Measuring Gaussian quantum information and correlations using the

Measuring Gaussian Quantum Information and Correlations Using the R´enyi Entropy of Order 2 Gerardo Adesso,1 Davide Girolami,1 and Alessio Serafini2 1

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, U.K. (Dated: August 17, 2012)

arXiv:1203.5116v3 [quant-ph] 23 May 2013

2

We demonstrate that the R´enyi-2 entropy provides a natural measure of information for any multimode Gaussian state of quantum harmonic systems, operationally linked to the phase-space Shannon sampling entropy of the Wigner distribution of the state. We prove that, in the Gaussian scenario, such an entropy satisfies the strong subadditivity inequality, a key requirement for quantum information theory. This allows us to define and analyze measures of Gaussian entanglement and more general quantum correlations based on such an entropy, which are shown to satisfy relevant properties such as monogamy. PACS numbers: 03.67.-a, 03.65.Ta, 03.65.Ud, 42.50.Dv

In quantum information theory, the degree of information contained in a quantum state ρ is conventionally quantified via the von Neumann entropy of the state S(ρ) = −tr(ρ ln ρ), that is the direct counterpart to the Shannon entropy in classical information theory [1]. The most fundamental mathematical implications and physical insights in quantum information theory, ranging from the Holevo bound [2] all the way to the whole genealogy of quantum communication protocols [3, 4], rely on a key property satisfied by the von Neumann entropy, the strong subadditivity inequality [5, 6] S(ρAB ) + S(ρBC ) ≥ S(ρABC ) + S(ρB ) ,

(1)

for an arbitrary tripartite state ρABC . The strong subadditivity inequality implies in particular that the mutual information I(ρA:B ) = S(ρA ) + S(ρB ) − S(ρAB ) ,

(2)

which measures the total correlations between the subsystems A and B in the bipartite state ρAB , is always nonnegative. However, in classical as well as quantum information theory, several other entropic quantities have been introduced and studied. In particular, R´enyi-α entropies [7] are an interesting family of additive entropies, whose interpretation is related to derivatives of the free energy with respect to temperature [8], and which have found applications especially in the study of channel capacities [9], work value of information [10], and entanglement spectra in many-body systems [11]. They are defined as Sα (ρ) = (1 − α)−1 ln tr(ρα ) ,

(3)

and reduce to the von Neumann entropy in the limit α → 1. R´enyi entropies are powerful quantities for studying quantum correlations in multipartite states. For a bipartite pure state ρAB , any of the entropies in Eq. (3) evaluated on the reduced density matrix of one subsystem only, say ρA , is an entanglement monotone [12], dubbed R´enyi-α entanglement [13]. Any such measure can be extended to mixed states via conventional convex roof techniques [14]. Using the α = 2 instance, S2 (ρ) = − ln tr(ρ2 ), the ensuing R´enyi-2 measure of entanglement has been defined and proven to satisfy the important ‘monogamy’ inequality [15, 16] for multiqubit states [13, 17]. Nonetheless, R´enyi-α entropies for α , 1 are not in

general subadditive [1]; this entails, e.g., that if one replaces S by Sα,1 in (2), the corresponding quantity can become negative, i.e., meaningless as a correlation measure; see [18] for explicit two-qubit instances of this fact when α = 2. In this Letter, we focus our attention on multimode quantum harmonic oscillators, and analyze the informational properties of general Gaussian states as measured by the R´enyi2 entropy. Gaussian states constitute versatile resources for quantum communication protocols with continuous variables [19, 20] and are important testbeds for investigating the structure of quantum correlations [21], whose role is crucial in fields as diverse as quantum field theory, solid state physics, quantum stochastic processes, and open system dynamics. Gaussian states naturally occur as ground or thermal equilibrium states of any physical quantum system in the ‘smalloscillations’ limit [22, 23], and can be very efficiently engineered, controlled and detected in various experimental setups, including light, atomic ensembles, trapped ions, nano/opto-mechanical resonators, and hybrid interfaces thereof [20]. Given the special role played by Gaussian states and operations, which are formally ‘extremal’ with respect to several quantum primitives [23], a Gaussian-only theory of quantum information is actively pursued [24, 25]. In particular, a key conjecture which —despite its disproof for general channels [26]— still stands within the Gaussian scenario, is the socalled minimum output entropy conjecture for bosonic channels [24, 25, 27, 28]. Interestingly, the conjecture is verified (for all phase-insensitive channels) if using R´enyi-α entropies with α ≥ 2 [29], but to date is resisting an analytical proof for α → 1 [30]. This somehow raises the question whether the von Neumann entropy is indeed the most natural one within the Gaussian scenario [31]. Here we show that the R´enyi-2 entropy should be regarded as a specially meaningful choice to develop a Gaussian theory of quantum information and correlations. We prove that the R´enyi-2 entropy S2 satisfies the strong subadditivity inequality (1) for all Gaussian states, is operationally linked to the Shannon entropy of Gaussian Wigner distributions (providing a natural measure of state distinguishability in phase space), and can be employed to define valid measures of Gaussian entanglement [32, 33] and general discord-like quantum correlations [34–38], whose monogamy properties [15, 16, 39–42] we investigate in detail. In particular, we obtain a truly bona

2 fide measure of genuine tripartite entanglement for threemode Gaussian states, based on S2 . Our study allows us to explore within a unified framework the various facets of nonclassicality in the Gaussian realm. R´enyi-2 entropy for Gaussian states.— We consider a nmode continuous variable system; we collect the quadrature operators in the vector Rˆ = (qˆ 1 , pˆ 1 , qˆ 2 , pˆ 2 , . . . , qˆ n , pˆ n )T ∈ R2n and write the canonical commutation  0 1 relations compactly as
[Rˆ j , Rˆ k ] = i ω⊕n j,k with ω = −1 0 being the symplectic matrix. A Gaussian state ρ is described by a positive, Gaussianshaped Wigner distribution in phase space, Wρ (ξ) =

πn

1
exp − ξT γ−1 ξ , p det γ

(4)

where ξ ∈ R2n , and γ is the covariance matrix (CM) of elements γ j,k = tr[ρ{Rˆ j , Rˆ k }+ ], which (up to local displacements) completely characterizes the Gaussian state ρ. Let us now evaluate different entropic quantities on the Gaussian state ρ. All the measures defined below are invariant under local unitaries, so we will assume our states to have ˆ = 0, without loss of zero first moments, h Ri R generality. The purity is easily computed as tr ρ2 = (2π)n R2n Wρ2 (ξ) d2n ξ = 1 (det γ)− 2 . Hence the R´enyi-2 entropy of an arbitrary n-mode Gaussian state is 1 S2 (ρ) = ln(det γ) , 2

(5)

ranging from 0 on pure states (det γ = 1) and growing unboundedly with increasing mixedness (i.e., temperature) of the state. The von Neumann entropy is instead a more complicated function that depends on the local temperatures of all the n normal modes of ρ, that is, on the full symplectic spectrum of γ [21, 31, 43, 44]. On the other hand, since the Wigner function is a valid probability distribution providing a fully equivalent description of ρ, we can alternatively compute the Shannon entropy of Wρ to obtain an alternative quantifier of the informational content of any Gaussian state. Such an entropy has a clear interpretation in terms of phasespace sampling via homodyne detections [45]. The continuous (Boltzmann–Gibbs–)Shannon entropyRfor a probability distribution P(x) is defined as H(P) = − P(x) ln P(x) dx [46]. It can be shown that [47] (see the Appendix [48] for a derivation) Z Wρ (ξ) ln[Wρ (ξ)] d2n ξ H(Wρ ) = − R2n

= S2 (ρ) + n(1 + ln π) .

(6)

Interestingly, we see that this sampling entropy coincides (modulo the additional constant) with the R´enyi-2 entropy rather than the von Neumann one for Gaussian states. We can now take an extra step and introduce a ‘relative sampling entropy’, to quantify the phase-space distinguishability of two n-mode Gaussian quantum states ρ1 and ρ2 (with CMs γ1 and γ2 , respectively), defined as the relative Shannon entropy (also known as Kullback-Leibler divergence [1, 6]) between

their respective Wigner distributions Wρ1 and Wρ2 , yielding [48] ! Z Wρ1 (ξ) 2n
Wρ1 (ξ) ln H Wρ1 kWρ2  d ξ (7) Wρ2 (ξ) R2n " !  # det γ2 1 −1 = ln + tr γ1 γ2 − n . 2 det γ1 Let us evaluate Eq. (7) when ρ1 ≡ ρAB is a generic Gaussian state of a composite system, partitioned into two subsystems A and B (of nA and nB modes respectively, with nA + nB = n), and ρ2 ≡ ρA ⊗ ρB is the tensor product of the two marginals of ρAB . Writing the CMs in block form, ! γA ς AB γ1 ≡ γAB = T , γ2 ≡ γA ⊕ γB , (8) ςAB γB   = 2(nA + nB ) = 2n, which entails we have tr γ1 γ−1 2 H WρAB kWρA ⊗ρB




= H(WρA ) + H(WρB ) − H(WρAB ) ! det γA det γB 1 = ln 2 det γAB = S2 (ρA ) + S2 (ρB ) − S2 (ρAB )  I2 (ρA:B ) .

(9)

The above equation defines the ‘Gaussian R´enyi-2 (GR2) mutual information’ I2 for an arbitrary bipartite Gaussian state ρAB . It follows that I2 (ρA:B ) ≥ 0, as it coincides exactly with the Shannon continuous mutual information of the Wigner function of ρAB , which is positive semidefinite. The expression in Eq. (9), which is analogous to the more familiar von Neumann one [Eq. (2)], has thus a precise operational interpretation as the amount of extra discrete information (measured in natural bits) that needs to be transmitted over a continuous variable channel to reconstruct the complete joint Wigner function of ρAB rather than just the two marginal Wigner functions of the subsystems [45, 46]; in short, I2 (ρA:B ) measures the total quadrature correlations of ρAB . We can then enquire whether a more general property such as the strong subadditivity inequality (1) —which would independently imply the nonnegativity of I2 — holds in general for S2 in the Gaussian scenario. Let ρABC be a tripartite Gaussian state whose subsystems encompass arbitrary number of modes. Writing its CM in block form as in Eq. (8), and using the definition (5), we have the following Theorem 1 The R´enyi-2 entropy S2 satisfies the strong subadditivity inequality for all Gaussian states ρABC , S2 (ρAB ) + S2 (ρBC ) − S2 (ρABC ) − S2 (ρB ) ! det γAB det γBC 1 = ln ≥ 0. 2 det γABC det γB

(10)

Proof. The result follows by applying a particular norm compression inequality to the CM γABC . Given a positive Hermitian matrix A ∈ Mm , and given any two index sets α, β ⊆ N = {1, . . . , m}, the Hadamard-Fisher inequality [49]

3 states that det Aα∪β det Aα∩β ≤ det Aα det Aβ . Recalling that any CM γABC is a positive real symmetric matrix [50], the claim follows upon identifying α with the indices of modes AB and β with the indices of modes BC.  Beyond its apparent simplicity, Theorem 1 has profound consequences. It yields that the core of quantum information theory can be consistently reformulated, within the Gaussian scenario [24, 25], using the simpler and physically natural R´enyi-2 entropy in alternative to the von Neumann one. In the rest of this Letter we will focus on defining GR2 quantifiers of entanglement and other correlations for Gaussian states. Gaussian R´enyi-2 measures of correlations.— The GR2 entanglement E2 can be defined as follows. Given a n-mode bipartite Gaussian state ρAB with CM γAB , E2 (ρA:B ) 

inf

(

σAB : ) ±iω⊕n ≤σ AB ≤γ AB , det σAB =1

1 ln (det σA ) . 2

(11)

For a pure Gaussian state ρAB = |ψAB ihψAB |, the minimum is saturated by σAB = γAB , so that E2 (ψA:B ) = S2 (ρA ) = 1 2 ln(det γA ), where γA is the reduced CM of subsystem A. For a generally mixed state, Eq. (11) —where the minimization is over pure n-mode Gaussian states with CM σAB smaller than γAB — amounts to taking the Gaussian convex roof of the pure-state R´enyi-2 entropy of entanglement, according to the formalism of [21, 32]. Closed formulae for E2 can be obtained for special classes of two-mode Gaussian states [33, 51, 52] exploiting the same procedure adopted for the Gaussian entanglement of formation [32, 48]. Like the latter measure (and all Gaussian convex-roof entanglement measures), it follows from the results of [32] that the GR2 entanglement is in general monotonically nonincreasing under Gaussian local operations and classical communication, and is additive for twomode symmetric Gaussian states. We can also introduce a ‘GR2 measure of one-way classical correlations’ in the spirit of Henderson and Vedral [35]. We define J2 (ρA|B ) as the maximum decrease in the R´enyi-2 entropy of subsystem A, given a Gaussian measurement has been performed on subsystem B, where the maximization is over all Gaussian measurements—i.e. those that map Gaussian states into Gaussian states [37, 38]. Any such measurement on, say, the nB -mode subsystem B = (B1 . . . BnB ), is described by a positive operator valued measure (POVM) [53] QB ˆ† QB ˆ WB j (η j )] WB j (η j )]ΛΠB [ nj=1 of the form ΠB (η) = π−nB [ nj=1 † ∗ ˆ ˆ ˆ B (η j ) = exp(η j b − η b j ) is the Weyl operator, bˆ j = where W j R j √ (qˆ B j +i pˆ B j )/ 2, π−nB ΠB (η)d2nB η = 11, and ΛΠB is the density matrix of a (generally mixed) nB-mode Gaussian state with CM ΓΠB which denotes the seed of the measurement. The conditional state ρA|η of subsystem A after the measurement ΠB (η) has been performed on B has a CM γ˜ ΠA independent of the outcome η and given by the Schur complement [54] γ˜ ΠA = γA − ς AB (γB + ΓΠB )−1 ςTAB ,

(12)

where the original bipartite CM γAB of the n-mode state ρAB has been written in block form as in (8). We have then   1  det γA   .  (13) J2 (ρA|B)  sup ln  det γ˜ ΠA ΓΠB 2

The one-way classical correlations J2 (ρB|A ), with measurements on A, can be defined accordingly by swapping A ↔ B. We can now define a Gaussian measure of quantumness of correlations based on R´enyi-2 entropy. Following the seminal study by Ollivier and Zurek [34], and the recent analysis of Gaussian quantum discord using von Neumann entropy [37, 38], we define the ‘GR2 discord’ as the difference between mutual information (9) and classical correlations (13), D2 (ρA|B )  I2 (ρA:B ) − J2 (ρA|B )   1  det γB det γ˜ ΠA   . = inf ln  det γAB ΓΠB 2

(14)

For the case of A and B being single modes, that is, ρAB being a general two-mode Gaussian state, closed formulae can be obtained for Eqs. (13,14) thanks to the results of [37]. We report them in the Appendix [48] for completeness. We remark that Theorem 1 is crucial to guarantee the nonnegativity and the faithfulness of the GR2 discord [55–57]. Notice also that 1 2 I2 (ρA:B ) = J2 (ρA|B ) = D2 (ρA|B ) = S2 (ρA ) for pure bipartite Gaussian states ρAB . A trade-off relation between the entanglement E2 and the classical correlations J2 can be written for arbitrary tripartite pure Gaussian states ρABC [37], following Koashi and Winter [59]. One can essentially exploit the fact that all possible Gaussian POVMs on B induce all possible Gaussian purestate decompositions of the subsystem AC (see also [16]), which implies S2 (ρA ) = J2 (ρA|B ) + E2 (ρA:C ) .

(15)

This relation can be manipulated to express the “conservation” of different types of correlations in a generic pure tripartite Gaussian state ρABC , along the lines of [60]. We now look at monogamy properties of the GR2 measures. For an entanglement monotone E and a n-partite state ρA1 A2 ...An , the monogamy relation (choosing party A1 as the focus), which constrains the distribution of bipartite entanglement among different splits, can be written as [15] P E(ρA1 :A2 ...An ) − nj=2 E(ρA1 :A j ) ≥ 0 . The R´enyi-2 entanglement measure [13, 17], as well as the tangle (squared concurrence) [15, 16], satisfy this inequality for general n-qubit states. A Gaussian version of the tangle (based on squared negativity) has been defined that obeys the inequality for all n-mode Gaussian states [40]. We now show that E2 does too. Theorem 2 The GR2 entanglement defined in Eq. (11) is monogamous for all n-mode Gaussian states ρA1 A2 ...An , E2 (ρA1 :A2 ...An ) −

Pn

j=2 E2 (ρA1 :A j )

≥ 0,

(16)

where each A j comprises one mode only. Proof. The structure of the proof follows closely the one for the tangle of n-qubit systems [16]. It suffices to prove the inequality for tripartite Gaussian states ρA1 A2 A′3 where A′3 comprises n − 2 modes (with n arbitrary), as iterative applications to E2 (ρA1 :A′3 ) would then imply Eq. (16). It is further enough to prove the inequality on pure states, as it would then extend to mixed ones by convexity [15, 16, 39, 40]. Exploiting

4 the phase-space Schmidt decomposition [21, 43, 61], when ρA1 A2 A′3 is pure, the state of subsystem A′3 is locally equivalent to a 2-mode state, tensored by n − 4 irrelevant vacuum modes. The problem reduces to proving that E2 (ρA:BC ) ≥ E2 (ρA:B ) + E2 (ρA:C ) for an arbitrary pure Gaussian state ρABC , where A and B are single modes, while C groups two modes. Noting that E2 (ρA:BC ) = S2 (ρA ), and exploiting Eq. (15), we see that Eq. (16) is verified if one establishes that J2 (ρA|B ) ≥ E2 (ρA:B )

(17)

holds for all (mixed) two-mode Gaussian states ρAB . The inequality (17) is proven in the Appendix [48], which concludes the proof of the Theorem.  Let us analyze in detail the case of ρA1 A2 A3 being a pure 3-mode Gaussian state, whose CM is characterized up to local unitaries by three parameters (local symplectic invariants) a j ≥ 1, with S2 (ρA j ) = ln a j ( j = 1, 2, 3) [48, 52]. We define the residual entanglement emerging from the monogamy inequality as E2 (ρA1 :A2 :A3 ) = E2 (ρA1 :A2 A3 )−E2(ρA1 :A2 )−E2 (ρA1 :A3 ). This quantity, which can be calculated exactly [39, 52], depends in general on the focus mode (say A1 ) chosen for the decomposition of the bipartite entanglements. Remarkably, we find that for all pure three-mode Gaussian states which are fully inseparable and display entanglement in all global and reduced bipartitions (E2 (ρAi :A j ) > 0, E2 (ρAi :A j Ak ) > 0, ∀i , j , k), the residual GR2 entanglement E2 (ρA1 :A2 :A3 ) is invariant under mode permutations, thus representing (to the best of our knowledge) the first and only known intrinsically bona fide measure of genuine tripartite Gaussian entanglement [62]. We report its explicit formula here, while a derivation is provided in [48], E2 (ρA1 :A2 :A3 ) = ln (8a1 a2 a3 ) (18) √  P3 P  − ln − 1 − δ + i=1 (2a2i − a4i ) + 3i, j=1 (a2i a2j ) ,

Q
with δ = 1µ,ν=0 (a1 + (−1)µ a2 + (−1)ν a3 )2 − 1 . The formula 1 (18) holds when |ai − a j | + 1 < ak < (a2i + a2j − 1) 2 . We finally remark that, importantly, the GR2 discord Eq. (14) also turns out to be a monogamous measure of quantum correlations for arbitrary pure three-mode Gaussian states ρA1 A2 A3 (unlike the von Neumann entropy-based discord [37, 38, 41]). By using Eq. (15), one finds (see also [59, 60]) D2 (ρA1 :A2 :A3 )  D2 (ρA2 |A2 A3 ) − D2 (ρA1 |A2 ) − D2 (ρA1 |A3 ) = E2 (ρA1 :A2 :A3 ). In other words, the residual tripartite discord equates the residual tripartite entanglement for pure ρA1 A2 A3 . This extends to the multipartite case the equivalence between entanglement and general quantum correlations valid for pure bipartite states [34, 64, 65], and places D2 as the only known

[1] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006). [2] A. S. Holevo, Probl. Inf. Transm. 9, 177 (1973). [3] M. Horodecki, J. Oppenheim, and A. Winter, Nature 436, 673

measure of quantumness beyond entanglement in continuous variable systems that fulfills monogamy [66]. Conclusions.— In this Letter we planted the seeds for a full Gaussian quantum information theory [25] using the R´enyi-2 entropy S2 . This is possible thanks to the fact, proven in Theorem 1, that such an entropy satisfies the strong subadditivity inequality for arbitrary Gaussian states of quantum harmonic systems. We employed S2 to define valid measures of entanglement, total, classical, and quantum correlations, highlighting their properties. The R´enyi-2 mutual information is intimately related to Wigner distribution sampling by homodyne detections in phase space. The residual R´enyi-2 entanglement measure allows for a quantification of genuine tripartite entanglement in three-mode fully inseparable pure Gaussian states, which is invariant under mode permutations. We argue that the measures defined in this Letter should be adopted as privileged tools to address the quantification of relevant correlations in Gaussian states. A very recent application to relativistic quantum information has been reported [68]. By conception, this work has been biased towards Gaussian states and operations. However, the approach pursued here can be extended to arbitrary, even non-Gaussian n-mode states ρ of continuous variable systems, provided one chooses the Wehrl entropy [6] to quantify the informational content of ρ. Such an entropy is operationally associated to phasespace sampling via heterodyne detections [45] as it corresponds to the continuous Boltzmann–Gibbs–Shannon entropy of the Husimi Q distribution of ρ, Q(α) = π−n hα|ρ|αi, which is a valid (nonnegative) probability distribution for all quantum states [1]. One can then adopt a distance measure between any two ρ1 and ρ2 in terms of the relative Shannon entropy between their respective Q distributions, define ensuing correlation measures, and so on. Such a formalism could also accommodate a measure of non-Gaussianity of quantum states [69], and one might then naturally compare Gaussian with non-Gaussian operations for the realization of specific tasks [27] such as extracting classical correlations [70, 71], maximizing information-disturbance trade-off [72] and performing optimal cloning of coherent states [73]. This will be the subject of further study. Acknowledgments.— We acknowledge enlightening discussions with I. Apicella, V. P. Belavkin, N. Cerf, F. P. Cimino Wood, A. Datta, N. Datta, J. Eisert, V. Giovannetti, M. Guta, M. van Horssen, L. Miˇsta Jr., M. Mosonyi, S. Pirandola, A. Winter. GA thanks the University of Nottingham for financial support through an Early Career Research and Knowledge Transfer Award. AS thanks Jessica & Dave for moral and financial support.

(2005); Comm. Math. Phys. 268, 107 (2007). [4] I. Devetak, A. W. Harrow, and A. Winter, Phys. Rev. Lett. 93, 230503 (2004); I. Devetak, A. W. Harrow, and A. Winter, IEEE Trans. Inf. Th. 54, 4587 (2008); A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465, 2537, (2009).

5 [5] E. H. Lieb and M. B. Ruskai, J. Math. Phys. 14, 1938 (1973); M. A. Nielsen and D. Petz, Quant. Inf. Comput. 5, 507 (2005). Volume 5 Issue 6, September 2005 [6] A. Wehrl, Rev. Mod. Phys. 50, 221 (1978). [7] A. R´enyi, “On measures of information and entropy”, Proc. of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, pp. 547-561 (1960). [8] J. C. Baez, arXiv:1102.2098 (2011). [9] R. Alicki and M. Fannes, arXiv:quant-ph/0407033 (2003); M. M. Wolf and J. Eisert, New J. Phys. 7, 93 (2005); M. Mosonyi and F. Hiai, IEEE Trans. Inf. Th. 57, 2474 (2011). [10] R. K¨onig, R. Renner, and C. Schaffner, IEEE Trans. Inf. Th. 55, 4337 (2009); O. C. O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, New J. Phys. 13, 053015 (2011) [11] F. Franchini, A. R. Its, and V. E. Korepin, J. Phys. A: Math. Theor. 41, 025302 (2008); N. Datta and R. Renner, IEEE Trans. Inf. Th. 55, 2807 (2009); S. M. Giampaolo, S. Montangero, F. Dell’Anno, S. De Siena, F. Illuminati, arXiv:1208.0735 (2012). [12] G. Vidal, J. Mod. Opt. 47, 355 (2000). [13] J. S. Kim and B. C. Sanders, J. Phys. A: Math. Theor. 43, 445305 (2010). [14] T. J. Osborne, Quant. Inf. Comput. 7, 209 (2007); R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [15] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000). [16] T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006). [17] M. F. Cornelio and M. C. de Oliveira, Phys. Rev. A 81, 032332 (2010). [18] G. Adesso, F. Illuminati, and S. De Siena, Phys. Rev. A 68, 062318 (2003). [19] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005) [20] N. Cerf, G. Leuchs, and E. S. Polzik (eds.), Quantum Information with Continuous Variables of Atoms and Light (Imperial College Press, London, 2007). [21] G. Adesso and F. Illuminati, J. Phys. A: Math. Theor. 40 7821, (2007). [22] N. Schuch, J. I. Cirac, and M. M. Wolf, Commun. Math. Phys. 267, 65 (2006). [23] M. M. Wolf, G. Giedke, and J. I. Cirac, Phys. Rev. Lett. 96, 080502 (2006). [24] J. Eisert and M. M. Wolf, “Gaussian quantum channels”, pp. 23-42 in [20]. [25] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 671 (2012). [26] M. B. Hastings, Nature Phys. 5, 255 (2009). [27] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, and J. H. Shapiro, Phys. Rev. A 70, 032315 (2004). [28] A. Serafini, J. Eisert, and M. M. Wolf, Phys. Rev. A 71, 012320 (2005). [29] V. Giovannetti and S. Lloyd, Phys. Rev. A 69, 062307 (2004); V. Giovannetti, S. Lloyd, L. Maccone, J. H. Shapiro, and B. J. Yen, Phys. Rev. A 70, 022328 (2004). [30] S. Lloyd, V. Giovannetti, L. Maccone, N. J. Cerf, S. Guha, R. Garcia-Patron, S. Mitter, S. Pirandola, M. B. Ruskai, J. H. Shapiro, and H. Yuan, arXiv:0906.2758v3 (2010); R. GarciaPatron, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, and N. J. Cerf, Phys. Rev. Lett. 108, 110505 (2012). [31] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 70, 022318 (2004). [32] M. M. Wolf, G. Giedke, O. Kr¨uger, R. F. Werner, and J. I. Cirac,

Phys. Rev. A 69, 052320 (2004). [33] G. Adesso and F. Illuminati, Phys. Rev. A 72, 032334 (2005). [34] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). [35] L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). [36] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, arXiv:1112.6238v2 (2012); to appear in Rev. Mod. Phys. [37] G. Adesso, and A. Datta, Phys. Rev. Lett. 105, 030501 (2010). [38] P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105, 020503 (2010). [39] G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006). [40] T. Hiroshima, G. Adesso, and F. Illuminati, Phys. Rev. Lett. 98, 050503 (2007). [41] G. L. Giorgi, Phys. Rev. A 84, 054301 (2011). [42] A. Streltsov, G. Adesso, M. Piani, and D. Bruss, Phys. Rev. Lett. 109, 050503 (2012). [43] A. S. Holevo and R. F. Werner, Phys. Rev. A 63, 032312 (2001). [44] A. Serafini, F. Illuminati, and S. De Siena, J. Phys. B: At. Mol. Opt. Phys. 37, L21 (2004). [45] V. Buˇzek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995). [46] C. E. Shannon, Bell Syst. Tech. J. 27, 623 (1948). [47] R. L. Stratonovich, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 8, 116 (1965) [Probl. Inf. Transm. 2, 35 (1966)]. [48] See the Supplemental Material for additional explicit expressions and proofs [EPAPS XXX]. [49] R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1990). [50] J. Williamson, Am. J. Math. 58, 141 (1936). [51] G. Giedke, M. M. Wolf, O. Kr¨uger, R. F. Werner, and J. I. Cirac, Phys. Rev. Lett. 91, 107901 (2003). [52] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 73, 032345 (2006). [53] J. Fiur´asˇek and L. Miˇsta, Jr., Phys. Rev. A 75, 060302(R) (2007). [54] J. Fiur´asˇek, Phys. Rev. Lett. 89, 137904 (2002); G. Giedke and J. I. Cirac, Phys. Rev. A 66, 032316 (2002). [55] A. Datta, arXiv:1003.5256v2 (2011); A. Datta, Phd Thesis (The University of New Mexico, 2008), arXiv:0807.4490; P. J. Coles, arXiv:1101.1717 (2011). [56] P. Hayden, R. Jozsa, D. Petz, and A. Winter, Commun. Math. Phys, 246, 359, (2004). [57] In particular, the only classically correlated Gaussian states ρAB (with D2 (ρA|B ) = 0) are those that saturate the strong subadditivity inequality (10) on a tripartite Gaussian state ρABC where C is an ancillary system realizing the measurements on B, as characterized in [37, 58]. [58] D. Petz and J. Pitrik, J. Math. Phys. 50, 113517 (2009). [59] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004). [60] F. F. Fanchini, M. F. Cornelio, M. C. de Oliveira, and A. O. Caldeira, Phys. Rev. A 84, 012313 (2011). [61] A. Botero and B. Reznik, Phys. Rev. A 67, 052311 (2003); G. Giedke, J. Eisert, J. I. Cirac, and M. B. Plenio, Quant. Inf. Comp. 3, 211 (2003). [62] The other currently known Gaussian entanglement measures that satisfy a monogamy inequality, namely the so-called contangle [39] and Gaussian tangle [40], do not enjoy this property: the genuine tripartite entanglement in those cases had to be defined by adding a minimum ad hoc over the three possible residual entanglements, each dependent on the focus mode used in the decomposition [52]. The GR2 residual entanglement behaves instead truly like the residual tangle for three pure qubits, which is invariant under permutations [15] being related to a hyperdeterminant [63].

6 [63] A. Miyake, Phys. Rev. A 67, 012108 (2003). [64] M. Piani and G. Adesso, Phys. Rev. A 85, 040301(R) (2012). [65] G. L. Giorgi, B. Bellomo, F. Galve, and R. Zambrini, Phys. Rev. Lett. 107, 190501 (2011). [66] A similar property is satisfied by the geometric discord [67] for pure three-qubit states [42], but with the residual geometric discord being partition-dependent and different from the residual tangle [15] in such a case. In general, measures of nonclassical correlations other than entanglement cannot be monogamous for all mixed multipartite states of arbitrary-dimensional systems [42]. [67] B. Daki´c, C. Brukner, and V. Vedral, Phys. Rev. Lett. 105, 190502 (2010); S. Luo and S. Fu, Phys. Rev. A 82, 034302 (2010). [68] G. Adesso, S. Ragy, and D. Girolami, Class. Quantum Grav. 29, 224002 (2012). [69] M. G. Genoni and M. G. A. Paris, Phys. Rev. A 82, 052341 (2010); J. S. Ivan, M. S. Kumar, and R. Simon, Quant. Inf. Process. 11, 853 (2012). [70] L. Miˇsta Jr., R. Tatham, D. Girolami, N. Korolkova, and G. Adesso, Phys. Rev. A 83, 042325 (2011). [71] R. Tatham, L. Miˇsta Jr., G. Adesso, and N. Korolkova, Phys. Rev. A, 85, 022326 (2012). [72] L. Miˇsta Jr., Phys. Rev. A 73, 032335 (2006). [73] N. J. Cerf, O. Kr¨uger, P. Navez, R. F. Werner, and M. M. Wolf, Phys. Rev. Lett. 95, 070501 (2005). [74] W. Karush, Minima of Functions of Several Variables with Inequalities as Side Constraints, M.Sc. Dissertation, Dept. of Mathematics, University of Chicago (1939). [75] H. W. Kuhn, and A. W. Tucker, “Nonlinear programming”, Proceeding of 2nd Berkeley Symposium, Berkeley, University of California press, pp. 481-492 (1951).

1 Supplemental Material

Measuring Gaussian Quantum Information and Correlations Using the R´enyi Entropy of Order 2 Gerardo Adesso, Davide Girolami, and Alessio Serafini

Appendix A: Derivation of formulae (6) and (7)

For the sake of completeness, here we provide the reader with an explicit derivation of Eqs. (6) and (7), on which the operational interpretation of the R´enyi-2 entropy is ultimately based. Direct substitution of Eq. (4) into (6) yields " # Z
1 1 exp − ξT γ−1 ξ ξT γ−1 ξ + n ln π + ln (detγ) d2n ξ . H(Wρ ) = (A1) p 2 R2n πn det γ

The first term in the square bracket is conveniently handled by performing the integration in phase space coordinates ξ that diagonalize the symmetric, positive definite matrix γ (whose eigenvalues will be denoted by γ j ), and by noting that, for any γ > 0, one has Z +∞ 1 γ (A2) exp(−γ−1 x2 )x2 dx = . √ πγ −∞ 2 Hence, by applying (A2) as well as the normalization of the Wigner function to (A1), one obtains Eq. (6): H(Wρ ) =

2n X γj + n ln π + S 2 (ρ) = n + n ln π + S 2 (ρ) . 2γ j j=1

Next, the substitution of (4) into (7) leads to: # " Z
2n 1 1 T −1 T −1
exp − ξ γ1 ξ ξ γ2 ξ + n ln π + ln detγ2 d ξ . H(Wρ1 kWρ2 ) = −H(Wρ1 ) + p 2 R2n πn det γ1

(A3)

(A4)

Once again, it is expedient to carry out the integration in phase space coordinates where γ2 is diagonal (with eigenvalues γ2, j , while the entries of γ1 will be denoted by γ1, jk ). Then, Eq. (A2) and the normalization of Wρ1 lead to (7) ! ! 2n X γ1, j j det γ2 det γ2 1 1 1 H(Wρ1 kWρ2 ) = ln −n+ − n + tr(γ1 γ−1 = ln 2 ), 2 det γ1 2γ 2 det γ 2 2, j 1 j=1

(A5)

where the last step follows from the invariance of the quantity tr(γ1 γ−1 2 ) under changes of basis. Appendix B: Explicit formulae for two-mode Gaussian correlations

Standard form.— The CM γAB of any two-mode Gaussian state ρAB can be transformed, by means of local unitary operations, into a standard form of the type [21]    a 0 c+ 0       0 a 0 c−    =  γAB (B1)  c+ 0 b 0  ,   0 c− 0 b i  h   where a, b ≥ 1, a2 − 1 b2 − 1 − 2c− c+ − abc2+ + c2− −ab + c2+ ≥ 0, and we can set c+ ≥ |c− | without losing any generality. These conditions ensure that the uncertainty relation γAB ≥ iω⊕2 is verified, which is a bona fide requirement for the CM γAB to be associated with a physical √ Gaussian state in a two-mode infinite-dimensional Hilbert space [21]. Recall that for pure Gaussian states, b = a, c+ = −c− = a2 − 1.   γ ς =  TA AB ς AB γB

2 All the formulae presented in the following will be written explicitly for standard form CMs for simplicity. However, they can be recast in a locally invariant form by expressing them in terms of the four local symplectic invariants of a generic twomode Gaussian state [44], I1 = det γA , I2 = det γB , I3 = det ς AB , I4 = det γAB . This is accomplished by inverting the relations I1 = a2 , I2 = b2 , I3 = c+ c− , I4 = (ab − c+ )(ab − c− ) so that the {I j }4j=1 appear explicitly in the formulae below [21]. The obtained expressions would then be valid for two-mode CMs in any symplectic basis, beyond the standard form. GR2 entanglement.— For generally mixed two-mode Gaussian states ρAB , the R´enyi-2 entanglement measure E2 (ρA:B ), defined by Eq. (11) in the main text, admits the following expression if the CM γAB is in standard form [32, 33], ! 1 E2 (ρA:B ) = ln inf mθ (a, b, c+, c− ) , (B2) θ∈[0,2π] 2 with qh  ih i2 mθ (a, b, c+, c− ) = 1 + c+ (ab − c2− ) − c− + cos θ a − b(ab − c2− ) b − a(ab − c2− ) v  u u i2 h  u  t         c+ (ab − c2− ) + c−   2 ab − c2− a2 + b2 + 2c+ c− + sin θ a2 − b2 ×  1− h ih i    a − b(ab − c2− ) b − a(ab − c2− )   −1 h        i   cos θ 2abc3− + a2 + b2 c+ c2− + 1 − 2b2 a2 + b2 c− − ab a2 + b2 − 2 c+    − . qh  ih i    2 2  a − b(ab − c− ) b − a(ab − c− )

(B3)

The optimal θ minimizing Eq. (B3) can be found numerically for general two-mode Gaussian states [32], and analytically for relevant subclasses of states (including symmetric states [51], squeezed thermal states, and so-called GLEMS—Gaussian states of partial minimum uncertainty [33]). GR2 classical correlations and discord.— For generally mixed two-mode Gaussian states ρAB , the R´enyi-2 measures of oneway classical correlations J2 (ρA|B) and quantum discord D2 (ρA|B ), defined by Eqs. (13) and (14) in the main text, respectively, admit the following expression if the CM γAB is in standard form [37] ! 1 Π (B4) J2 (ρA|B ) = ln a − ln inf det γ˜ A λ,ϕ , λ,ϕ 2 !
1 1 Π (B5) D2 (ρA|B ) = ln b − ln det γAB + ln inf det γ˜ A λ,ϕ , λ,ϕ 2 2

with λ ∈ (0, ∞), ϕ ∈ [0, 2π], and Π det γ˜ A λ,ϕ

=

      2a2 (b + λ)(1 + bλ) − a c2+ + c2− 2bλ + λ2 + 1 + 2c2+ c2− λ + a c2+ − c2− λ2 − 1 cos(2ϕ) 2(b + λ)(1 + bλ)

.

(B6)

The optimal values of λ and ϕ minimizing Eq. (B6) can be found analytically1 for all two-mode Gaussian states [37]. In particular, for standard form CMs, one gets          c2+  2 2 2 2 2 2 2 2  a a − , if ab c − c a + bc ab c − c a + bc < 0;  − + − + − +  b  Π √ 2 (B7) inf det γ˜ A λ,ϕ =  2 2 2 2 2 2 2 2 2  2|c− c+ | (a(b −1)−bc− )(a(b −1)−bc+ )+(a(b −1)−bc− )(a(b −1)−bc+ )+c− c+  λ,ϕ  , otherwise.  2 −1 2 b ( ) Inserting Eq. (B7) into Eqs. (B4,B5) one gets closed formulae for the one-way GR2 classical correlations and for the GR2 discord of general two-mode Gaussian states.

1

In general, given that S2 (ΛΠ B ) is concave on the convex hull of the set of Gaussian states —having inherited the property from the Shannon entropy of the corresponding Wigner distributions, via Eq. (6)— and given that every Gaussian state admits a convex decomposition into pure Gaussian states (see [28, 37, 70] for more details), it follows that the optimizations in Eqs. (13,14) for an arbitrary number of modes are always achieved by pure Gaussian seed elements, i.e., det(ΓΠ B ) = 1. This simplifies considerably the evaluation of GR2 one-way classical correlations and discord.

3 Appendix C: Proof that GR2 classical correlations exceed GR2 entanglement

Here we prove that the inequality J2 (ρA|B ) ≥ E2 (ρA:B ) ,

(C1)

holds for all two-mode Gaussian states ρAB . This is a central step in the proof of the general monogamy inequality for the GR2 entanglement measure, reported in Theorem 2 in the main text (see also [16]). Without loss of generality, we can assume the CM γAB in standard form. We observe from Eqs. (B2,B4) that setting any value of θ in (B3) provides an upper bound to E2 , while setting any value of λ, ϕ in (B6) provides a lower bound to J2 ,
1 ln mθ (a, b, c+, c− ) , 2  1  Π J2 (ρA|B ) ≥ ln a − ln det γ˜ A λ,ϕ , 2 E2 (ρA:B ) ≤

We will then set θ = π (similarly to what done = 0 (corresponding to heterodyne detections on B [38]),  in [40]), and λ = 1, ϕ
Π and proceed to prove that ln a − 12 ln det γ˜ A 1,0 ≥ 12 ln mπ (a, b, c+, c− ) , which implies (C1). We then want to prove the inequality 1 + F 2 /G ≤ J, where qh ih i a − b(ab − c2− ) b − a(ab − c2− ) , F = c+ (ab − c2− ) − c− +         2abc3− + a2 + b2 c+ c2− + 1 − 2b2 a2 + b2 c− − ab a2 + b2 − 2 c+   2 2 2 G = 2 ab − c− a + b + 2c+ c+ − , qh ih i a − b(ab − c2− ) b − a(ab − c2− ) J =

a2 (1 + b)2 . (a + ab − c2+ )(a + ab − c2− )

This is equivalent to show that, defining K = K(a, b, c+, c− )  J − F 2 /G − 1, we have min(a,b,c+ ,c− ) K = 0. Let us search for the absolute minimum of K. This function has no singularities apart from the trivial case of two-mode vacuum states (a = b = 1, c± = 0) for which we know that lim(a,b,c+ ,c− )→(1,1,0,0) K = 0. Thus, we can focus on the stationary points of K and on the values at the boundaries of its domain. We adopt the Karush-Kuhn-Tucker method [74, 75]. Given constraints fi = fi (a, b, c+, c− ) ≤ 0 and associated multipliers Λi ≥ 0, the coordinates (a∗ , b∗ , c∗+ , c∗− , Λ∗i ) associated to a minimum of K satisfy the following conditions:     P ∂ ∂ ∗ ,b∗ ,c∗ ,c∗ ,Λ∗ ) = ∇, (a) ∇, ∂Λ L(a, b, c , c , Λ )| (K + i Λi fi )|(a∗ ,b∗ ,c∗+ ,c∗− ,Λ∗i ) = 0 + − i (a + − ∂Λ i i i (b) Λi fi (a∗ , b∗ , c∗+ , c∗− ) = 0,

Λi ≥ 0 ∀i

(c) there exists a vector ~x = {xi } such that (∇K −

P

i

xi ∇ fi )|(a∗ ,b∗ ,c∗+ ,c∗− ) = 0.

Conditions (a) and (b) are necessary, while condition (c) is sufficient. In this case, we have f1 = a2 + b2 + 2c+ c− − 1 − (ab − c2+ )(ab − c2− ) , f2 = −a2 − b2 + 2c+ c− + 1 + (ab − c2+ )(ab − c2− ) , f3 = −c+ − c− . A bit of algebra reveals that condition (a) is verified by the solutions to the following system                     

∂K ∂ f1 ∂K ∂ f1 = ∂a ∂b ∂b ∂a ∂K ∂K = + ∂c− ∂c+

∂ f1 ∂K ∂ f1 ∂a ( ∂c− − ∂c+ ) ∂ f1 ∂a

(C2) + Λ2

! ∂ f1 ∂ f2 ∂ f1 ∂ f2 . + − − ∂c+ ∂c+ ∂c− ∂c−

It follows that, among the stationary points, the absolute minimum of K is again reached at a = b = 1, c± = 0, yielding K = 0: it is immediate to verify that the conditions (b) and (c) are satisfied as well at (1, 1, 0, 0). Finally, we have to check the values of K

4 √ at the boundaries of its domain. The relevant cases are a → ∞, b → ∞, c+ = ab − 1, c+ = |c− |, c− = 0, and somewhat tedious yet straightforward analysis reveals that K ≥ 0 always holds for all those cases. We therefore conclude that the function K is always nonnegative, which proves Eq. (C1), thereby proving that the entanglement measure E2 satisfies the general monogamy inequality [Eq. (16) of the main text] for all n-mode Gaussian states of continuous variable quantum systems. 

Appendix D: Residual tripartite GR2 entanglement for pure three-mode Gaussian states

Up to local unitaries, the CM γA1 A2 A3 of any pure three-mode Gaussian state can be written in the following standard form [52]

γ A1 A2 A3

where

      =     

a1 0 c+3 0 c+2 0

0 a1 0 c−3 0 c−2

c+3 0 a2 0 c+1 0

0 c−3 0 a2 0 c−1

c+2 0 c+1 0 a3 0

0 c−2 0 c−1 0 a3

          

(D1)

p [(ai − 1)2 − (a j − ak )2 ][(ai + 1)2 − (a j − ak )2 ] ± [(ai − 1)2 − (a j + ak )2 ][(ai + 1)2 − (a j + ak )2 ] , = √ 4 a j ak and |a j − ak | + 1 ≤ ai ≤ a j + ak − 1 ,

c±i

p

with {i, j, k} being all possible permutations of {1, 2, 3}. The GR2 entanglement in the two-mode reduced state with CM γAi A j is E2 (ρAi :A j ) =

1 ln gk , 2

(D2)

with [33] q    1, if a ≥ a2i + a2j − 1;  k    q   β    if αk < ak < a2i + a2j − 1;  2, gk =  8a    2k   2 2     ai − a j        a2 − 1  , if ak ≤ αk .

(D3)

k

Here we have set

αk =

v u u u t

q 2(a2i + a2j ) + (a2i − a2j )2 + |a2i − a2j | (a2i − a2j )2 + 8(a2i + a2j ) 2(a2i + a2j )

,

√ β = −1 + 2a21 + 2a22 + 2a23 + 2a21 a22 + 2a21 a23 + 2a22 a23 − a41 − a42 − a43 − δ , δ = (−1 + a1 − a2 − a3 )(1 + a1 − a2 − a3 )(−1 + a1 + a2 − a3 )(1 + a1 + a2 − a3 ) × (−1 + a1 − a2 + a3 )(1 + a1 − a2 + a3 )(−1 + a1 + a2 + a3 )(1 + a1 + a2 + a3 ) . The residual GR2 entanglement, with respect to the focus mode Ai , is E2 (ρAi :A j :Ak ) = E2 (ρAi :A j Ak ) − E2 (ρAi :A j ) − E2 (ρAi :Ak )   1  a2i   . ln  = 2 gk g j

(D4)

5 In general, this expression is dependent on the choice of the focus mode. However, let us consider the relevant case of a fully inseparable three-mode pure Gaussian state, for which entanglement is nonzero for all global splittings and for all reduced two-mode bipartitions, E2 (ρAi :A j Ak ) > 0, E2 (ρAi :A j ) > 0, ∀{i, j, k}. In our parametrization, this occurs when [33] |ai − a j | + 1 < ak
αk ∀k = 1, 2, 3. In this case, exploiting Eq. (D3), the residual GR2 entanglement becomes  2 2 2 1  64ai a j ak  E2 (ρAi :A j :Ak ) = ln  (D6)  , 2 β2 which is manifestly invariant under mode permutations, as reported in Eq. (18) in the main text. This symmetry is broken on states for which some of the reduced two-mode bipartitions become separable.