Efficient Tomography of Quantum-Optical Gaussian Processes Probed with a Few Coherent States

PHYSICAL REVIEW A 88, 022101 (2013)

Efﬁcient tomography of quantum-optical Gaussian processes probed with a few coherent states

Xiang-Bin Wang,1,2,3,* Zong-Wen Yu,1,4 Jia-Zhong Hu,1 Adam Miranowicz,2,5 and Franco Nori2,6 1Department of Physics and State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China 2CEMS, RIKEN, Saitama 351-0198, Japan 3Jinan Institute of Quantum Technology, Shandong Academy of Information Technology, Jinan, China 4Data Communication Science and Technology Research Institute, Beijing 100191, China 5Faculty of Physics, Adam Mickiewicz University, PL-61-614 Poznan,´ Poland 6Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Received 2 October 2010; revised manuscript received 24 April 2013; published 1 August 2013) An arbitrary quantum-optical process (channel) can be completely characterized by probing it with coherent states using the recently developed coherent-state quantum process tomography (QPT) [Lobino et al., Science 322, 563 (2008)]. In general, precise QPT is possible if an infinite set of probes is available. Thus, realistic QPT of infinite-dimensional systems is approximate due to a finite experimentally feasible set of coherent states and its related energy-cutoff approximation. We show with explicit formulas that one can completely identify a quantum-optical Gaussian process just with a few different coherent states without approximations like the energy cutoff. For tomography of multimode processes, our method exponentially reduces the number of different test states, compared with existing methods.

DOI: 10.1103/PhysRevA.88.022101 PACS number(s): 03.65.Wj, 42.50.Dv

I. INTRODUCTION Any physical process can be described by a completely positive map ε. Such a process is fully characterized if the One of the basic problems of quantum physics is to evolution of any input state ρ is predictable: ρ = ε(ρ ). predict the evolution of a quantum system under certain in out in In general, QPT is very difficult to implement in high- conditions. For an isolated system with a known Hamiltonian, dimensional spaces, and, more challengingly, in an infinite- the evolution is characterized by a unitary operator determined by the Schrodinger¨ equation. However, the system may interact dimensional space, such as a Fock space [9,12]. Recently, with its environment, and the total Hamiltonian of the system Ref. [12] described QPT in a Fock space for continuous plus the environment is in general not completely known. The variable (CV) states. Two conclusions can be drawn [12]: (i) If evolution can then be regarded as a “black-box process” [1–3] the output states of all coherent input states are known, then one which maps the input state into an output state. An important can predict the output state of any input state; (ii) by taking the problem here is how to characterize an unknown process by photon-number-cutoff (or energy-cutoff) approximation, one testing the black box with some specific input states, which can then characterize an unknown process with a finite number is referred to as quantum process tomography (QPT) (for of different input coherent states (CSs). reviews, see Refs. [4,5]). It is an interesting question to identify an exact QPT with QPT can be understood as the tomography of a quantum a finite number of coherent states. If the process is completely channel since any physical operation describing the dynamics unknown, then QPT with a finite number of coherent states is of a quantum state can be considered as a channel [6]. In impossible. However, if some of the constraints of the quantum contrast, the goal of quantum state tomography (QST) is the process are known, then QPT can be simplified and, thus, reconstruction of an unknown state (i.e., its density matrix) by effective. Gaussian maps are the most common for quantum- a series of measurements on multiple copies of the state (for a optical processes. In this article, we show that if a certain review, see Ref. [4]). Both QPT and QST are essential tools in quantum process is known to be Gaussian, then an exact QPT quantum engineering and quantum-information processing. can be performed with only a few different coherent states. A few methods for QPT were developed, including the It is worth noting that there is an analogy between QPT standard QPT [1,2], ancilla-assisted QPT [3,7–9], direct char- and QST, especially for quantum-optical Gaussian processes acterization of quantum dynamics [10,11], and coherent-state (channels) [6,36] and Gaussian states [37,38]. This analogy QPT [12]. There are dozens of proposals and experimental re- can be seen, e.g., by comparing correlations between observ- alizations of QPT for systems with a few qubits. These include ables encoded in the covariance matrices, which completely the estimation of quantum-optical gates [8,11,13–19], liquid describe a Gaussian object (either a quantum state or process). nuclear-magnetic-resonance gates [20–22], superconducting Thus, tomographies of quantum Gaussian systems are effec- gates [23–28] (for a review, see Ref. [29]) and other solid-state tively finite-dimensional with their covariance matrix having gates [30–32], ion-trap gates [33,34], or the estimation of the a physical meaning analogous to a finite-dimensional density dynamics of atoms in optical lattices [35]. In contrast, there matrix. are only a very few experimental demonstrations of QPT for As has been shown in Ref. [3], QPT can be performed infinite-dimensional systems (see, e.g., Ref. [12]). with two-mode squeezed vacuum (TMSV) for any unknown process. However, TMSVs are not so easy to manipulate in practice, especially because this involves quantum tomography *[email protected] of entangled states, which is not an easy task.

Here, we show that based on existing results [3], by using be written as the standard quantum-optical Husimi Q representation, one ∗ + + ∗ (|ψψ|)b = aψ | |ψ a can perform QPT with only a few CSs without entangled ∗ ∗ + + ancillas for quantum-optical Gaussian processes. The method = tra(|ψ ψ |⊗I| |), (7) described here has several advantages. First, it presents | ∗ = ∗| and ψ a k ck k a is a single-mode state for mode a explicit formulas without any approximations, such as the (sometimes we omit the subscript a or b for simplicity). We photon-number-cutoff approximation. Second, it requires only obtain the output state a few different states to characterize a process, rather than all ∗ ∗ ∗ ∗ CSs. Third, for multimode Gaussian process tomography, the ρψ = aψ |ρ|ψ a = tra(|ψ ψ |⊗Iρ) number of input CSs increases polynomially with the number ∗ k ∗ k = tra[|ψ ({ck/q })ψ ({ck/q })|⊗Iq ] of modes, rather than exponentially. Fourth, it uses the Husimi = ∗ { k} | | ∗ { k} Q functions only, which is always well-defined for any state a ψ ( ck/q ) q ψ ( ck/q ) a. (8) without any higher-order singularities in the calculation. More explicit expressions can be obtained by using the Husimi The paper is organized as follows: We review the existing Q function. If the single-mode input state in mode b is a results about QPT based on entangled ancillas in Sec. II.In coherent state |α, the output state then becomes Sec. III,theQPTwithout ancillas is proposed for single-mode ∗ ∗ ∗ −1 −1 ∗ Gaussian processes. A simple illustrative example of the ρα =α |ρε|α =α |T (q) ⊗ Iq T (q) ⊗ I|α . (9) method is discussed in Sec. III A. A generalization of our QPT Note that the state |α∗ here is a single-mode coherent state in for a multimode case is presented in Sec. IV. We conclude in mode a. Using the property of T (q) and the definition of CSs, Sec. V. a|α∗=α∗|α∗, we easily find −1 ⊗ | ∗=N | ∗ II. ANCILLA-ASSISTED QUANTUM PROCESS T (q) I α q (α) α /q , (10) 2 2 TOMOGRAPHY where the factor Nq (α) = exp[−|α| (1 − 1/q )/2]/cq , and | ∗ | ∗ = First, we review the existing result of the ancilla-assisted α /q is a coherent state in mode a defined by a α /q ∗ | ∗ QPT with TMSV [3] to show some similarities but also crucial (α /q) α /q . Thus, the output state of mode b is 2 ∗ ∗ differences in comparison to our proposal of ancilla-free QPT, ρα =|Nq (α)| aα /q|q |α /qa. (11) which will be described in Sec. III. | = † † | Let |Z ,Z be a two-mode coherent state defined by A TMSV is defined by χ(q) cq exp(qa b ) 00 , where a b 2 (a,b)|Za,Zb=(Za,Zb)|Za,Zb, where Za,Zb are complex cq = 1 − q , and q is real. The (unnormalized) maximally entangled state here is amplitudes. Then, the Husimi Q function for q can be defined as ∞ + † † ∗ ∗ = | | | =lim exp(qa b )|00= |kk, (1) Qq (Za ,Zb ,Za,Zb) Za,Zb q Za,Zb , (12) q→1 k=0 and the corresponding density operator is the following where a† (b†) is the creation operator for mode a (b). Note that normally ordered operator entanglement is not required for the ancilla-assisted QPT, but it † † q = : Q (a ,b ,a,b):, (13) makes it more efficient. In particular, the use of the maximally q entangled states can make the QPT experimentally optimal which is simply the operator functional obtained by replacing ∗ ∗ † † with regard to perfect nonlocal correlations [8]. the variables (Za ,Zb ,Za,Zb) with (a ,b ,a,b)intheQ Assume now that the black box process acts only in mode function given by Eq. (12), analogously to Eq. (16). Therefore, b of the bipartite state |χ(q). After the process, we obtain a using Eq. (11) and the normally ordered form of q ,wehave two-mode state q . One can define the projection operator the following simple form for the Husimi Q function: ∗ 2 ∗ ∗ † Qρ (Zb ,Zb) =|Nq (α)| Q (α/q,Zb ,α /q,Zb) (14) T (q) = cq exp[(ln q)a a], (2) α q which has the property [40] of the output state ρα. Equations (11)–(14) are the explicit expressions of the output state for the input of any coherent † − † T (q)(a,a ) T 1(q) = (a/q,qa ). (3) state |α. According to Ref. [12], if we know the output states for all input CSs, then we know the output states of all states in |χ q The TMSV ( ) can be written as Fock space. In this approach, given any input state |ψ, we can |χ(q)=T (q) ⊗ I|+. (4) write it in its linear superposition form in the coherent-state basis, and then obtain the Q function of its output state by According to Eq. (4),wehave using Eq. (14). These results can be generalized for a multimode q = T (q) ⊗ IρεT (q) ⊗ I, (5) QPT. To apply the Jamiolkowski isomorphism [41], + + where ρε = I ⊗ ε(| |). Naturally, we consider k pairs of maximally entangled states, | += = −1 ⊗ −1 ⊗ each in modes a1,b1,a2,b2,...,ak,bk. Explicitly, ρε T (q) Iq T (q) I. (6) | + | + ···| + | + = † † | φ 1 φ 2 φ k. Here φ i limq→1 exp(qai bi ) 00 We now also formulate the output state of any single-mode indicates a maximally entangled state in modes ai ,bi . Sub- | { } = | | input state ψ( ck ) k ck k of mode b. Obviously it can spaces a and b each are now k mode. Any state ψ in subspace

022101-2 EFFICIENT TOMOGRAPHY OF QUANTUM-OPTICAL ... PHYSICAL REVIEW A 88, 022101 (2013)

† b can still be written in the form of Eq. (7), with the new The quadratic functional terms (R,R ,R0) in the exponent in + definitions for |ψ and | .UsingEq.(6), it is obvious that Eq. (18) are independent of u; these terms must be the same the output state of these k pairs of TMSV fully characterizes for the output states from any input CSs. Therefore, these can the process. be known by testing the map with one coherent state. Thus, we do not need to consider these terms below. III. ANCILLA-FREE GAUSSIAN PROCESS Now suppose that we test the process with six different CSs, TOMOGRAPHY WITH A FEW COHERENT STATES |αi , and i = 1,2,...,6. Assume also that the detected Husimi Q function of the output states is Now we present the main result of this paper, which is an † efficient tomography of Gaussian processes probed with only Q (Z∗,Z ) = exp(c + D + D + R + R† + R ), (20) ραi b b i i i 0 a few single-mode coherent states without the assistance of where D = d Z is the detected (hence known) linear term. ancillas. i i b We note that there are available efficient methods of Gaussian As shown in Ref. [12], if we only use CSs in the QST based on homodyne detection, which enable the estima- test, the tomography of an unknown process in Fock space tion of the Wigner function or, equivalently, the Husimi Q requires tests with all CSs. Though this problem can be function for Gaussian states [38]. According to Eq. (18),the solved by taking the photon-number-cutoff approximation, in Q function of the output state from the initial state |α of a quantum-optical process associated with intense light, one i mode b must be still needs a huge number of different CSs for the test. Here we show that the most important process in quantum optics, Q (Z∗,Z ) = Q (α ,Z∗,α ∗,Z ). (21) ραi b b ρε i b i b the Gaussian process [6,36], can be exactly characterized with Therefore, we can derive self-consistent equations by using only a few CSs in the test. the detected data from ρ and setting u = α in Eq. (18), A Gaussian process maps Gaussian states into Gaussian αi i = = states [39]. Therefore, the Husimi Q function of the operator Li Di ,cαi ci , (22) ρε must be Gaussian: = where Li , cαi are just Lu, cu, respectively, after setting u ∗ ∗ = + + † + + † + Qρε (Za ,Zb ,Za,Zb) exp(c0 L L S S S0), αi in Eqs. (18) and (19); Di and ci are known from tests. (15) Explicitly, where = + ∗ + Li ( b αi Xab αi Yab)Zb. (23) Za 1 Za L = ( , ) ,S= (Z ,Z )X , The first part of Eq. (22) gives a b Z 2 a b Z b b T K ( b,Xab,Yab) = d, (24) Z X X = ∗ ∗ a = T = aa ab S0 (Za ,Zb )Y ,X X , where Zb Xba Xbb ⎛ ⎞ ⎛ ⎞ ∗ 1 α1 α1 d1 † Yaa Yab ⎝ ∗ ⎠ ⎝ ⎠ = = K = 1 α2 α2 ,d= d2 . Y Y . ∗ Yba Ybb 1 α3 α3 d3

Before testing the map, all these are unknowns. The normally There are three unknowns ( b, Xab, and Yab) with three ordered form of the density operator ρε is equations now. We ﬁnd = † † T = −1 ρε : Qρε (a ,b ,a,b) : (16) ( b,Xab,Yab) K d. (25) ∗ ∗ corresponding to Eq. (15) but with variables (Za ,Zb ,Za,Zb) If the Gaussian process is known to be trace-preserving, † † replaced by (a ,b ,a,b). The normal order notation : ...: then Eq. (25) completes the tomography: up to a numerical indicates that any term inside it is reordered by placing the factor, we can deduce all the output states of the other input † † = creation operator in the left. For example, : aba b a : CSs, |αi ,fori = 4,5,6. The term ci can be ﬁxed through † † a b a2b. normalization, which is determined by the quadratic and linear The output state from any single-mode input coherent state functional terms in the exponent of the Q functions. Knowing | u (in mode b)is these {ci }, one can construct ρε completely as shown below. | ρ = tr [(|u∗u∗|) ⊗ Iρ ], (17) For any map, ci can be known from tests with αi for u a a ε i = 1,2,...,6. We then have where ρ is given by Eq. (16). Its Husimi Q function is ε ∗ ∗ T = ∗ ∗ ∗ J (c0, a, a ,Xaa,Xaa,Yaa) c, (26) Qρ (Z ,Zb) = Qρ (u,Z ,u ,Zb) u b ε b where = exp(c + L + L† + R + R† + R ), (18) ⎛ ⎞ ⎛ ⎞ u u u 0 ∗ 1 ∗2 1 2 | |2 1 α1 α1 2 α1 2 α1 α1 c1 where ⎜ ∗ 1 ∗2 1 2 | |2 ⎟ ⎜ ⎟ ⎜ 1 α α2 α α α2 ⎟ ⎜ c2 ⎟ ∗ ⎜ 2 2 2 2 2 ⎟ ⎜ ⎟ Lu = ( b + u Xab + uYab)Zb, ⎜ ∗ 1 ∗2 1 2 2 ⎟ ⎜ ⎟ 1 α α3 α α |α3| c3 ∗ = ⎜ 3 2 3 2 3 ⎟ = ⎜ ⎟ = = J ⎜ ∗ ∗ ⎟ ,c ⎜ ⎟ , R ZbXbbZb/2,R0 Zb YbbZb, ⎜ 1 α α 1 α 2 1 α2 |α |2 ⎟ ⎜ c ⎟ ⎜ 4 4 2 4 2 4 4 ⎟ ⎜ 4 ⎟ ∗ ∗ and cu is determined by c0, a, Xaa, and Yaa. Explicitly, ⎝ 1 2 1 2 | |2 ⎠ ⎝ c ⎠ 1 α5 α5 2 α5 2 α5 α5 5 ∗ ∗ ∗ ∗ ∗ ∗ = + + + 1 2 1 2 | |2 c6 cu c0 Re(2 au u Xaau uYaau ). (19) 1 α6 α6 2 α6 2 α6 α6

022101-3 WANG, YU, HU, MIRANOWICZ, AND NORI PHYSICAL REVIEW A 88, 022101 (2013) for the second part of Eq. (22). Thus As a result, ∗ ∗ T −1 ∗ (c0, a, ,Xaa,X ,Yaa) = J c. (27) = H − H + H + H a aa Qρξ (Zb ,Zb) C exp( 1 2 3 4), (34) Theorem:GivenK and J defined by Eqs. (24)–(26), then the where C is the normalization factor, and QPT of any single-mode Gaussian process in Fock space can be performed with six input CSs, when det K = 0 and det J = H =| |2 2 2 − 1 Zb (tanh r sin θ 1)/g, 0. The QPT of any trace-preserving single-mode Gaussian H = Z2 + Z∗2 tanh r cos2 θ/(2g), process in Fock space can be executed with three input CSs, 2 b b ∗ 2 when det K = 0. H3 = Zb cos θ(Z − Z tanh r sin θ)/(g cosh r), For example, one can simply choose α = 0, α = 1, α = H = ∗ − ∗ 2 1 2 3 4 Zb cos θ(Z Z tanh r sin θ)/(g cosh r), i, α4 =−1, α5 =−i, and α6 = 1 + i. One finds and g = 1 − tanh2 r sin4 θ. This is the same result obtained c = c , = d , 0 1 b 1 from direct calculations using Eq. (29). = 1 + − − a 4 (c2 ic3 c4 ic5), = 1 − + + + Xab 2 [ (1 i)d1 d2 id3], IV. EFFICIENT MULTIMODE-GAUSSIAN QPT = 1 − − + − Yab 2 [ (1 i)d1 d2 id3], (28) Multimode Gaussian QPT has many important applica- Y = 1 (c + c + c + c ) − c , aa 4 2 3 4 5 1 tions. For example, it applies to a complex linear optical circuit = 1 − + − + − − + Xaa 4 [c2 c3 c4 c5 2i(c1 c2 c3 c6)], with BSs, squeezers, homodyne detections, linear losses, Gaussian noises, and so on. Consider now a Gaussian process where {di } and {ci } are defined in Eq. (20). actingonak-mode input state (in modes b1,b2,...,bk), with outcome also a k-mode state. Even though other methods [12] A. Example: Output state of a beam-splitter process can also be extended to the multimode case, the number of As a check of our conclusions, we calculate the output state input states required there increases exponentially with the of a beam-splitter (BS) process as shown in Fig. 1. The BS has number of modes k, because the number of ket-bra operators input modes b and c and output modes b and c . Regarding |{ni }{mi }| in Fock space increases exponentially with k. this as a black-box process, the only input is mode b and the As shown below, the number of input states in our method only output is mode b . We set mode c to be the vacuum. The increases polynomially. BS transforms the creation operators of modes b and c by A k-mode QPT can be tested with k-mode CSs if the process † † −1 † † is Gaussian. The main Eqs. (25)–(27) still hold after redefining UBS(b ,c )U = (b ,c )MBS, (29) BS the notations there. First, a, b, u, αi , di , Za, and Zb are now = cos θ sin θ k-mode vectors. For example, where MBS ( − sin θ cos θ ). If we test such a process with | =| | | =| a coherent state αi , we shall find ραi αi cos θ αi cos θ . αi αi1,αi2,...,αik , = ∗ = Comparing this with Eq. (20),wehavedi αi cos θ and ci d = (d ,d ,...,d ),Z= (Z ,Z ,...,Z ), 2 i i1 i2 ik b b1 b2 bk −|αi cos θ| . Using Eqs. (25)–(27), we find X × =− = =− 2 and so on. Following Eq. (15), xy is now a k k matrix, Ybb 1,Xab cos θ, Yaa cos θ, X = = = (30) for X or Y with x a,b; y a,b. We still apply { } { } a = b = Yab = Xaa = Xbb = c0 = 0. Eqs. (25)–(27) to calculate b,Xab,Yab and a,Xaa,Yaa , respectively, but keep in mind that the matrices K,J and Therefore, symbols d,c are now redefined. There are (2k + 1)k unknowns † † † 2 † + ρε = :exp(a b cos θ − a a cos θ − b b + ab cos θ):. (31) in ( B ,Xab,Yab). We need (2k 1) different CSs of k mode to fix these unknowns. Now we have With this we can predict the output state of any input state, ⎛ ∗ ⎞ for example the squeezed coherent state (squeezed displaced 1 α1 α1 vacuum) ⎜ ∗ ⎟ ⎜ 1 α2 α2 ⎟ = ⎜ ⎟ r † † ∗ K ⎜ . . . ⎟ , (35) |ξ(r,Z)=exp (b2 − b 2) exp(Zb − Z b)|0, (32) ⎝ . . . ⎠ 2 ∗ 1 α + α2k+1 where r is real. According to Eq. (8), 2k 1 ∗ ∗ which is a (2k + 1) × (2k + 1) matrix, since each α here is ρ = tr {[|ξ(r,Z )ξ(r,Z )|] ⊗ I ρ }. (33) i ξ a a b ε a k-mode row vector. Moreover, d is here a (2k + 1) × k ma- T = T T T = trix as d (d1 ,d2 ,...,d2k+1), with di (di1,di2,...,dik). Similarly, ⎛ ⎞ ∗ 1 ∗2 1 2 | |2 1 α1 α1 α1 α1 α1 ⎜ 2 2 ⎟ ⎜ 1 α∗ α 1 α∗2 1 α2 |α |2 ⎟ ⎜ 2 2 2 2 2 2 2 ⎟ J = ⎜ ⎟ , (36) ⎝ ...... ⎠ FIG. 1. Gaussian map constructed by a beam-splitter. Here, we ...... assume that the input mode c is in the vacuum. ∗ 1 ∗2 1 2 | |2 1 αN αN 2 αN 2 αN αN

022101-4 EFFICIENT TOMOGRAPHY OF QUANTUM-OPTICAL ... PHYSICAL REVIEW A 88, 022101 (2013) which is now an N × N matrix, and N = (k + 1)(2k + problem of Gaussian QPT to Gaussian QST, for which efﬁcient 2 | |2 2 = 1), since αi and αi here are row vectors of αi methods are experimentally available [38]. We have extended 2 (Ei1,Ei2,...,Eik) and |αi | = (E˜i1,E˜i2,...,E˜ik), and each our results to multimode Gaussian QPT and demonstrated that element of Eim (or E˜im) is a vector with (k − m + 1) modes the number of test states required increases only polynomially (or k modes), as with the number of modes. = 2 2 Eim (αim,αimαi,m+1,αimαi,m+2,...,αimαk,αk ), ˜ = ∗ ∗ ∗ ∗ ACKNOWLEDGMENTS Eim (αimαi1,αimαi2,...,αimαi,k−1,αimαik). Obviously, c is a column vector with N elements. Therefore, X.B.W. is supported by the National High-Tech Program we conclude with the following: of China, Grants No. 2011AA010800, No. 2011AA010803, Corollary.Anyk-mode Gaussian QPT can be performed and No. 2006AA01Z420, NSFC Grants No. 11174177 and with (k + 1)(2k + 1) different CSs of k mode, or with (2k + 1) No. 60725416, and the 10 000 Plan of Shandong province. different CSs of k mode if the process is trace-preserving. A.M. acknowledges support from the Polish National Science Centre under Grant No. DEC-2011/03/B/ST2/01903. F.N. acknowledges partial support from the ARO, RIKEN iTHES V. CONCLUSIONS Project, MURI Center for Dynamic Magneto-Optics, JSPS- In summary, we have presented explicit formulas for the RFBR Contract No. 12-02-92100, a Grant-in-Aid for Scientiﬁc tomography of quantum-optical Gaussian processes probed Research (S), MEXT Kakenhi on Quantum Cybernetics, and with only a few different coherent states. We have reduced the FIRST (Funding Program for Innovative R&D on S&T).

[1]J.F.Poyatos,J.I.Cirac,andP.Zoller,Phys.Rev.Lett.78, 390 [18] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, (1997). Phys. Rev. Lett. 95, 210505 (2005). [2]I.L.ChuangandM.A.Nielsen,J. Mod. Opt. 44, 2455 (1997). [19] X. S. Ma et al., Nature (London) 489, 269 (2012). [3] G. M. D’Ariano and P. Lo Presti, Phys.Rev.Lett.86, 4195 [20] A. M. Childs, I. L. Chuang, and D. W. Leung, Phys.Rev.A64, (2001). 012314 (2001). [4] Quantum State Estimation,editedbyM.G.A.Parisand [21] N. Boulant, T. F. Havel, M. A. Pravia, and D. G. Cory, Phys. J. Rehacek (Springer, Berlin, 2004). Rev. A 67, 042322 (2003). [5] M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Phys. Rev. A 77, [22] Y. S. Weinstein, T. F. Havel, J. Emerson, N. Boulant, 032322 (2008). M. Saraceno, S. Lloyd, and D. G. Cory, J. Chem. Phys. 121, [6] J. Eisert and M. M. Wolf, in Quantum Information with 6117 (2004). Continuous Variables of Atoms and Light (Imperial College [23] Y. X. Liu, L. F. Wei, and F. Nori, Europhys. Lett. 67, 874 (2004); Press, London, 2007), pp. 23–42. Phys. Rev. B 72, 014547 (2005). [7] D. W. Leung, J. Math. Phys. 44, 528 (2003). [24] A. M. Zagoskin, S. Ashhab, J. R. Johansson, and F. Nori, Phys. [8] J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, Rev. Lett. 97, 077001 (2006). R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Phys. [25] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, Rev. Lett. 90, 193601 (2003). E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. [9] G. M. D’Ariano and P. LoPresti, Phys.Rev.Lett.91, 047902 Martinis, Nat. Phys. 4, 523 (2008). (2003). [26] J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch, L. S. Bishop, [10] M. Mohseni and D. A. Lidar, Phys.Rev.A75, 062331 (2007). A. A. Houck, B. R. Johnson, L. Frunzio, S. M. Girvin, and R. J. [11] Z.-W. Wang, Y.-S. Zhang, Y.-F. Huang, X.-F. Ren, and G.-C. Schoelkopf, Phys. Rev. Lett. 102, 090502 (2009). Guo, Phys. Rev. A 75, 044304 (2007). [27] R. C. Bialczak et al., Nat. Phys. 6, 409 (2010). [12] K. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, [28] J. M. Chow et al., Phys.Rev.Lett.109, 060501 (2012). and A. I. Lvovsky, Science 322, 563 (2008); S. Rahimi-Keshari, [29] J. Q. You and F. Nori, Nature (London) 474, 589 (2011); Phys. A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Today 58(11), 42 (2005). Sanders, New J. Phys. 13, 013006 (2011). [30] H. Kampermann and W. S. Veeman, J. Chem. Phys. 122, 214108 [13] M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. (2005). Steinberg, Phys. Rev. Lett. 91, 120402 (2003). [31] M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, [14] F. DeMartini, A. Mazzei, M. Ricci, and G. M. D’Ariano, Phys. and J. Wrachtrup, New J. Phys. 8, 33 (2006). Rev. A 67, 062307 (2003). [32] D. Burgarth, K. Maruyama, and F. Nori, New J. Phys. 13, 013019 [15]J.L.O’Brien,G.J.Pryde,A.Gilchrist,D.F.V.James,N.K. (2011). Langford, T. C. Ralph, and A. G. White, Phys.Rev.Lett.93, [33] M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. 080502 (2004). Korber, W. Hansel, H. Haffner, C. F. Roos, and R. Blatt, Phys. [16] Y.Nambu and K. Nakamura, Phys.Rev.Lett.94, 010404 (2005). Rev. Lett. 97, 220407 (2006). [17] N. K. Langford, T. J. Weinhold, R. Prevedel, A. Gilchrist, [34] T. Monz et al., Phys. Rev. Lett. 102, 040501 (2009). J. L. O’Brien, G. J. Pryde, and A. G. White, Phys. Rev. Lett. [35] S. H. Myrskog, J. K. Fox, M. W. Mitchell, and A. M. Steinberg, 95, 210504 (2005). Phys. Rev. A 72, 013615 (2005).

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[36] A. S. Holevo and V. Giovannetti, Rep. Prog. Phys. 75, 046001 [39] J. Eisert, S. Scheel, and M. B. Plenio, Phys.Rev.Lett.89, 137903 (2012). (2002); G. Giedke and J. I. Cirac, Phys. Rev. A 66, 032316 [37] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum (2002). Theory (Springer, Berlin, 2011). [40] W. H. Louisell, Quantum Statistical Properties of Radiation [38] J. Rehacek, S. Olivares, D. Mogilevtsev, Z. Hradil, M. G. A. (Wiley, New York, 1973). Paris, S. Fornaro, V. D’Auria, A. Porzio, and S. Solimeno, Phys. [41] A. Jamiolkowski, Rep. Math. Phys. 3, 275 Rev. A 79, 032111 (2009). (1972).

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