ and ξ2 = ξ sin φ

(12)

For a π/2 phase shift, the entanglement of the output state is maximized because the single-mode squeezed operators do not appear on the right-hand side of Eq. (12) [33,34]. The squeezed state deﬁned in terms of orthonormal basis |k, μ> can be described under SU(1, 1) Lie group transformation [35,36]. The two-mode squeezed operator is described in terms of SU(1, 1) Lie group transformation and applied to the initial vacuum state: k ⎞ ⎛ |ζ > = exp(ζk−)exp⎜(1 − ζ 2 ) k z ⎟exp(ζk +)|0 > = ⎝ ⎠ † †

†

†

(13)

where k− = ab , k+ = a b and kz = (a a + bb /2) are SU(1, 1) Lie group operators. Thus, the squeezed state can be described in the

Please cite this article as: M. Ali Jafarizaeh, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.10.072i

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Reference

Fig. 4. A plot of the measure of entanglement E (k) of two-mode squeezed state as a function of photon number difference and squeezed parameter r = 0.3.

following way: k

|ζ > = (1 − ζ 2 )

⎛ Γ(2k + m) ⎞1/2 ⎜ ⎟ (ζ)m|k, k + m> Γ ! (2 k ) m ⎝ ⎠ m=0 ∞

∑

(14)

An arbitrary two-mode number state |na , nb> is equivalent to SU(1, 1) weight states |k, μ> with μ = k + m =

na + nb + 1 2

,

|na − nb | +1 2

and ζ = tanh r . As irrep label k is proportional to k= the photon number difference and irrep spaces are invariant under SU(1, 1), the photon number difference is conserved for twomode squeezed operators [35,36]. In particular, the same number of photon in the each mode (or vacuum initial state, k = 1/2) are in the squeezed state. The above Eq. (14) in terms of the number states is obtained as follows:

|ξ >2 =

1 cosh r

∞

∑ (−1)neiθ tanhn r|n, n > . n= 0

(15)

Because of the symmetry between the two modes, the average photon number in each mode is the same and can be easily shown to be

lytically calculated as

E = cosh2 r log2(cosh2 r) − sinh2 r log2(sinh2 r).

(16)

As the squeezed parameter r increases, the above entropy and phase sensitivity rises. Since it is not possible to determine entanglement amount analytically for the general case (every value of k), it is needed to determine the von Newumann entropy numerically. In Fig. 4, this entropy against the photon number difference k is plotted. As Fig. 4 shows, as the photon number difference k increases, the degree of entanglement rises.

5. Conclusion It was shown that achieving more amounts of the photon entanglement in each speciﬁc case of twin-input or N00N states leads to much more phase sensitivity of the interferometer. In addition, achieving more amount of the entanglement between spatial modes depends on the phase shift applied to the MZI and displayed interference of the quantum states. It was found that the increase in squeezed parameter of the two-mode squeezed states is followed by the increase in the entanglement amount between two modes ‘A’ and ‘B’ as well as the phase sensitivity. Finally, it was shown that for the squeezed states which are non-vacuum initial states, the entanglement amount between two modes is increased.

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