Quantum coherence of Gaussian states
Daniela Buono, Gaetano Nocerino, Giuseppe Petrillo, Gianpaolo Torre, Giuseppe Zonzo, Fabrizio Illuminati
QQuantum coherence of Gaussian states
D. Buono , G. Nocerino , G. Petrillo , G. Torre , G. Zonzo and F. Illuminati , , ∗ Dipartimento di Ingegneria Industriale, Universit`a degli Studi di Salerno,Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy Dipartimento di Fisica, Universit`a degli Studi di Salerno,Via Giovanni Paolo II, I-84084 Fisciano (SA), Italy Consiglio Nazionale delle Ricerche, Istituto di Nanotecnologia, Rome Unit, I-00195 Roma, Italy and INFN, Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy (Dated: October 1, 2018)We introduce a geometric quantification of quantum coherence in single-mode Gaussian states andwe investigate the behavior of distance measures as functions of different physical parameters. In thecase of squeezed thermal states, we observe that re-quantization yields an effect of noise-enhancedquantum coherence for increasing thermal photon number.
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I. INTRODUCTION
The superposition principle is the essential propertythat discriminates the intrinsically linear and coherentquantum mechanics from the essentially nonlinear andchaotic classical mechanics. Quantum coherence is thekey ingredient in all quantum phenomena, from quan-tum optics and quantum information [1–4] to condensedmatter physics and quantum thermodynamics [5–10].In recent years, the scientific community has starteda significant effort towards the rigorous exploration ofquantum coherence, including its qualification, quantifi-cation, and operational significance, along conceptualand mathematical lines analogous to the ones previouslyfollowed for the rigorous qualification and quantificationof quantum entanglement and quantum correlations.So far, the investigation have been limited to states offinite-dimensional quantum systems. In this respect, fol-lowing the seminal identification of entropic and geomet-ric quantifiers and the axiomatic properties that mustbe satisfied by any proper measure of quantum coher-ence [15], important progress has included the classifica-tion of single-qubit incoherent quantum operations [14],the characterization of quantum coherence in two-qubitBell-diagonal states [16], and the first results on the op-erational significance of quantum coherence for quantumtechnology protocols [17]. In particular, it has recentlybeen shown that quantum coherence can be exploited toactivate quantum entanglement and other useful quan-tum correlations [18].On the other hand, continuous variable (CV) states ofinfinite-dimensional systems are of fundamental impor-tance both from theoretical and experimental point ofview; in particular a lot of work has been dedicated toGaussian states and Gaussian channels, as they are boththeoretically easy to manage and experimentally easy toproduce. Moreover, although non-Gaussian CV states ∗ Corresponding author: fi[email protected] have been proved quite useful in particular cases, such asCV quantum teleportation [19–21], Gaussian states con-stitute the foundation of every quantum processing tasksin the CV setting. As a consequence, it is important togeneralize the quantification of quantum coherence andthe classification of quantum incoherent operations to thecase of Gaussian states.In this work we follow the geometric approach pio-neered by Baumgratz, Cramer, and Plenio [15], and wecompute different distance measures in detail for themain classes of single-mode Gaussian states.The article is organized as follows. In Sec. II webriefly review the basic formalism of single-mode Gaus-sian states. In Sec. III we classify the set of incoherentsingle-mode Gaussian states, namely those states thathave diagonal density matrix in the Fock basis, show-ing that they correspond to purely thermal states withzero displacement and zero squeezing. In Sec. IV we de-fine quantum coherence as the minimal distance fromthe set of thermal states and we introduce two such geo-metric measures, respectively in terms of the Bures andHellinger metric. In Sec. V we explicitly compute thegiven measures for the most significant classes of single-mode Gaussian states as functions of different physicalparameters such as displacement, squeezing, and ther-mal noise. In particular, we show that squeezed ther-mal states feature a remarkable phenomenon of noise-enhanced quantum coherence with increasing number ofthermal photons at fixed squeezing. In Sec. VI, we drawour conclusions and discuss future research on the exten-sion to multimode Gaussian states and the applicationto CV quantum technologies. In the App. A we includevarious mathematical and calculational details leading tothe results presented in the main text.
II. SINGLE-MODE GAUSSIAN STATES
In this section we briefly review the mathematical for-malism aiming at describing one-mode Gaussian states.Every single-mode Gaussian state can be expressed in the a r X i v : . [ qu a n t - ph ] S e p following form [22–24] (cid:37) G = D ( β ) S ( ξ ) ν th S † ( ξ ) D † ( β ) , (1)where D ( β ) = exp { βa † − β ∗ a } is the displacement oper-ator with complex amplitude β , S ( ξ ) = exp { ξ ( a † ) − ξ ∗ a } is the squeezing operator with squeezing param-eter ξ = re iψ and ν th in the one-mode thermal state: ν th = 11 + N th ∞ (cid:88) n =0 (cid:18) N th N th (cid:19) n | n (cid:105) (cid:104) n | . (2)Due to the Gaussian form of the Wigner function rep-resenting these states in the phase space, they are charac-terized uniquely by the first moments (the displacementvector): R ≡ ( (cid:104) X (cid:105) , (cid:104) P (cid:105) ) = √ Re [ β ] , Im [ β ]) (3)and second moments (the covariance matrix): σ G = (cid:20) a cc b (cid:21) . (4)with a = 1 + 2 N th r ) + cos ψ sinh(2 r )) ,b = 1 + 2 N th r ) − cos ψ sinh(2 r )) , (5) c = 1 + 2 N th ψ sinh(2 r )) . This parametrization will be useful for the study ofquantum coherence for the classes of Gaussian state con-sidered in Sec. V.
III. INCOHERENT SINGLE-MODE GAUSSIANSTATES
In this section we identify the set of one-mode Gaussianincoherent states. Indeed, we obtain the following result:
Theorem 1.
A single mode Gaussian state is incoherentif and only if the covariance matrix σ is diagonal, thereis not displacement and squeezing.Proof. By definition [15], a single mode incoherent stateis represented by a diagonal density matrix (cid:37) I , (cid:37) I = (cid:88) n p n | n (cid:105) (cid:104) n | (6)It is easy to verify that the corresponding CM σ I isdiagonal with elements σ jj = (cid:88) n p n n + 12 (7) with j = 1 ,
2, while the first moments (cid:104) X (cid:105) and (cid:104) P (cid:105) arenull, so that the displacement is zero. So if the state isincoherent, that is described by the density matrix (cid:37) I ,the corresponding CM σ I is such that the elements haveto satisfy the following conditions: (cid:104) X (cid:105) = 0 = (cid:104) P (cid:105) , (8) σ = σ = (cid:88) n p n n + 12 , (9) σ = σ = 0 . (10)Referring explicitly to the generic single mode Gaussianstate Eq. (1), the condition of incoherence Eq. (8) is sat-isfied by null values of the complex amplitude β . Indeedthe vector of the first moments R Eq. (3) is null just for β = 0 (no displacement). Similarly, the conditions Eq.s(9) and (10) are satisfied by the values r = 0 and ψ = 0(no squeezing), for which a = b = N th cosh(2 r ) and c = 0. This means that a single mode Gaussian state isincoherent when there is no squeezing and no displace-ment thermal state. Under these conditions the genericGaussian state Eq. (1) becomes thermal one. So, we haveshown that a single mode Gaussian state is incoherent ifit is a thermal state, (cid:37) I ≡ ν th with p n = N nth (1+ N th ) ( n +1) .Let consider now a Gaussian state described by a diago-nal CM and without squeezing and displacement. It easyto verify that only in this case the density matrix describ-ing the state is diagonal, (cid:104) m | (cid:37) G | n (cid:105) = 0. Indeed a densitymatrix represented by a diagonal CM and by at least nonull value between β and r isn’t diagonal, (cid:104) m | (cid:37) | n (cid:105) (cid:54) = 0just as a the density matrix of a state with β = r = 0and not diagonal CM .In conclusion, the only incoherent single mode Gaus-sian states are the thermal states, described by the CM: σ i = 12 (cid:20) N th,i
00 1 + 2 N th,i (cid:21) . (11)This class of states will be the reference set from whichit is necessary to calculate the minimum distance. IV. QUANTUM COHERENCE OF GAUSSIANSTATES: DISTANCE FROM THE SET OFINCOHERENT STATES
We extend the coherence measure C x introduced in [15]to Gaussian states (cid:37) G Eq. (1), considering the minimumof the distance, induced by the x -metric, from the set ofthe incoherent states Eq. (6): C x = 12 min (cid:37) I d x ( (cid:37) G , (cid:37) I ) . (12)In particular, we consider the Bures and the
Hellinger metric.The Bures measure is then defined in terms of the Buresdistance as C Bu ≡
12 min (cid:37) I d Bu ( (cid:37) G , (cid:37) I ) = 1 − max (cid:37) I (cid:112) F ( (cid:37) G , (cid:37) I ) , (13)where F ( (cid:37) G , (cid:37) I ) is the Uhlmann fidelity. For Gaussianstates, the Uhlmann fidelity [25] is found to be: F = e − δR T ( σ G + σ I ) − δR ( √ ∆ + Λ − √ Λ) − , (14)where δR is the difference between the first moments vec-tors of the relative states (cid:37) G and (cid:37) I . In the previous sec-tion we showed that the class of the incoherent states hasnull first moments; hence δR ≡ R Eq. (3). The explicitexpressions of Λ, and ∆ are:∆ = det( σ G + σ I ) , (15)Λ = 4 det( σ G + i σ I + i , (16)where Ω is the symplectic matrix:Ω = (cid:20) − (cid:21) . (17)The measure Eq. (13) enjoys the main properties thata bona fide coherence measure must satisfy according to[15]. Indeed, from the properties of the Fidelity [26],the necessary requirements ( C (cid:48) ), ( C a ) and ( C
3) of [15]are automatically satisfied by the definition Eq. (13). Infact, C Bu = 0 iff (cid:37) G ≡ (cid:37) I ( C (cid:48) ); moreover C Bu is mono-tone under incoherent completely positive trace preserv-ing map ICPTP ( C a ); finally C Bu is convex because d Bu is contractive and d Bu is convex ( C C He ≡
12 min (cid:37) I d He ( (cid:37) G , (cid:37) I ) = 1 − max (cid:37) I (cid:112) A ( (cid:37) G , (cid:37) I ) , (18)where A ( (cid:37) G , (cid:37) I ) is the Affinity [27]. For Gaussian states,the Affinity assumes the simple form: A ( (cid:37) G , (cid:37) I ) = 2 e − δR T ( σ G + σ I ) − δR [det( σ G ) det( σ I )] [det( σ G + σ I )] . (19)It is possible to show, following the same line of reasoninggiven for the Bures distance, that the Hellinger distanceenjoys the required properties. V. RESULTS
In this section we analyse the behaviour of the coher-ence measures Eqs. (13) and (18) for the principal classesof one-mode Gaussian states, obtained from Eqs. (4)and (5) for particular values of parameters.
A. Squeezed thermal states
Let us consider, the set of squeezed thermalstates (STS), with real squeezing parameter ξ ≡ r . Such states is described by the CM (4)with a = (cid:0) N th (cid:1) (cosh(2 r ) + sinh(2 r )), b = (cid:0) N th (cid:1) (cosh(2 r ) − sinh(2 r )) and c = 0. C x N th = N th = N th = C x N th = N th = N th = FIG. 1: (color online) Behavior of the coherencemeasures Eqs. (13) and (18) for STSs, as a function ofthe squeezing parameter r for fixed N th = 0 (orange), N th = 1 (green), N th = 2 (blue). The solid and dashedlines represent the Bures and Hellinger measuresrespectively. The case β = 0 is shown in the upperpanel, while in the lower panel the case β = 1 isreported.In Fig. (1) is reported the behaviour of the coherencemeasures Eqs. (13) and (18) as a function of the squeez-ing parameter r for the squeezed vacuum state ( N th = 0)(orange line), and for fixed N th = 1 (green line) and N th = 2 (blue line) for the cases β = 0 (upper panel) and β = 1 (lower panel). The coherence measures increaseswith r , due to the fact that the state becomes more andmore distinguishable form the set of the matrices propor-tional to the identity matrix, namely, the set of thermal(incoherent) states. Therefore: C d ( ˜ r, N th ) → r → ∞ . (20)In Fig. (2) is reported the behaviour of quantum co-herence measures Eqs. (13) and (18) as a function of themean thermal photon number N th of the STS for r = 0(orange line), for r = 1 (green line) and for r = 2 (blueline). The value of the coherence tends asymptotically toa constant value, as shown in (A). As you can see from in N th C x r = = = N th C x r = = = FIG. 2: (color online) Behavior of the coherence, forSTSs, as a function of the thermal photon number N th for fixed r = 0 (orange), r = 1 (green) and r = 2 (blue).The solid and dashed lines stand for Bures andHellinger measures respectively. In the upper panel isshown the case β = 0, in the lower panel β = 1.Fig. (2), the value of C d tends asymptotically to a con-stant (in the case of the Hellinger, is constant for each N th ). Apparently, this may seem abnormal behavior,since the elements a and b of the CM always differ moreincreasing of N th . However, we have shown analytically(see Appendix A) that the C d value tends asymptoticallyto a constant for N th → ∞ . Indeed, although the terms a and b differ more and more, for very large N th bothtend to infinite of the same order. Therefore: C d ( ˜ r, N th ) → const for N th → ∞ (21)Squeezed Thermal State C x N sq N th C Bu − C He − TABLE I: Example values of the parameters N sq and N th for the coherence measures Eq. (13), and Eq. (18)corresponding to 99% of quantum coherence.In Tab.(I) some value of parameters corresponding to99% of quantum coherence are reported for the quantumcoherence measures considered in the text. B. Coherent thermal states
We consider, at first, coherent thermal states CTSswith real amplitude of the displacement so that the firstmoments Eq. (3) are R = √ β, σ CT S = 12 (cid:20) N th
00 1 + 2 N th (cid:21) . (22) β C x N th = N th = N th = FIG. 3: (color online) Behavior of coherence, for a CTS,as a function of the displacement β , for different valuesof the mean thermal photon number: N th = 0 (orange), N th = 1 (green) and N th = 2 (blue). The solid anddashed lines stand for Bures and Hellinger measuresrespectively.In Fig (3) it is reported the behaviour of quantum co-herence measures Eqs. (13) and (18) as a function of β for CTS with fixed thermal photon number N th = 0 (or-ange line), representing the ideal coherent states, N th = 1(green line), and N th = 2 (blue line). As expected,the coherence increases at the increasing of the displace-ment. In particular, fixing the thermal contribution to N th ≡ ˜ N th , it results: C x ( β, ˜ N th ) → β → ∞ . (23)In Fig. (4) is reported the behaviour of quantum co-herence measures Eqs. (13) and (18) as a function of thethermal contribution N th for three different values of thecoherent amplitude: β = 0 (orange line), that representsthermal (incoherent) state, β = 1 (green line) and β = 2(blue line). The measures are decreasing at the increas-ing of N th . We can see that, more in general, for a genericfixed value of the coherent amplitude β ≡ ˜ β , C d ( ˜ β, N th ) → N th → ∞ . (24)This is due to the fact that the contribution of thedisplacement β to the quantum coherence becomes moreand more negligible at N th increasing, namely the num-ber of incoherent thermal photons N th,i , that minimize N th C x β = β = β = FIG. 4: (color online) Behavior of coherence, for a CTS,as a function of the thermal photon number N th , fordifferent values of displacement: β = 0 (orange), β = 1(green) and β = 2 (blue). The solid and dashed linesstand for Bures and Hellinger measures respectively.the measures, tends to the number of thermal photons ofthe state. Coherent Thermal State C x N coh N th C Bu . ×
22 never C He
22 neverTABLE II: Example values of the parameters N coh and N th for the coherence measures Eq. (13), and Eq. (18)corresponding to 99% of quantum coherence.In Tab.(II) some value of parameters corresponding to99% of quantum coherence are reported for the quantumcoherence measures considered in the text. C. Thermal squeezed states
In conclusion, we consider thermal squeezed statesTSSs with real squeezing parameter ξ ≡ r . Such state isdescribed by the CM σ T SS = (cid:20) N th + e r N th + e − r (cid:21) . (25)In Fig. (5) is reported the behaviour of quantum co-herence measures Eqs. (13) and (18) as a function of thethermal contribution N th for three different values of thesqueezing parameter: r = 0 (orange line), r = 1 (greenline) and r = 2 (blue line). The measures are decreasingat the increasing of N th . Indeed, fixing the value r of thesqueezing, increasing of the thermal photons N th causesthe diagonal elements of the covariance matrix Eq. (25)to become less distinguishable. As a consequence, in thelimit N th → ∞ the TSS become incoherent state. Hencewe have: C d (˜ r, N th ) → N th → ∞ . (26) N th C x r = = = FIG. 5: (color online) Behavior of coherence, for a TSS,as a function of the thermal photon number N th , fordifferent values of squeezing: r = 0 (orange), r = 1(green) and r = 2 (blue). The solid and dashed linesstand for Bures and Hellinger measures, respectively. C x N th = N th = N th = FIG. 6: (color online) Behavior of coherence, for a TSS,as a function of the squeezing r , for different values ofthermal photon number: N th = 0 (orange), N th = 1(green) and N th = 2 (blue). The solid and dashed linesstand for Bures and Hellinger measures, respectively.In Fig. (6) is reported the behaviour of quantum co-herence measures Eqs. (13) and (18) as a function of thesqueezing parameter r for three different values of ther-mal photon number: N th = 0 (orange line), N th = 1(green line) and N th = 2 (blue line). The measures areincreasing at the increasing of r .At N th fixed, increasing r causes the diagonal elementsof σ T SS to become more distinguishable. As a result thecovariance matrix defining the state becomes more andmore different from the matrices that define the incoher-ent state. Consequently, it is never possible to find avalue of N th,i that minimizes the distance between σ T SS and σ I . Therefore we have: C d ( r, ˜ N th ) → r → ∞ . (27)Thermal Squeezed State C x N sq N th C Bu − C He − N sq and N th for the coherence measures Eq. (13), and Eq. (18)corresponding to 99% of quantum coherence.In Tab.(II) some value of parameters corresponding to99% of quantum coherence are reported for the quantumcoherence measures considered in the text. VI. CONCLUSIONS
Summarizing we have defined a geometric-basedapproach for the quantification of quantum coherencefor one-mode Gaussian states as the minimal distanceof the given state form the set of incoherent (thermal)states. Furthermore, we have studied the behaviourof the quantum coherence for the main classes of thequantum Gaussian states.As expected, the increase of the displacement β causesthe increase of the quantum coherence measures. In-stead, in general, increasing the number of thermal pho-tons, has a detrimental effect on the quantum coherence.Surprisingly, we observe that, for the squeezed thermalstates, except for the case r = 0, the quantum coherencetends to a constant value different from zero for N th (cid:29) r this asymptotic value is greater than the initial value( N th = 0). This counterintuitive behaviour is the same asobserved for other geometric quantum correlations mea-sure [28].This work represents the first step in understandingthe usefulness of quantum coherence for the quantumtechnologies. The natural next step will be to extend themeasure to the case of two-mode Gaussian states [29] andto classify the Gaussian incoherent quantum operations. Appendix A: Asymptotic study of the Buresmeasure for the σ STS
In this section we show that, for the class of squeezedthermal states, the value of coherence, when the num-ber of thermal photons is sufficiently high, tends to asqueezed-dependent value.We consider, without loss of generality, a covariancematrix of the form σ p = (cid:20) C (cid:0) N th (cid:1) C (cid:0) N th (cid:1)(cid:21) , (A1) as Eq. (A1) simulates perfectly the state σ ST S in the caseof fixed squeezing r , putting: C = (sinh( r ) + cosh( r )) C = (cosh( r ) − sinh( r )) . (A2)We consider, furthermore, the covariance matrix Eq. (11)of the reference states.For ease of calculation we fix the displacement β =0; we have verified numerically that the more generalcase presents the same behaviour. With this choice, theexponential in the fidelity is equal to one, namely Eq. (14)reduces to: F = 1 √ ∆ + Λ − √ Λ ; (A3) N th C d r = Log [ ] r = Log [ ] r = Log [ ] FIG. 7: (color online) Asymptotic behavior of the Buresmeasure for squeezed thermal state for different valuesof the squeezing parameter r .Now we need to maximize the fidelity on the set ofincoherent states, namely, we have to maximize on N th,i .Remember that the number of thermal photons N th and N th,i are greater or equal than 0, it is possible to showthat the maximized fidelity is given by: F a,r = N ( r ) a = 0 PZ a,r [1] 0 < a < ¯ a ( r ) PZ a,r [3] ¯ a ( r ) ≤ a < ¯ a ( r ) PZ a,r [2] a ≥ ¯ a ( r ) , (A4)where 0 ≤ N ( r ) ≤ r -dependent constant and PZ a,r [ i ] are the i -th solution of the following polynomial PZ a,r ( ζ ): PZ a,r ( ζ ) = ( K ( r ) a + K ( r ) a + K ( r ) a + K ( r ) a + K ( r ) a + K ( r ) a + K ( r ) a + K ( r ) a + K ( r )) ζ + ( I ( r ) a + I ( r ) a + I ( r ) a + I ( r ) a + I ( r ) a + I ( r ) a + I ( r ) a + I ( r ) a + I ( r )) ζ + ( J ( r ) a + J ( r ) a + J ( r ) a + J ( r ) a + J ( r ) a + J ( r ) a + J ( r ) a + J ( r ) a + J ( r )) ζ + ( L ( r ) a + L ( r ) a + L ( r ) a + L ( r ) a + L ( r ) a + L ( r ) a + L ( r )) ζ + M ( r ) a + M ( r ) a + M ( r ) a + M ( r ) a + M ( r ) (A5)From the explicit expression Eqs. (A4) and (A5) it is pos-sible to show that the asymtotic value of the Fidelity withrespect the number of thermal photons is a r -dependentpositive constant:lim a →∞ F a,r = lim a →∞ PZ a,r [2] = const . (A6) To better clarify the result, in Fig. (7) we show the be-haviour of the Bures measure as a function of N th fordifferent values of the squeezing parameter r . [1] R.J. Glauber, Phys. Rev. , 2766 (1963)[2] M.O. Scully, Phys. Rev. Lett. , 1855 (1991)[3] A. Albrecht, J. Mod. Opt. , 2467 (1994)[4] M. Nielsen and I. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000), ISBN 9781139495486.[5] B. Deveaud-Pl´edran, A. Quattropiani and P. Schwendi-mann, eds.,
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