Quantum discord in photon added Glauber coherent states of GHZ -type
aa r X i v : . [ qu a n t - ph ] D ec Quantum discord in photon added Glauber coherent states of
GHZ -type
M. Daoud a,b,c , W. Kaydi d,e and H. El Hadfi d,e a Max Planck Institute for the Physics of Complex Systems, Dresden, Germany b Abdus Salam International Centre for Theoretical Physics, Miramare, Trieste, Italy c Department of Physics , Faculty of Sciences, University Ibnou Zohr, Agadir, Morocco d LPHE-Modeling and Simulation, Faculty of Sciences, University Mohammed V, Rabat, Morocco e Centre of Physics and Mathematics (CPM), University Mohammed V, Rabat, Morocco
Abstract
We investigate the influence of photon excitations on quantum correlations in tripartite Glaubercoherent states of Greenberger-Horne-Zeilinger type. The pairwise correlations are measured by meansof the entropy-based quantum discord. We also analyze the monogamy property of quantum discordin this class of tripartite states in terms of the strength of Glauber coherent states and the photonexcitation order. email: m − [email protected] email: [email protected] email: hanane.elhadfi@gmail.com Introduction
In the context of information processing and transmission, several theoretical and experimental re-sults confirm the advantages of quantum protocols compared to their classical counterparts (see forinstance [1, 2, 3]). Quantum technology exploiting the intriguing phenomena of quantum world, suchas entanglement, offers secure ways for communication [4, 5] and potentially powerful algorithms inquantum computation [6]. Originally, quantum information processing focused on discrete (finite-dimensional) entangled states like the polarizations of a photon or discrete levels of an atom. But, theextension from discrete to continuous variables has been also proven beneficial in coding and manip-ulating efficiently quantum information. Coherent states, which constitute the prototypical instanceof continuous-variables states, are expected to play a central role in this context. They are appealingfor their mathematical elegance (continuity and over-completion property) and closeness to classicalphysical states (minimization of Heisenberg uncertainty relation). Implementing a logical qubit en-coding by treating entangled coherent states as qubits in a two dimensional Hilbert space has beenshown a promising strategy in performing successfully various quantum tasks such as quantum tele-portation [7, 8], quantum computation [9, 10, 11], entanglement purification [12] and errors correction[13]. In view of these potential applications, a special attention was paid, during the last years, to theidentification, characterization and quantification of quantum correlations in bipartite coherent statessystems (see for instance the papers [14, 15, 16] and references therein). The bipartite treatment wasextended to superpositions of multimode coherent states [17, 18, 19, 20, 21] which exhibit multipartiteentanglement. One may quote for instance entanglement properties in GHZ (Greenberger-Horne-Zeilinger), W (Werner) states discussed in [22, 23] and entangled coherent state extensions of clusterqubits investigated in [24, 25, 26]. To quantify quantum correlations beyond entanglement in coherentstates systems, measures such as quantum discord [27, 28] and its geometric variant [29] were used.Explicit results were derived for quantum discord [30, 31, 32, 33, 34, 35, 36, 37] and geometric quantumdiscord [38, 39, 40, 41] for some special sets of coherent states.On the other hand, decoherence is a crucial process to understand the emergence of classicalityin quantum systems. It describes the inevitable degradation of quantum correlations due to exper-imental and environmental noise. Various decoherence models were investigated and in particularthe phenomenon of entanglement sudden death was considered in a number of distinct contexts (seefor instance [42] and reference therein). For optical qubits based on coherent states, the influence ofthe environment, is mainly due to energy loss or photon absorption. The photon loss or equivalentlyamplitude damping in a noisy environment can be modeled by assuming that some of field energyand information is lost after transmission through a beam splitter [36, 43]. Interestingly, it has beenshown that a beam spitting device with a coherent in the first input and a number state in the secondinput generates photon-added coherent states [44]. Henceforth, understanding the influence of photonexcitations might be useful to develop the adequate strategies in improving the performance of noise2eduction in quantum processing protocols involving coherent states. In this sense, some authorsconsidered the concurrence as measure of the entanglement in bipartite and tripartite photon addedcoherent states [23, 45].In this work, we derive the analytical expression of pairwise quantum discord in a three modessystem initially prepared in a tripartite Glauber coherent state of GHZ-type. In particular, we shallconsider the influence of photon excitation of a single mode on the dynamics of pairwise quantumcorrelations. Mathematically, this process is represented by the action of a suitable creation operatoron the state of the first subsystem. Another important issue in photon added GHZ-type coherentstates concerns the distribution of quantum discord between the different parts of the whole system.In fact, we study the shareability of quantum correlations which obeys a restrictive inequality termedin the literature as the monogamy property [46] (see also [47, 48, 49, 50, 51, 52]).This paper is organized as follows. In section 2, basic definitions and equations related to photonadded coherent states are presented. We also consider the quantum correlations as measured bythe entanglement of formation in quasi-Bell states. In particular, we introduce an encoding map topass from continuous variables (coherent states) to discrete variables (logical quantum bits). Alongthe same line of reasoning, this qubit encoding is extended, in section 3, to tripartite photon addedcoherent states of GHZ-type. The pairwise quantum discord quantifying the amount of quantumcorrelations existing in the system is analytically derived. In section 4, we study the monogamyproperty of quantum discord. Numerical illustrations of the monogamy inequality are presented insome special cases. Concluding remarks close this paper.
The basic objects in this work are the Glauber coherent states | α i and | − α i where α is a complexnumber which determines the coherent amplitude of the electromagnetic field. Mathematically, asingle-mode quantized radiation field is represented by the harmonic oscillator algebra spanned by thecreation a + and annihilation a − operators. The process of adding m photons to coherent states of type | α i and | − α i is usually represented by the action of the operator ( a + ) m ( m is a non negative integer)[53]. Several experimental as well theoretical studies were devoted to the generation and nonclassicalproperties of photon-added coherent states [54] (for a recent review see [55]). Explicitly, m successiveactions of creation operator a + on the Glauber coherent states | α i = e − | α | ∞ X n =0 α n √ n ! | n i , (1)3ead to the un-normalized states || α, m i = (cid:0) a + (cid:1) m | α i = e − | α | ∞ X n =0 α n n ! p ( n + m )! | n + m i . (2)The vectors | n i denote the Fock-Hilbert states of the harmonic oscillator. The normalized m -photonadded coherent states are defined by | α, m i = ( a + ) m | α i p h α | ( a − ) m ( a + ) m | α i , (3)where the quantity h α | ( a − ) m ( a + ) m | α i = m ! L m ( −| α | ) , (4)involves the Laguerre polynomial of order m defined by L m ( x ) = m X n =0 ( − n m ! x n ( n !) ( m − n )! . (5)Photon added coherent states interpolate between electromagnetic field coherent states (quasi-classicalstates) and Fock states | n i (purely quantum states). Furthermore, they exhibit non-classical featuressuch as squeezing, negativity of Wigner distribution and sub Poissonian statistics [55]. Their ex-perimental generation using parametric down conversion in a nonlinear crystal was reported in [54].Photon-coherent states | α, m i and | − α, m i , of the same amplitude and phases differing by π , are notorthogonal to each other. Indeed using the expression h− α | ( a − ) m ( a + ) m | α i = e − | α | m ! L m ( | α | ) , (6)it is simply verified that the overlap between the two states is h− α, m | α, m i = e − | α | L m ( | α | ) L m ( −| α | ) . (7)Considering the nonorthogonality property (7), the identification of photon added coherent states | α, m i and | − α, m i as basis of a logical qubit is only possible for | α | large ( | α | ≥ | + , m i and |− , m i defined by |± , m i = 1 p ± κ m e − | α | ( | α, m i ± | − α, m i ) (8)where κ m ≡ κ m ( | α | ) := L m ( | α | ) L m ( −| α | ) . (9)Clearly, for m = 0, one has κ = 1 and the logical qubits (8) reduce to |±i = 1 p ± e − | α | ( | α i ± | − α i ) , (10)4hich coincide with even and odd Glauber coherent states providing the qubit encoding schemeintroduced in [11]. This qubit encoding is important in dealing with quantum correlation in photonadded coherent states and to investigate the influence of the photon excitations processes. To illustratethis, we shall first consider the entanglement in quasi-Bell states which are very interesting in quantumoptics and serve as valuable resource for quantum teleportation and many other quantum computingoperations. The quasi-Bell states | B k ( α ) i = N k ( α ) (cid:2) | α i ⊗ | α i + e ikπ | − α i ⊗ | − α i (cid:3) , (11)with k = 0 (mod 2) (resp. k = 1 (mod 2)) stands for even (resp. odd) quasi-Bell states and thenormalization factor N k ( α ) is N − k ( α ) = 2 + 2 e − | α | cos kπ. (12)By repeated actions of the creation operator on the first mode, the resulting excited quasi-Bell statesare || B k ( α, m ) i = N k ( α ) (cid:20)(cid:2) ( a + ) m ⊗ I (cid:3) | α i ⊗ | α i + e ikπ (cid:2) ( a + ) m ⊗ I (cid:3) | − α i ⊗ | − α i (cid:21) (13)are un-normalized ( I stands for the unity operator). Using the norm of the vectors || B k ( α, m ) i givenby h B k ( α, m ) || B k ( α, m ) i = m ! L m ( −| α | ) + e − | α | L m ( | α | ) cos kπ e − | α | cos kπ , (14)we introduce the normalized photon-added quasi-Bell states as | B k ( α, m ) i = || B k ( α, m ) i p h B k ( α, m ) || B k ( α, m ) i . (15)They can be rewritten as | B k ( α, m ) i = N k ( α, m ) (cid:2) | m, α i ⊗ | α i + e ikπ | m, − α i ⊗ | − α i (cid:3) , (16)in terms of the normalized photon added coherent state (3). The normalization factor in (16) is N − k ( α, m ) = 2 + 2 κ m e − | α | cos kπ, (17)which reduces for m = 0 to (12) and the quasi-Bell states (11) are recovered. Using the qubit mapping (8) for the first mode and (10) for the second, the bipartite state (16) isconverted in the two qubit state | B k ( α, m ) i = N k ( α, m ) X i = ± X j = ± C ij | i, m i ⊗ | j i , (18)where the vectors | i, m i (resp. | j i ) are defined by (8) (resp. (10)) and the expansion coefficients aregiven by C ++ = c + m c + (1 + e ikπ ) , C − + = c + c − m (1 − e ikπ ) , C + − = c + m c − (1 − e ikπ ) , C −− = c − c − m (1 + e ikπ ) , c ± m = s ± κ m e − | α | c ± = s ± e − | α | . In a pure bipartite system, the quantum discord coincides with entanglement of formation (see forinstance [30, 31, 32]). Thus, to discuss the effect of the photon excitations of quasi-Bell states (16),we quantify the quantum correlations by means of the entanglement of formation. We recall that for ρ the density matrix for a pair of qubits 1 and 2 which may be pure or mixed, the entanglement offormation is defined by [56] E ( ρ ) = H ( 12 + 12 p − | C ( ρ ) | ) , (19)where H ( x ) = − x log x − (1 − x ) log (1 − x ) is the binary entropy function. The concurrence C ( ρ )is given by C ( ρ ) = max { λ − λ − λ − λ , } (20)for λ ≥ λ ≥ λ ≥ λ the square roots of the eigenvalues of the ”spin-flipped” density matrix ̺ ≡ ρ ( σ y ⊗ σ y ) ρ ⋆ ( σ y ⊗ σ y ) where the star stands for complex conjugation and σ y is the usualPauli matrix. In the state (16), it easy to check that the concurrence (20) gives C = 2 N k ( α, m ) | C ++ C −− − C + − C − + | , (21)which rewrites explicitly as C = p − e − | α | p − κ m e − | α | κ m e − | α | cos kπ (22)in terms of the coherent states amplitude | α | and the excitation order m . This result coincides withone obtained in [45] using a different qubit encoding. It follows that entanglement of formation is E = H "
12 + e − | α | (1 + κ m cos kπ )2 + 2 κ m e − | α | cos kπ . (23)For m = 0, one has C = 1 − e − | α | e − | α | cos kπ . (24)To illustrate the influence of the photon excitation on the quantum correlation between the modes ofthe quasi-Bell state (11), we report in the figures 1 and 2 the behavior of the entanglement of formation E (23) versus Glauber coherent states amplitude | α | and the overlap p = h α | − α i = e − | α | fordifferent values of m . We note that for | α | large ( | α | ≥ . m . Indeed, from equation (23), one gets E = 1for | α | −→ ∞ . Note that, in this limit, the Glauber coherent states | α i and |− α i tends to orthogonalityand an orthogonal basis can be constructed such that | i ≡ | α i and | i ≡ | − α i . Thus, in the strongregime | α | −→ ∞ , the quasi-Bell states (11) become maximally entangled | B k ( ∞ ) i = 1 √ (cid:2) | i ⊗ | i + e ikπ | i ⊗ | i (cid:3) . α (weak regime).For α −→
0, the symmetric ( k = 0 (mod 2))-quasi-Bell state (11) reduces to the separable state | i⊗| i and by adding m photons it becomes | m i ⊗ | i . No quantum correlation is created by the photonexcitation ( E = 0). This result can be also obtained from (23) for | α | −→
0. As depicted in thefigure 2, the situation is completely different for anti-symmetric quasi-Bell states ( k = 1 (mod 2)) (11).For α approaching zero, the entanglement of formation decreases as the photon excitation number m increases. For | α | −→
0, the Laguerre polynomial (5) can be approximated by L m ( | α | ) ≃ − m | α | and the quantity (9) writes κ m ( | α | ) ≃ − m | α | . (25)Reporting (25) in (23), one gets E (B (0 , m )) ≃ H (cid:18) m + 1 m + 2 (cid:19) . (26)It is interesting to note that in the situation when | α | −→
0, the anti-symmetric quasi-Bell states (11)reduce to the maximally entangled two qubit state of W-type | B (0) i = 1 √ (cid:2) | i ⊗ | i + | i ⊗ | i (cid:3) (27)where | i and | i denote the Fock number states. The action of the operator ( a + ) m on the state | B (0) i gives | B (0 , m ) i = 1 √ m + 2 (cid:2) | m i ⊗ | i + √ m + 1 | m + 1 i ⊗ | i (cid:3) . In this case, the concurrence is C (B (0 , m )) = 2 √ m + 1 m + 2 , (28)which agrees with the result (26). Clearly, adding photons to maximally entangled states of W-type(27) diminishes the amount of pairwise quantum correlations. For the intermediate regime, corre-sponding to | α | ranging between 0 and 1.5, the entanglement of formation increases as the Glaubercoherent state amplitude α increases. We note that adding photon process induces a quick activationof the creation of quantum correlation for the symmetric quasi-Bell states (figure 1). Similarly, theresults presented in figure 2 show that increasing the amplitude of anti-symmetric quasi-Bell statestends to compensate the quantum correlation loss due to photon excitations in states of W type. The analysis and results derived in the previous section are useful in investigating pairwise quantumcorrelations in tripartite states involving Glauber coherent states. In this respect, we consider thequasi-GHZ coherent states defined by | GHZ k ( α ) i = C k ( α )( | α, α, α i + e ikπ | − α, − α, − α i ) . (29)7 igure 1. The entanglement of formation E versus | α | and p = e − | α | for k = 0 and different valuesof photon excitation number m . Figure 2.
The entanglement of formation E versus | α | and p = e − | α | for k = 1 and different valuesof photon excitation number m . where the normalization constant C k is given by C − k ( α ) = 2 + 2 e − | α | cos kπ. (30)The excitation of the first mode by adding m photon leads to the tripartite state || GHZ k ( α, m ) i = (( a + ) m ⊗ I ⊗ I ) | GHZ k ( α ) i , (31)from which we introduce the normalized photon added quasi-GHZ coherent states as | GHZ k ( α, m ) i = || GHZ k ( α, m ) i p h GHZ k ( α, m ) || GHZ k ( α, m ) i . (32)Using the expressions (4) and (6), the vector (32) rewrites as | GHZ k ( α, m ) i = C k ( α, m )( | m, α i ⊗ | α i ⊗ | α i , + e ikπ | m, − α i ⊗ | − α i ⊗ | − α i ) . (33)where the normalization factor is C − k ( α, m ) = 2 + 2 κ m e − | α | cos kπ. (34)For m = 0, the state | GHZ k ( α, m ) i (33) reduces to | GHZ k ( α ) i (29). It is also important to note thatfor | α | large, the overlap between Glauber coherent states | α i and | − α i approaches zero and then8hey are quasi-orthogonal. In this case, the state | GHZ k ( α ) i (29) reduces to the usual GHZ threequbit state | GHZ k ( ∞ ) i = 1 √ | i ⊗ | i ⊗ | i + e ikπ | i ⊗ | i ⊗ | i ) , (35)where | i ≡ | α i and | i ≡ | − α i .In investigating the pairwise quantum discord in a tripartite system 1 − − | GHZ k ( α, m ) i , one needs the reduced density matrices describing the two qubit subsystems 1 −
2, 2 − −
3. Since only the first mode is affected by the photon excitations, it is simply seen that thereduced density matrices ρ = Tr ρ and ρ = Tr ρ are identical. The pure three mode densitymatrix ρ is given ρ = | GHZ k ( α, m ) ih GHZ k ( α, m ) | . (36)After some algebra, the reduced density matrices ρ and ρ can be written as ρ = ρ = C k ( α, m ) N k ( α, m ) "(cid:18) e − | α | (cid:19) | B k ( α, m ) ih B k ( α, m ) | + (cid:18) − e − | α | (cid:19) Z | B k ( α, m ) ih B k ( α, m ) | Z (37)in terms of photon added quasi-Bell states (16). The operator Z is the third Pauli generator definedby Z | B k ( α, m ) i = N k ( α, m )[ | m, α ) ⊗ | α i − e ikπ | m, − α ) ⊗ | − α i ] . Similarly, by tracing out the first mode, the reduced matrix density ρ takes the form ρ = C k ( α, m ) N k ( α, "(cid:18) κ m e − | α | (cid:19) | B k ( α, ih B k ( α, | + (cid:18) − κ m e − | α | (cid:19) Z | B k ( α, ih B k ( α, | Z . (38)To derive the pairwise correlation between the components of the subsystems 1 −
2, 2 − − | m, ± α i = s κ m e − | α | | i ± s − κ m e − | α | | i . (39)This coincides with the encoding scheme (8) introduced in the previous section to study the entangle-ment in quasi-Bell states. For the second and third modes, we consider the qubits defined by | ± α i = s e − | α | | i i ± s − e − | α | | i i , i = 2 , . (40)Substituting (39) and (40) in (37) (resp. (38)), one can express the density matrix ρ (resp. ρ ) inthe two qubit basis {| i ⊗ | i , | i ⊗ | i , | i ⊗ | i , | i ⊗ | i } (resp. {| i ⊗ | i , | i ⊗ | i , | i ⊗| i , | i ⊗ | i } ). The resulting density matrices have non-vanishing entries only along the diagonaland the anti-diagonal. 9 .2 Bipartite measures of quantum discord The state | GHZ k ( α, m ) i (33) has rank two reduced density matrices (37) and (38). For these two qubitstates, the Koashi-Winter relation which provides the connection between the quantum discord andthe entanglement of formation, can be exploited to obtain the relevant pairwise quantum correlations.It is important to note that for two qubit states with rank larger than two, the derivation of quantumdiscord involves optimization procedures that are in general complicated to achieve analytically.The total correlation in the subsystem 1 − I = S + S − S , (41)where S is the von Neumann entropy of the quantum state ρ (37) ( S ( ρ ) = − Tr ρ ln ρ ) and S (resp. S ) is the entropy of the reduced state ρ = Tr ( ρ ) (resp. ρ = Tr ( ρ )) of the mode 1( resp. 2). Themutual information I contains both quantum and classical correlations. The classical correlations C can be determined by a local measurement optimization procedure. To remove the measurementdependence, a maximization over all possible measurements is performed and the classical correlationwrites C = S − e S min , (42)where e S min denotes the minimal value of the conditional entropy [57, 58] (for more details, see therecent review [59]). Thus, the quantum discord, defined as the difference between total correlation I and classical correlation C [57, 58], writes D = I − C = S + e S min − S . (43)The main difficulty in deriving the analytical expression of bipartite quantum discord (43), in arbitrarymixed state, arises in the minimization process of conditional entropy. This explains why the explicitexpressions of quantum discord were obtained only for few exceptional two-qubit quantum states,especially ones of rank two. One may quote for instance the results obtained in [31, 32] (see also[36, 37, 41]). Since the density matrix ρ (37) is of rank two, the derivation of the analytical expressionof e S min in equation (42), can be performed by purifying the density matrix ρ and making use ofKoashi-Winter relation [60] (see also [33]). This relation establishes the connection between theclassical correlation of a bipartite state ρ and the entanglement of formation E of its complement ρ in the pure state ρ (36). The minimal value of the conditional entropy coincides with theentanglement of formation of ρ [60]: e S min = E . (44)The Koaschi-Winter relation and the purification procedure provide us with a computable expressionof quantum discord in the bipartite state ρ D = S − S + E (45)10hen the measurement is performed on the subsystem 1. The von Neumann entropy of the reduceddensity ρ = Tr ρ is S = H (cid:18)
12 (1 + κ m e − | α | )(1 + e − | α | cos kπ )1 + κ m e − | α | cos kπ (cid:19) , (46)and the entropy of the bipartite density ρ is explicitly given by S = H (cid:18)
12 (1 + κ m e − | α | cos kπ )(1 + e − | α | )1 + κ m e − | α | cos kπ (cid:19) . (47)It is important to note that the entanglement of formation measuring the entanglement of the subsys-tem 2 with the ancillary qubit, required in the purification process to minimize the conditional entropy,is exactly the entanglement of formation measuring the degree of intricacy between the optical modes2 and 3. Using the definition of Wootters concurrence (20), one gets C = κ m e − | α | (1 − e − | α | ) κ m e − | α | cos kπ (48)and subsequently the corresponding entanglement of formation writes E = H (cid:18)
12 + 12 s − κ m e − | α | (1 − e − | α | ) (1 + κ m e − | α | cos kπ ) (cid:19) . (49)Reporting (46), (47) and (49) in (45), the quantum discord in the state ρ is explicitly given by D = H (cid:18)
12 (1 + κ m e − | α | )(1 + e − | α | cos kπ )1 + κ m e − | α | cos kπ (cid:19) (50) − H (cid:18)
12 (1 + κ m e − | α | cos kπ )(1 + e − | α | )1 + κ m e − | α | cos kπ (cid:19) + H (cid:18)
12 + 12 s − κ m e − | α | (1 − e − | α | )(1 − e − | α | )(1 + κ m e − | α | cos kπ ) (cid:19) , The pairwise quantum discord existing in the mixed states ρ (38) can be computed along the sameprocedure. As result, when the measurement is performed on the subsystem 2, the quantum discordis D = S − S + E . (51)The von Neumann entropy of the reduced density ρ = Tr ρ is S = H (cid:18)
12 (1 + e − | α | )(1 + κ m e − | α | cos kπ )1 + κ m e − | α | cos kπ (cid:19) , (52)and the entropy of the bipartite density ρ is S = H (cid:18)
12 (1 + e − | α | cos kπ )(1 + κ m e − | α | )1 + κ m e − | α | cos kπ (cid:19) . (53)In purifying the state ρ to derive the minimal amount of conditional entropy, it is simple to show herealso that the entanglement of formation measuring the entanglement of the mode 3 with an ancillary11ubit, is exactly the entanglement of formation measuring the degree of intricacy between the opticalmodes 1 and 3. From (20), the concurrence between the modes 1 and 3 takes the following form C = e − | α | p (1 − κ m e − | α | )(1 − e − | α | )1 + κ m e − | α | cos kπ , (54)from which one gets E = H (cid:18)
12 + 12 s − e − | α | (1 − κ m e − | α | )(1 − e − | α | )(1 + κ m e − | α | cos kπ ) (cid:19) . (55)Finally, the expression of quantum discord in the state ρ is D = H (cid:18)
12 (1 + e − | α | )(1 + κ m e − | α | cos kπ )1 + κ m e − | α | cos kπ (cid:19) (56) − H (cid:18)
12 (1 + e − | α | cos kπ )(1 + κ m e − | α | )1 + κ m e − | α | cos kπ (cid:19) + H (cid:18)
12 + 12 s − e − | α | (1 − κ m e − | α | )(1 − e − | α | )(1 + κ m e − | α | cos kπ ) (cid:19) . In order to analyze the influence of the photon excitation number m on the bipartite quantum discord D (50) and D (56), we first give the figures 3 and 4 representing respectively D and D forsymmetric states ( k = 0). Figure 3.
The quantum discord D versus | α | and p = e − | α | for k = 0 and different values ofphoton excitation number m . We can see from figure 3 that the quantum discord D ( | α | ) between the optical modes 1 and 2, inthe symmetric case ( k = 0), exhibits peaks which move to the left-hand when the photon excitationnumber m increases. It must be noticed that the height of peaks, D max12 ( m ), increases with increasingthe number of added photons. We observe also that on the left-hand side of the peak (weak regime),the quantum discord D rises rapidly with increasing the optical strength | α | . This indicates thatthe photon excitation of Glauber coherent states, in the weak regime, induces an activation of thecorrelations between the modes 1 and 2. In the strong regime ( | α | large), the quantum discord tends12 igure 4. The quantum discord D versus | α | and p = e − | α | for k = 0 and different values ofphoton excitation number m . to zero quickly as m increases. The behavior of D ( | α | ) in symmetric quasi-GHZ coherent states( k = 0), depicted in figure 4, shows that the maximal amount of quantum discord D max23 ( m ) is obtainedfor m = 0 and | α | ∼ .
5. In contrast with D , D max23 ( m ) decreases as m increases (figure 5). Thus, theincrease of the quantum discord D is accompanied by a decrease of D when the photon excitationnumber m increases. Remark also that for symmetric states ( k = 0), the photon excitation does notaffect the amount of pairwise quantum correlations D and D in the limiting situations | α | −→ | α | −→ ∞ . This is no longer valid for antisymmetric states k = 1 especially for | α | approachingzero (see figures 5 and 6). Indeed, the quantum discord D and D decreases for α −→ Figure 5.
The quantum discord D versus | α | and p = e − | α | for k = 1 and different values ofphoton excitation number m . The behavior of quantum discord D and D in anti-symmetric states ( k = 1), when | α | ap-proaches zero, can be confirmed analytically. In fact, using (25), one shows that for | α | −→ D (50) and D (56) are given by D = D = H (cid:18) m + 3 (cid:19) − H (cid:18) m + 2 m + 3 (cid:19) + H (cid:18)
12 + 12 p ( m + 1)( m + 5) m + 3 (cid:19) , (57)and D = H (cid:18) m + 2 m + 3 (cid:19) − H (cid:18) m + 3 (cid:19) + H (cid:18)
12 + 12 √ m + 2 m + 5 m + 3 (cid:19) . (58)13 igure 6. The quantum discord D versus | α | and p = e − | α | for k = 1 and different values ofphoton excitation number m . It is interesting to note that the antisymmetric photon added GHZ-type coherent states | GHZ ( α, m ) i (33) reduces, for | α | −→
0, to | GHZ (0 , m ) i = 1 √ m + 3 ( √ m + 1 | m + 1 , , i + | m, , i + | m, , i ) (59)which coincides with the usual three qubit W states for m = 0 [61]. The state (59) is expressed inthe Fock-Hilbert basis. Hence, according to the results plotted in figures 5 and 6, one concludes thatphoton excitations diminish the pairwise quantum correlations existing in three qubit states of Wtype. GHZ -type
Having investigated the pairwise quantum discord in the state | GHZ k ( α, m ) i (33), we shall considerthe distribution of quantum correlations among its three optical modes. It is well established that ina multi-qubit quantum system, the monogamy property imposes restrictive constraints for the qubitsto share freely quantum correlations. Now, it is well established that, unlike the square of concurrenceand the squashed entanglement, the quantum discord does not follow the monogamy relation. In thissection, we investigate the influence of photon excitation number m on monogamy relation of quantumentropy-based quantum discord in tripartite state of type | GHZ k ( α, m ) i (33).The entropy-based quantum discord, in the three modes states GHZ k ( α, m ), is monogamous if, andonly if, the quantum discord deficit defined by∆ = ∆ ( m, | α | ) = D | − D − D , (60)is positive. This condition reflects that the monogamy property is satisfied when the quantum discord D | between the first mode and the modes 2-3 (viewed as a single subsystem) exceeds the sum ofpairwise quantum discord D and D . We recall that the concept of monogamy was originally in-troduced by Coffman, Kundu and Wootters in 2001 [46] in analyzing the distribution of entanglement14n a tripartite qubit system. After, several works considered the monogamy of other quantum correla-tions quantifiers. In this section, we shall determine the conditions under which the quantum discordsatisfies the monogamy property and a special attention will be devoted to the influence of photonexcitations of Glauber coherent states. For this end, one has to determine the pairwise quantumdiscord D | in the pure state | GHZ k ( α, m ) i (33). In pure states the quantum discord coincides withthe entanglement of formation. Hence, to compute the entanglement between qubit (1) and the jointqubits (23), we introduce the orthogonal basis {| i , | i } defined by | i = | α, m i + | − α, m i p κ m e − | α | ) , | i = | α, m i − | − α, m i p − κ m e −| α | ) , (61)for the first subsystem. For the modes (23), grouped into a single subsystem, we introduce theorthogonal basis {| i , | i } given by | i = | α, α i + | − α, − α i p e − | α | ) | i = | α, α i − | − α, − α i p − e − | α | ) . (62)Inserting (61) and (62) in | GHZ k ( α, m ) i , we get the expression of the pure state | GHZ k ( α, m ) i in thebasis {| i ⊗ | i , | i ⊗ | i , | i ⊗ | i , | i ⊗ | i } . Explicitly, it is given by | GHZ k ( α, m ) i = X α =0 , X β =0 , C α,β | α i ⊗ | β i (63)where the coefficients C α,β are C , = C k ( α, m )(1 + e ikπ ) c +1 c +23 , C , = C k ( α, m )(1 − e ikπ ) c +1 c − ,C , = C k ( α, m )(1 − e ikπ ) c +23 c − , C , = C k ( α, m )(1 + e ikπ ) c − c − , in terms of the quantities c ± = s ± κ m e − | α | , c ± = s ± e − | α | . The concurrence between the two logical qubits 1 and 23 is given by C | = p (1 − κ m e − | α | )(1 − e − | α | )1 + κ m e − | α | cos kπ , (64)from which we obtain D | = E | = H (cid:18)
12 + 12 κ m e − | α | + e − | α | cos kπ κ m e − | α | cos kπ (cid:19) . (65)Inserting D | (65) and D = D (45) in (60), one gets the explicit expression of the quantumdiscord deficit ∆ . The corresponding behavior as function of | α | (and p = e − | α | ) for variousvalues of photon excitation order m is displayed in the figures 7 and 8.It can be inferred that the photon excitation of symmetric quasi GHZ-coherent states ( k = 0) doesnot affect the monogamy property of quantum discord. The quantum discord deficit ∆ is always15 igure 7. The quantum discord deficit ∆ versus | α | and p = e − | α | for k = 0 and different valuesof photon excitation number m . Figure 8.
The quantum discord deficit ∆ versus | α | and p = e − | α | for k = 1 and different valuesof photon excitation number m . positive. The situation is slightly different for antisymmetric quasi GHZ-coherent states ( k = 1).In absence of photon excitation ( m = 0), the quantum discord violates the monogamy inequality for | α | < . p > . m ≥ is almost identical in particular for highvalues of | α | ( | α | ≥ . | α | large the photon added three mode coherent states | GHZ k ( α, m ) i tend to the usual Greenberger-Horne-Zeilinger three qubit states (35). This indicatesthat, in this case, photon addition process does not affect the distribution of the quantum correlations.Another special limiting situation concerns Glauber states with amplitude approaching zero. Forsymmetric states | GHZ ( α = 0 , m ), it is easy to verify from the equations (45) and (65) that ∆ = 0for any photon excitation order m . For the antisymmetric states | GHZ ( α = 0 , m ), which coincidewith three qubit states of W-type (59), the monogamy discord deficit increases as m increases (seefigure 8). This result can be recovered analytically. Indeed, for k = 1 and | α | −→
0, one shows that D | −→ H (cid:18) m + 3 (cid:19) , (66)16nd using the result (58) one has∆ −→ H (cid:18) m + 2 m + 3 (cid:19) − H (cid:18)
12 + 12 p ( m + 1)( m + 5) m + 3 (cid:19) − H (cid:18) m + 3 (cid:19) . (67)The behavior ∆ near the point α = 0, plotted in the figure 8, reflects that the photon additiontends to increase the quantum deficit ∆ and subsequently to reduce the violation of monogamyrelation in states of W-type. In multipartite quantum systems, the monogamy is probably one of the most important relation whichimposes severe restriction on the structure of entanglement distributed among many parties. In thiscontext, the main interest of this paper was the monogamy property of quantum discord in three qubitsystems where the information is encoded in even and odd Glauber coherent states. In particular, weinvestigated the influence of photon excitations on the shareability of quantum discord between thethree optical modes of a quantum of GHZ-type. We derived the quantum discord deficit by evaluat-ing analytically the pairwise correlations in terms of the photon excitation number and the opticalstrength of Glauber coherent states. The symmetric quasi-GHZ coherent states follow the monogamyproperty for any photon excitation order. We have also shown that the photon excitation of antisym-metric quasi-GHZ coherent states reduces the violation of the monogamy property especially in statesinvolving Glauber coherent states with small amplitudes.Finally, the investigation of the influence of photon excitations on the monogamy of quantum corre-lations in the states of GHZ-type using geometric based quantifiers such as Hilbert-Schmidt norm ortrace distance would be interesting. On the other hand, another significant issue which deserves to beexamined concerns the dynamics of quantum discord under the effect of subtracting photons on thepairwise correlations in multipartite coherent states.
References [1] M.A. Nielsen and I. L. Chuang,
Quantum Computation and Quantum Information , CambridgeUniversity Press, Cambridge, U.K., 2000.[2] N.D. Mermin,
Quantum Computer Science: An Introduction , Cambridge University Press, U.K.,2007.[3] Mark M. Wilde,
Quantum Information Theory , Cambridge Univ Press, U.K., 2013.[4] C. H. Bennett and G. Brassard.
Quantum cryptography: Public key distribution and coin tossing .In Proc. IEEE Int. Conf. on Computers, Systems, and Signal Proc essing,p. 175., (1984)175] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres and W. K. Wooters, Phys. Rev.Lett. (1993) 1895.[6] P.W. Shor, Algorithms for quantum computation: discrete logarithms and factoring . In Proc.35th Annual Symp. on Foundations of Computer Science. IEEE Press, (1994).[7] S.J. van Enk and O. Hirota, Phys. Rev. A (2001) 022313 .[8] H. Jeong, M.S. Kim and J. Lee, Phys. Rev. A (2001) 052308 .[9] H. Jeong and M.S. Kim, Phys. Rev. A (2002) 042305 .[10] T.C. Ralph, W.J. Munro, and G.J. Milburn, Proceedings of SPIE (2002) 1;quant-ph/0110115.[11] T.C. Ralph, A. Gilchrist, J. Milburn, W.J. Munro and S. Glancy, Phys. Rev. A (2003) 042319.[12] H. Jeong and M.S. Kim, Quant. Inf. Comput. (2002) 208.[13] P.T. Cochrane, G.J. Milburn and W.J. Munro, Phys. Rev. A (1999) 2631; S. Glancy, H.Vasconcelos and T.C. Ralph, quant-ph/0311093.[14] B.C Sanders, J. Phys. A: Math. Theor. (2012) 244002.[15] B.C. Sanders, Phys. Rev. A (1992) 2966.[16] X. Wang, B.C. Sanders and S.H. Pan, J. Phys. A (2000) 7451.[17] I. Jex, P. T¨orm¨a and S. Stenholm, J. Mod. Opt. (1995) 1377.[18] S.-B. Zheng, Quant. Semiclass. Opt. B: J. European Opt. Soc. B (1998) 691.[19] X. Wang and B.C. Sanders, Phys. Rev. A (2001) 012303.[20] M. Daoud, A. Jellal, E.B. Choubabi and E.H. El Kinani, J. Phys. A: Math. Theor. (2011)325301.[21] M. Daoud and E.B. Choubabi, Int. J. Quant. Inf. (2012) 1250009.[22] H. Jeong and N.B. An, Phys. Rev. A (2006) 022104 .[23] H.-M. Li, H.-C. Yuan and H.-Y. Fan, Int. J. Theor. Phys. (2009) 2849.[24] P.P. Munhoz, F.L. Semi˜ao and Vidiello, Phys. Lett. A (2008) 3580.[25] W.-F. Wang, X.-Y. Sun and X.-B. Luo, Chin. Phys. Lett. (2008) 839.[26] E.M. Becerra-Castro, W.B. Cardoso, A.T. Avelar and B. Baseia, J. Phys. B: At. Mol. Opt. Phys. (2008) 085505. 1827] H. Ollivier and W.H. Zurek, Phys. Rev. Lett. (2001) 017901.[28] L. Henderson and V. Vedral, J. Phys. A (2001) 6899; V. Vedral, Phys. Rev. Lett. (2003)050401; J. Maziero, L. C. Cel´eri, R.M. Serra and V. Vedral, Phys. Rev A (2009) 044102.[29] B. Dakic, V. Vedral and C. Brukner, Phys. Rev. Lett. (2010) 190502.[30] S. Luo, Phys. Rev. A (2008) 042303; Phys. Rev. A (2008) 022301.[31] M. Ali, A.R.P. Rau and G. Alber, Phys. Rev. A (2010) 042105.[32] G. Adesso and A. Datta, Phys. Rev. Lett. (2010) 030501.[33] M. Shi, W. Yang, F. Jiang and J. Du, J. Phys. A: Math. Theor. (2011) 415304.[34] D. Girolami and G. Adesso, Phys. Rev. A (2011) 052108.[35] M. Shi, F. Jiang, C. Sun and J. Du, New J. Phys. (2011) 073016.[36] M. Daoud and R. Ahl Laamara, J. Phys. A: Math. Theor. (2012) 325302.[37] M. Daoud and R. Ahl Laamara, Int. J. of Quant. Inf. (2012) 1250060.[38] G. Adesso and A. Datta, Phys. Rev. Lett. (2010) 030501; G. Adesso and D. Girolami, Int.J. Quant. Info. (2011) 1773.[39] P. Giorda and M.G.A. Paris, Phys. Rev. Lett. (2010) 020503.[40] X. Yin, Z. Xi, X-M Lu, Z. Sun and X. Wang, J. Phys. B: At. Mol. Opt. Phys. (2011) 245502.[41] M. Daoud and R. Ahl Laamara, Phys. Lett. A (2012) 2361.[42] T. Yu and J.H. Eberly, Quant. Inf. Comput. (2007) 459.[43] R. Wickert, N.K. Bernardes and P. van Loock, Phys. Rev. A (2010) 062344.[44] M. Dakna, L. Kn¨oll and D-G Welsch, Opt. Comm., (1998) 309.[45] L. Xu and L-M. Kuang, J. Phys. A: Math. Gen. (2006) L191.[46] V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev. A (2000) 052306.[47] G.L. Giorgi, Phys. Rev. A (2011) 054301.[48] R. Prabhu, A. K. Pati, A.S. De and U. Sen, Phys. Rev. A (2012) 052337.[49] Sudha, A.R. Usha Devi and A.K. Rajagopal, Phys. Rev. A (2012) 012103.[50] M. Allegra, P. Giorda and A. Montorsi, Phys. Rev. B (2011) 245133.1951] X.-J. Ren and H. Fan, Quant. Inf. Comp. Vol. (2013) 0469.[52] A. Streltsov, G. Adesso, M. Piani and D. Bruss, Phys. Rev. Lett. (2012) 050503.[53] G.S. Agarwal and K. Tara, Phys. Rev. A (1991) 492.[54] A. Zavatta, S. Viciani and M. Bellini, Science (2004) 660.[55] M.S. Kim, J. Phys. B, At. Mol. Opt. Phys. (2008) 133001.[56] W.K. Wootters, Phys. Rev. Lett. (1998) 2245.[57] L. Henderson and V. Vedral, J. Phys. A. Math. Gen. (2001) 6899.[58] H. Ollivier and W. Zurek, Phys. Rev. Lett. (2002) 017901 .[59] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral, Rev. Mod. Phys. (2012) 1655.[60] M. Koachi and A. Winter, Phys. Rev. A (2004) 022309.[61] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A62