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

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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. 1050-2947/2013/88(2)/022101(6)022101-1 ©2013 American Physical Society WANG, YU, HU, MIRANOWICZ, AND NORI PHYSICAL REVIEW A 88, 022101 (2013) 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.
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