Maximal Entanglement Achievable By Controlled Dynamics - Www.oloscience.com

Maximal entanglement achievable by controlled dynamics Alessio Serafini1 and Stefano Mancini2

arXiv:0910.2205v1 [quant-ph] 12 Oct 2009

1

Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom 2 Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy (Dated: October 13, 2009) We consider the feedback control of quantum systems comprised of any number of bosonic degrees of freedom. We derive a general upper bound for the logarithmic negativity achievable, at steady state, with continuous Gaussian measurements on the environment and linear driving on the system. Our results apply to rotating wave system-bath couplings and to any quadratic system’s Hamiltonian. Furthermore, we apply this upper bound to parametric processes, show it to be tight, and compare it to feedback strategies limited to local measurements. PACS numbers: 03.67.Bg, 02.30.Yy, 42.50.Dv

The field of quantum control is central in the current rise of quantum technologies [1, 2]. In particular, the control of the coherent resources of quantum states is of crucial practical interest. Most valuable, and delicate, among such resources is certainly quantum entanglement, whose control is a primary requisite for quantum information and communication [3, 4]. In this paper, we focus on bosonic quantum systems subject to quadratic Hamiltonians and losses, and derive bounds on the maximal entanglement achievable, between specific bipartitions, through continuous feedback based on general Gaussian measurements and linear driving [5]. The class of dynamics and feedback strategies covered in our study is of great practical relevance in quantum optics, and is applicable to more general continuous variable systems. Because entanglement is not a linear figure of merit in the quantum state’s parameters, one cannot address the questions we are about to tackle with standard optimisation tools, like semi-definite programming [6], but rather requires the more detailed, specific analysis we shall present. Notation – We consider systems of N degrees of freedoms described by pairs of canonical operators:
⊤ definˆ ˆ ˆ ˆ ing a vector of operators x ˆ = q , p , ..., q , p , one has 1 1 N N i h xˆ j , xˆ k = iΩ jk , where Ω is the (2N) × (2N) symplectic     form: Ω jk = δ j+1,k 1 − (−1) j /2 − δ j,k+1 1 + (−1) j /2, in √ terms of Kroneker deltas δ j,k . Also, a j = (xˆ j + ipˆ j )/ 2. For a system with such a phase-space structure we can define “Gaussian states” as the states with a Gaussian Wigner function. These states are completely determined by the vector of means hˆxi, and by the covariance  matrix (CM) σ, with entries σ jk = h∆xˆ j ∆xˆ k i + h∆xˆ j ∆xˆ k i , ˆ The – always necwhere ∆oˆ = (oˆ − hˆoi) for operator o. essary – Robertson-Schrodinger ¨ uncertainty relation is also sufficient for Gaussian states to be physical [7]: σ + iΩ ≥ 0 .

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We will consider Hamiltonians Hˆ that are at most of the second-order in xˆ , so that their resulting free evolutions

are affine in phase-space: Hˆ = (1/2)ˆx⊤Hxˆ − xˆ ⊤ ΩBu(t), where the “Hamiltonian matrix” H is real and symmetric and B is real. The second term of Hˆ is a ‘linear driving’ proportional to a time-dependent input u(t): this term will describe the control exerted over the system. The system is considered to be open and such that each degree of freedom has its own channel to interact with the environment. Though thermal noise can also be treated along the lines we will present here, in this study we specialise for simplicity to pure losses, which are the main source of decoherence in quantum optical settings. We will thus assume a beam splitter-like (“rotating wave”) interaction a j b†j + a†j b j between each mode and the associated mode of the bath, with ladder operator b j . Under the conditions set out above, the first moments of the canonical operators evolve according to dhˆxi/dt = Ahˆxi + Bu(t), while the second moments obey dσ/dt = Aσ + σA⊤ + 1.

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Here, the “drift matrix” A = (ΩH−1)/2, and 1 is the identity matrix with dimension clear from the context. We will only address stable systems, for which (A + AT ) < 0. Note that, for Gaussian states, these equations describe the complete dynamics of the system. As customary in the context of feedback control, we will now assume that the degrees of freedom of the environment can be continuously monitored on time-scales which are short with respect to the system’s response time [8]. Because of the rotating wave interaction between system and environment, instantaneous Gaussian measurements on the environment degrees of freedom   ⊤ results into measuring the operators aˆ 1 + aˆ † Υ , where the vector aˆ = (a1 , . . . , aN )⊤ contains all the annihilation operators of the system, and the complex matrix Υ parametrises the Gaussian measurement. In turn, Υ defines the so called “unravelling matrix” U, given by ! 1 1 + Re [Υ] Im [Υ] . (3) U≡ Im [Υ] 1 − Re [Υ] 2 The only conditions on Υ are that U be symmetric and

2 positive semi-definite. The outcome of the measurements on the environment is recorded as a “current” y = √ 1/2 ¯ , where C = 2U Chˆxi+ dw C and C¯ jk = (δ2 j−1,k +δ2(j−N),k ) dt for j, k ∈ [1, . . . , 2N]. Finally, dw is a vector of real Wiener increments satisfying dwdw⊤ = 1dt. The conditional evolution of the moments under such continuous measurements can also be derived by standard techniques (Itoˆ calculus). It amounts to a diffusive equation with a stochastic component for the first moments hˆxi, and to a deterministic Riccati equation for the second moments [6]. In our reasonings to follow, we will not make use of the details of such equations directly. We will be interested in stable systems, and will determine the maximal entanglement achievable at steady state. Hence, all we need to remark is that the set of CMs {σ∞ } that are stabilising solutions [9] of the Riccati equation for the second moments can be characterised as follows [6]: Aσ ∞ + σ ∞ AT + 1 ≥ 0 .

Lemma 1 (Bound on smallest symplectic eigenvalue) The smallest partially transposed symplectic eigenvalue ν˜ − of a generic CM σ is bounded from below as follows ν˜ 2− ≥ λ↑1 λ↑2 ,

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λ↑1 and λ↑2 being the two smallest eigenvalues of σ. Next, the uncertainty principle entails: Lemma 2 (Uncertainty relation for CMs’ eigenvalues) Let { λ↑j } and { λ↓j } be, respectively, the 2N increasinglyordered and decreasingly-ordered eigenvalues of an N-mode CM σ. Then one has: λ↑j λ↓j ≥ 1 for 1 ≤ j ≤ N.

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As an immediate corollary of Lemma 2, one obtains (4)

Together with Inequality (1), this relationship completely determines the set of stabilising solutions of our conditional dynamics. The final ingredient to be added is the possible dependence of the linear drive u(t) on the history of the measurement record y(s) for s < t, which affects both first and second moments of the unconditional, ‘average’, evolution (whereas second moments of the conditional states are unaffected by the linear driving), and closes the control loop. We will denote the unconditional state by ̺. Note that, for our class of dynamics, ̺ is a statistical mixture of states with the same conditional CM σ ∞ , obeying Inequality (4), and varying first moments. For Gaussian states, this implies that ̺ can be obtained from a Gaussian state ̺0 with CM σ ∞ and vanishing first moments by local operations and classical communication alone: ̺ = L(̺0 ), where L is some LOCC map. The typical aim of control over some time interval is to minimise the expected value of a cost function [1, 9]. Our cost function will be the entanglement of Gaussian multi-mode steady states for bipartitions of 1 versus (N − 1) modes and ‘bi-symmetric’ bipartitions (i.e., invariant under the permutation of local modes). Such an entanglement can be quantified by the logarithmic negativity EN , which is in turn determined, for a Gaussian state with CM σ, as − log2 ν˜ − , where ν˜ 2− is the smallest ˜ Ω ˜ T ), being Ω ˜ the (skew-symmetric) eigenvalue of (σ Ωσ partial transposition of Ω [10, 11]. Clearly, ν˜ − is not a quadratic cost function (i.e., it is not linear in σ). This is why, albeit dealing with linear systems with Gaussian noise, we cannot resort to optimisation methods mutuated from classical LQG control problems [6]. General results – Hereafter, we present our main findings as three lemmas leading to a final proposition. Proofs of these statements may be found in appendix. Our investigation starts off from a corollary of Ref. [12]:

λ↑1 λ↑2 ≥

1 λ↓1 λ↓2

.

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Lemma 3 (Bound on eigenvalues of steady state CMs) Let σ ∞ be a conditional CM at steady state obtained under continuous Gaussian measurements, pure losses and a Hamiltonian matrix H. The product of the two largest eigenvalues λ↓1 and λ↓2 of σ ∞ is bounded as follows: λ↓1 λ↓2 ≤

1 α↑1 α↑2

,

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where {α↑j } are the (strictly positive) eigenvalues of (−A − A⊤)

in increasing order, and A = 12 (ΩH − 1).

The chain of Inequalities (5), (7) and (8) leads to ν˜ 2− ≥ α↑1 α↑2 ,

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which constrains the maximal logarithmic negativity achievable for states conditioned by Gaussian measurements. Further, and more generally, one has: Proposition 1 (Maximal unconditional entanglement) The logarithmic negativity EN (̺) of any 1 versus (N − 1) modes or bi-symmetric bipartition of an unconditional steady state achievable by continuous Gaussian feedback and linear driving is bounded by:   1 EN (̺) ≤ max 0, − log2 (α↑1 α↑2 ) . 2

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Applications – Clearly, the eigenvalues {α↑j } can be analytically or numerically determined for general quadratic Hamiltonians, describing a wide variety of systems of practical interest (bosonic atoms, trapped ions, nanomechanical resonators and Josephson junctions, just to

3 mention a few). Here, we will focus on yet another interesting case where our results apply and can be handled analytically: the case of parametric interactions, which are the state of the art technology to generate continuous variable entanglement in quantum optics. The parametric interaction between modes j and k is described by the Hamiltonian χ(xˆ j pˆk + pˆ j xˆ k ) [13]. We will assume equal interaction strengths χ between any pair of modes, consider a (m+n)-mode bipartition, and describe analytically the scaling of the control of the entanglement with the number of modes m and n (we have m + n = N). In point of fact, we will see that our bounds in this case are tight, and yield the actual optimal entanglement achievable by Gaussian filtering. Due to the symmetry of the system under the exchange of any two modes, the entanglement between the m- and the n-modes subsystems can be reduced to two-mode entanglement [14]: a local symplectic transformation exists that turns the ma¯ plus trix A into an equivalent two-mode drift matrix A, a direct sum of decoupled single-mode matrices that are irrelevant for the entanglement. The matrix A¯ reads: √   mnχ 0   (m − 1)χ √0    1  mnχ 0 −(m − 1)χ 0 −   − . A¯ =  √ mnχ 0 (n − 1)χ 0  2  √   0 − mnχ 0 −(n − 1)χ (11) 1 . For the system to be stable one must require: χ < 2(N−1) As A is symmetric and invertible, the ‘free’ steady state CM σ f can be promptly determined from Eq. (2): σ f = −A−1 /2. Its smallest partially transposed symplectic eigenvalue, which determines the asymptotic entanglement in the absence of control, is given by 1/[(1 + 2χ)(1 + 2(N − 1)χ)]. Instead, the bound of Inequality (9) for any steady state CM σ ∞ with Gaussian feedback control reads ν˜ 2− ≥ (1 − 2χ) [1 − 2(N − 1)χ] .

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This lower bound is attained by the CM RT diag(α4 , α3 , α1 , α2 )R, where R is the orthogonal transformation that diagonalises A¯ and {α j } are the eigenvalues of −A¯ in increasing order. This solution also saturates the Inequalities (4) and (1). Both the free asymptotic entanglement and the optimal one under Gaussian filtering have thus been obtained analytically. Quite remarkably, in the symmetric situation considered, none of these quantities depends on the chosen bipartition, but only on the total number of modes involved: the same amount of entanglement can be obtained regardless of the bipartition considered. Note also that the ‘optimal unravelling’, that is the matrix U granting maximal entanglement, may also be straightforwardly derived from the optimal state [6]. Local control – Such an optimal entanglement is in general achieved by filtering the system through global measurements on the environment, as no restrictions were

assumed for the unravelling matrix U. This applies to situations where the environments of the two local subsystems can be combined before being measured (like, e.g., for a parametric crystal in a cavity). We intend now to provide a lower bound on the entanglement achievable under local control, where the environmental degrees of freedom pertaining to the separate subsystems cannot be combined, and compare it to the upper bound we obtained above. To this end, we will adopt direct (Markovian) feedback [5] and set u(t) = Fy(t). The unconditional evolution of the system is then described by dσ/dt = A′ σ + σA′T + D′ ,

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with drift and diffusion matrices modified as A′ = A¯ + BFC and D′ = 1 − CT FT BT − BFC + 2BFFT BT . We also choose a specific form of U and BF. Since in the free dynamics, governed by the drift matrix of Eq. (11), the quadratures p1 and p2 are less noisy than q1 and q2 , it is advantageous to measure locally p1 and p2 and drive with the respective currents the quadratures q2 and q1 . However, due to the possible asymmetry of the two subsystems for m , n, we have to consider different driving amplitudes µ1 and µ2 for their quadratures. All this √ corresponds to setting U = U = 1, 2(BF)24 = µ2 , 33 44 √ 2(BF)43 = µ1 , and all other entries of U and BF vanishing. We can then find the steady state solution of Eq. (13) as a function of the two feedback amplitudes µ1 and µ2 , and evaluate its logarithmic negativity. It turns out that the maximum logarithmic negativity at steady state is attained for µ2 = µ1 n/m. Hence, we are left with the entanglement depending on one parameter, over which we minimise numerically in the stable region, determined by (A′ + A′T ) < 0. As a case of study, we have considered a system of 6 modes and summarised the results in Fig. 1. Local control is very close to optimal global control in the case of a balanced bipartition. However, the more unbalanced the bipartition, the more degraded the control, although numerics indicate that infinite entanglement can always be retrieved close to instability. Conclusion – We have derived bounds on the entanglement achievable, at steady state and for various bipartitions, in multimode linear bosonic systems subjected to continuous feedback control. Our investigation not only applies to quantum information processing and state engineering, as shown, but also yields a technique for optimisation problems lying beyond the LQG scenario. Appendix – Proofs of mathematical statements. Henceforth, |vi will stand for a unit vector in the phase space Γ and hv| will be its dual under the Euclidean scalar product. Also, note that the uncertainty relation (1) is equivalent to the two following conditions [12, 15]: σ 1/2 ΩT σΩσ 1/2 ≥ 1 ,

and σ > 0 .

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Proof of Lemma 1. The squared symplectic eigenvalue ˜ T σ Ωσ ˜ 1/2 : ν˜ 2− is the smallest eigenvalue of the matrix σ 1/2 Ω

4

∼ ν2

infhw|vi=0 hw|K|wihv|K|vi = λ↑1 λ↑2 , where λ↑j is the j-th smallest eigenvalue of the positive matrix K.

1.0

−

0.8

0.6

0.4

0.2

0.0 0.00

0.02

0.04

0.06

0.08

χ

0.10

FIG. 1: Squared symplectic eigenvalue ν˜ 2− at steady state for a system of 6 modes (ν˜ − → 0 implies infinite entanglement). Green (lighter) curves depict ν˜ − in the absence of control (from top to bottom: 1:5, 2:4, and 3:3 modes bipartition); bleu (darker) curves refer to numerically optimised local feedback (from top to bottom: 1:5, 2:4, and 3:3 modes bipartition); the red curve is the analytical lower bound (12) achievable by global control.

˜ T σ Ωσ ˜ 1/2 |vi. For each |vi, one can ν˜ 2− = min|vi hv|σ1/2 Ω √ ˜ 1/2 |vi/ hv|σ|vi, such that define the unit vector |wi = Ωσ ˜ and hv|σ1/2 |wi = 0 (due to the antisymmetry of Ω) ν˜ 2− = minhv|σ|vihw|σ|wi ≥ min hv|σ|vihw|σ|wi = λ↑1 λ↑2 . |vi

|vi,|wi

The last equality is easily verified once hv|σ1/2 |wi = 0 and σ > 0 are enforced, and completes the proof. Proof of Lemma 2. Once √ again, for any |vi ∈ Γ one can define |wi = Ωσ 1/2 |vi/ hv|σ|vi, so that the Robertson Schrodinger ¨ Inequality (14) can be recast as hv|σ|vihw|σ|wi ≥ 1 ∀ |vi ∈ Γ. We will now denote by |v j i the eigenvectors corresponding to the increasingly ordered eigenvalues of σ: σ|v j i = λ↑j |v j i. Let us consider a vector |vi belonging to the subspace, which we shall denote Γk , spanned by the k smallest eigenvectors of σ {|v ji}, for j ≤ k. Clearly one has hv|σ|vi ≤ λ↑k . The inequality above then leads to λ↑k hw|σ|wi ≥ hv|σ|vihw|σ|wi ≥ 1

∀ |vi ∈ Γk ,

which must be satisfied by all the vectors |wi belonging to the k-dimensional linear subspace ΩΓk (defined as the subspace spanned by the k orthogonal vectors Ω|vk i): λ↑k hw|σ|wi ≥ 1 ∀|wi ∈ ΩΓk . By Poincar´e Inequality [16], a vector |wi must exist in ΩΓk for which hw|σ|wi ≤ λ↓k , such that λ↑k λ↓k ≥ 1. Proof of Lemma 3. This statement is a consequence of Inequality (4) applied to the two eigenvectors corresponding to λ↓1 and λ↓2 , and of the relationship:

Proof of Proposition 1. As seen previously, ̺ = L(̺0 ), where L is a LOCC operation and ̺0 a Gaussian state with a CM which is a stabilisinghsolution of (2). Thus EN (̺) = i ↑ ↑ EN (L(̺0 )) ≤ EN (̺0 ) ≤ max 0, − log2 (α1 α2 )/2 , where (9), the formula EN = − log(ν˜ − ) (holding for 1 − (N − 1) and bi-symmetric bipartitions), and the monotonicity of EN under LOCC [17] have been invoked.

We acknowledge financial support from the EU through the FET-Open Project HIP (FP7-ICT-221899).

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