Quantum process tomography with coherent states
Saleh Rahimi-Keshari, Artur Scherer, Ady Mann, Ali T. Rezakhani, A. I. Lvovsky, Barry C. Sanders
aa r X i v : . [ qu a n t - ph ] S e p Quantum process tomography with coherent states
Saleh Rahimi-Keshari , Artur Scherer , Ady Mann , , Ali T.Rezakhani , , A. I. Lvovsky , and Barry C. Sanders Institute for Quantum Information Science and Department of Physics and Astronomy,University of Calgary, Alberta, Canada T2N 1N4 Physics Department, Technion, Haifa 32000, Israel Department of Chemistry, Center for Quantum Information Science and Technology,University of Southern California, Los Angeles, California 90089, USA Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran,IranE-mail: [email protected]
Abstract.
We develop an enhanced technique for characterizing quantum optical processesbased on probing unknown quantum processes only with coherent states. Our methodsubstantially improves the original proposal [M. Lobino et al., Science , 563 (2008)], whichuses a filtered Glauber-Sudarshan decomposition to determine the effect of the process on anarbitrary state. We introduce a new relation between the action of a general quantum processon coherent state inputs and its action on an arbitrary quantum state. This relation eliminatesthe need to invoke the Glauber-Sudarshan representation for states; hence it dramaticallysimplifies the task of process identification and removes a potential source of error. The newrelation also enables straightforward extensions of the method to multi-mode and non-trace-preserving processes. We illustrate our formalism with several examples, in which we deriveanalytic representations of several fundamental quantum optical processes in the Fock basis.In particular, we introduce photon-number cutoff as a reasonable physical resource limitationand address resource vs accuracy trade-off in practical applications. We show that the accuracyof process estimation scales inversely with the square root of photon-number cutoff.PACS numbers: 03.65.Wj, 42.50.-p, 03.67.-a. uantum process tomography with coherent states
1. Introduction
Assembling a complex quantum optical information processor requires precise knowledge ofthe properties of each of its components, i.e., the ability to predict the effect of the componentson an arbitrary input state. This gives rise to a quantum version of the famous “black boxproblem”, which is addressed by means of quantum process tomography (QPT) [1, 2, 3]. InQPT, a set of probe states is sent into the black box (here an unknown completely-positive,linear quantum process E over the set of bounded operators B ( H ) on a Hilbert space H ) andthe output states are measured. From the effect of the process on the probe states it is possibleto predict its effect on any other state within the same Hilbert space.QPT exploits linearity of quantum process over its density operators. If the effect of theprocess E ( ρ i ) is known for a set of density operators { ρ i } , its effect on any linear combination ρ = P β i ρ i equals E ( ρ ) = P β i E ( ρ i ) . Therefore, if { ρ i } forms a spanning set within thespace L ( H ) of linear operators over a particular Hilbert space H , knowledge of {E ( ρ i ) } issufficient to extract complete information about the quantum process.However, practical implementations of QPT become demanding especially for systemswith large Hilbert spaces. For dim( H ) = d , dim (cid:0) L ( H ) (cid:1) = d , which implies that at least d unknown operators {E ( ρ i ) } , each with d unknown parameters, must be estimated. Thisprocedure requires preparation of at least { ρ i } d i =1 states, subjecting each to the unknownprocess E , and determining each element of {E ( ρ i ) } d i =1 through measurement (each with d unknown elements), thereby inferring an overall number of d parameters. Furthermore,in order to build up sufficient statistics for reliable estimates of the output states, eachmeasurement should be performed many times on multiple copies of the inputs. Thus, alarge number of realizations and measurements are required for complete tomography of E .An additional complication, especially for QPT of quantum optical processes, isassociated with preparation of the probe states. Typical optical QPT implementations dealwith systems consisting of one or more dual-rail qubits [4, 5, 6], which implies that the probestates are highly nonclassical, hence difficult to generate.These difficulties have been partially alleviated in the recently proposed scheme of“coherent-state quantum process tomography” (csQPT) [7]. This scheme is based on theobservation that the density operator ρ of a generic quantum state of every electromagneticmode can be expressed in the Glauber-Sudarshan representation [8, 9], ρ = Z C d α P ρ ( α ) | α ih α | , (1)where P ρ ( α ) is a quasi-probability distribution referred to as the quantum state’s “ P -function”and integrated over the entire complex plane [10]. Linearity hence implies that measuring | α ih α | 7→ ̺ E ( α ) ≡ E ( | α ih α | ) , (2)i.e., determining the effect of the unknown process on all coherent states enables a predictionof its effect upon any generic state ρ according to E ( ρ ) = Z C d α P ρ ( α ) ̺ E ( α ) . (3) uantum process tomography with coherent states P function for many nonclassicaloptical states exists only in terms of a highly singular generalized function [13, 14]. A remedytherefor is provided by Klauder’s theorem [15], which states that any trace-class operator ρ canbe approximated, to arbitrary accuracy, by a bounded operator ρ L ∈ B ( H ) whose Glauber-Sudarshan function P L is in the Schwartz class [16], so integration (3) can be performed. TheKlauder approximation can be constructed by low-pass filtering of the P function, i.e., bymultiplying its Fourier transform with an appropriate regularizing function equal to overa square domain of size L × L and rapidly dropping to zero outside this domain. Ref. [7]employs this method to implement csQPT.Practical implementation of Klauder’s procedure is however complicated, because itrequires finding the characteristic function of the input state and subsequently its regularized P function. This function features high-frequency, high-amplitude oscillations that limitthe precision in calculating the output state (3). Furthermore, Klauder’s approximation isambiguous in the choice of the particular filtering function as well as the cutoff parameter L [7].Here we improve csQPT to overcome the above problems. Specifically, we developa new method for csQPT that eliminates the explicit use of the Glauber-Sudarshanrepresentation and thus removes the inherent ambiguity associated with employing Klauder’sapproximation for csQPT. In Sec. 2.1, we obtain an expression for the process tensor inthe Fock (photon number) basis that can be directly calculated from the experimental data.Using this tensor, the process output for an arbitrary input can be calculated by simplematrix multiplication rather than requiring integration and high-frequency cut-offs. In thisway, transformations between the Fock and Glauber-Sudarshan representations, which werenecessary in Ref. [7], can be sidestepped. Using our new approach, we easily extend csQPTfrom its restrictive single-mode applicability to multi-mode processes and even to non-trace-preserving conditional processes. These extensions are particularly relevant for quantuminformation processing circuits, whose basic components are inherently multi-mode andconditional [17].Process tomography is successful if, for every input state, the estimate for the processoutput closely approximates the actual process output state, and the worst-case error of thisestimate, given by a distance between the actual and estimated process outputs, is less thana given tolerance. For states over infinite-dimensional Hilbert spaces, this concept of erroris however not meaningful because the finiteness of sampling implies that the process is uantum process tomography with coherent states subset of B ( H ) . This versionof process tomography can always be successful with a sufficiently large amount of sampling.Of particular practical interest is the subspace B ( ˜ H ) defined by an energy cut-off, i.e.,estimating the process without accessing any information about its high-energy behavior. Thisrestriction is naturally consistent with our choice to work in the Fock basis, because thenthe resulting process tensor is of finite size and with many practical settings (e.g. quantum-information processing with photonic qubits). In Sec. 2.2, we provide process error estimatesfor several input state subsets that extend beyond B ( ˜ H ) .Many interesting processes are phase symmetric; that is, an optical phase shift of theinput state results in the same phase shift of the output. This property dramatically simplifiesthe experiment because one needs to collect data only for coherent states whose amplitudes lieon the real axis rather than the entire complex plane. This prompts us to discuss, in Sec. 3, howto obtain the process tensor for phase-symmetric processes, which we test on the experimentaldata from Ref. [12]. Next, in Sec. 4, we illustrate our method by analytically deriving thesuperoperators for certain fundamental quantum optical processes using the Fock basis. Thepaper is concluded in Sec. 5 and is supplemented with two appendices.
2. Coherent state quantum process tomography
We study general quantum optical processes E acting on quantum states of light and beginwith the simplest case for which only a single electromagnetic field mode is involved. Anarbitrary quantum state ρ can be expressed in the Fock representation as ρ = ∞ X m,n =0 ρ mn | m i h n | . (4)Subjecting this state to an unknown process E , and imposing linearity, yields E ( ρ ) = ∞ X j,k,m,n =0 ρ mn E mnjk | j ih k | , (5)where E mnjk := h j |E ( | m ih n | ) | k i (6)is a rank– tensor, hereafter referred to as the “process tensor” (superoperator). Thus, byexpressing input and output states in the Fock basis, a quantum process can be uniquelyrepresented and characterized by its rank- tensor, which relates the matrix elements of theoutput and input states according to [ E ( ρ )] jk = X m,n ∈ N E mnjk ρ mn , (7)where N ≡ N ∪ { } . uantum process tomography with coherent states E ( | m ih n | ) for m, n over a finitedomain. Because h α | ( | m i h n | ) | α i = e −| α | α n ¯ α m √ m ! n ! (8)is in the Schwartz class, the Glauber-Sudarshan P representation | m ih n | = Z C d α P mn ( α ) | α ih α | (9)is guaranteed to exist for any operator | m ih n | ( m, n ∈ N ) [18]. The P function is P mn ( α ) = ( − m + n e | α | √ m ! n ! ∂ mα ∂ n ¯ α δ ( α ) (10)for ∂ mα := ∂ m /∂α m and α and its complex conjugate ¯ α treated as independent variables, and δ ( α ) ≡ δ (cid:0) Re( α ) (cid:1) δ (cid:0) Im( α ) (cid:1) . By inserting representation (9) into Eq. (6), and exploitinglinearity of the process, we obtain the process tensor E mnjk = Z C d α P mn ( α ) h j | ̺ E ( α ) | k i . (11)This expression can be simplified by using Eq. (10) and performing integration by parts: E mnjk = Z C d α δ ( α ) √ m ! n ! ∂ mα ∂ n ¯ α (cid:2) e | α | h j | ̺ E ( α ) | k i (cid:3) = 1 √ m ! n ! ∂ mα ∂ n ¯ α (cid:2) e | α | h j | ̺ E ( α ) | k i (cid:3)(cid:12)(cid:12)(cid:12) α =0 . (12)Thus we have eliminated the need to make use of the Glauber-Sudarshan representationfor quantum states. The process tensor is found by taking partial derivatives (with respect to α and ¯ α ) of the matrix elements of ̺ E ( α ) , which are estimated from experimental data andevaluated at α = 0 .The mathematical procedure defined by Eq. (12) is simpler and computationally faster(see Sec. 3) than employing Eq. (11) with a regularized version of P L,mn ( α ) replacing thetempered distribution P mn ( α ) described in Refs. [7, 12]. Equation (12) has been usedto determine the fidelity of quantum teleportation of a single-rail optical qubit based onmeasurements performed on coherent states (see supplementary material in Ref. [19]).Generalization to the multi-mode case is straightforward. In the M -mode case, let usintroduce the notation | n i := | n , n , . . . , n M i (with n ∈ N M ) for multi-mode Fock statesand | α i := | α , α , . . . , α M i (with α ∈ C M ) for multi-mode coherent states. Then the matrixelements of the output and input states with respect to the Fock basis are related to one anotherby the rank– M tensor [ E ( ρ )] jk ≡ h j |E ( ρ ) | k i = X m , n ∈ N M E mnjk ρ mn , (13)where E mnjk := h j |E ( | m ih n | ) | k i . (14) uantum process tomography with coherent states P representation for themulti-mode operator | m ih n | , with the overall P function being a product of the P functionsfor the constituent modes: P mn ( α ) = M Y s =1 e | α s | ( − m s + n s √ m s ! n s ! ∂ m s α s ∂ n s ¯ α s δ ( α s ) . (15)Multiple integration by parts yields E mnjk = Z C M d M α M Y s =1 δ ( α s ) √ m s ! n s ! ∂ m s α s ∂ n s ¯ α s h e | α s | h j | ̺ E ( α ) | k i i = M Y s =1 √ m s ! n s ! ∂ m s α s ∂ n s ¯ α s h e | α s | h j | ̺ E ( α ) | k i i (cid:12)(cid:12)(cid:12)(cid:12) α s =0 , (16)where ̺ E ( α ) ≡ E ( | α ih α | ) . (17)Equations (12) and (16) complete our coherent-state tomography formalism and show thatcoherent states provide a complete set of probe states for characterizing quantum opticalprocesses, insofar as the expression for ̺ E ( α ) completely determines the process tensor.The above formalism is not restricted to trace-preserving quantum processes. Indeed,trace preservation was not required in the derivation of our results. Thus, our method isapplicable to all quantum optical processes that are mathematically described by completely-positive maps, but may be trace-preserving, trace-reducing or even trace-increasing. Trace-nonpreserving quantum processes are either conditional processes or part of a larger process E = E + E , which is trace-preserving as a whole, but whose components E and E may increase or decrease the trace, respectively. A conditional process is a process that isconditioned on a certain probabilistic event; it may be heralded if the event is observed. Oneof the most notable examples of such a process is a probabilistic conditional- NOT gate (
CNOT ),which forms the basis for the Knill-Laflamme-Milburn linear-optical quantum computingscheme [17]. Other examples are photon-addition and photon-subtraction processes, whosesuperoperators are derived in Sec. 4.In experimental csQPT, states ̺ E ( α ) are determined using homodyne tomography [11].It is important to remember, however, that this procedure reconstructs a density matrixnormalized to unity trace: e ̺ E ( α ) = ̺ E ( α ) / Tr [ ̺ E ( α )] . When measuring non-trace-preservingprocesses, one must recover the trace information contained in ̺ E ( α ) . This is done bymeasuring the probability p α ( E ) = Tr [ ̺ E ( α )] of the process heralding event for all α ’s forwhich the measurements are performed. The state to be used in Eqs. (12) and (16) in place of ̺ E ( α ) is then e ̺ E ( α )Tr [ ̺ E ( α )] .An interesting feature of Eqs. (12) and (16) is that complete information about a quantumoptical process is contained in its action on an infinitesimally small compact set of all probecoherent states in the immediate vicinity of the vacuum state. From a mathematical point ofview, this feature can be understood by realizing that, for any j, k ∈ N , the matrix element uantum process tomography with coherent states h j | ̺ E ( α ) | k i is an entire function (see Appendix A), i.e., a complex-valued function in thevariables α, ¯ α that is holomorphic over the whole complex plane, and so is its product with theexponential e | α | . Hence, each term e | α | h j | ̺ E ( α ) | k i is infinitely differentiable over the wholecomplex plane and is identical to its Taylor series expansion in any point of C . Moreover,Eq. (12) implies that the process tensor is determined by the corresponding Taylor coefficientsat α = 0 . The same conclusion applies to the multi-mode case, in which we deal with entirefunctions on C M . As discussed in Sec. 1, the incompleteness of the information acquired in the experiment isaccommodated in csQPT by evaluating the process tensor over a restricted finite-dimensionalsubspace e H of the Hilbert space H with a fixed maximum number N of photons. The incurredexpense is that, through this reduced tomography, only approximate information about theprocess will be inferred: for a given input state ρ , the predicted output is not E ( ρ ) , but rather ˜ E ( ˜ ρ ) , where ˜ ρ = e Π ρ e ΠTr[ ρ e Π] (18)is the trace-normalized projection of ρ onto B ( e H ) and ˜ E ( ˜ ρ ) = e Π E ( ˜ ρ ) e Π (19)is the predicted output of the reconstructed process for input state ˜ ρ . In Eqs. (18) and (19), e Π is the projection operator onto e H .If the input state ρ is outside B ( e H ) , the process output estimation error kE ( ρ ) − ˜ E ( ˜ ρ ) k (where k ρ k = Tr p ρ † ρ is the trace norm) is generally unbounded. However, it is possible tobound the error for certain practically important classes of input states and processes.For example, all linear-optical processes involving only linear-optical elements(interferometers, attenuators, conditional measurements) do not generate additional photons,and thus map B ( e H ) onto itself, so ˜ E ( ˜ ρ ) = E ( ˜ ρ ) . For such processes, the error for a particularinput ρ can be estimated according to kE ( ρ ) − E ( e ρ ) k ≤ kE k k ρ − e ρ k , with the superoperatornorm defined as kE k ≡ sup {kE ( ˆ B ) k : ˆ B ∈ B ( H ) , k ˆ B k ≤ } [20]. If the process is knownto be trace-nonincreasing, we have kE k ≤ [21] so the error is bounded from above by kE ( ρ ) − E ( e ρ ) k ≤ k ρ − e ρ k . (20)Note that the above result is not sufficient for evaluating the error for a general process,because this error is given by the deviation of E ( ρ ) from e E ( e ρ ) rather than from E ( e ρ ) [Fig. 1].A further error bound can be obtained for the class of trace-preserving processes thatdo not increase the mean energy, acting on a set of input states whose mean energy does notexceed a certain value [22]. We illustrate this for a single optical mode ˆ a with frequency ω andHamilton operator ˆ H = ω (ˆ a † ˆ a + 1 / whose eigenvalues are denoted by h n = ( n + 1 / ω . uantum process tomography with coherent states ! ~ )( E )~( E )~(~ E )( HB )( HB )~( HB )~( HB Figure 1.
Errors associated with photon number cutoff. Restricting B ( H ) to B ( e H ) results inapproximation e ρ of the input state ρ . If the error of this approximation k ρ − e ρ k is known,the error of the images kE ( ρ ) − E ( e ρ ) k can be estimated according to Eq. (20). However, thedifference between E ( ρ ) and e E ( e ρ ) in the cutoff space remains generally unknown. Suppose that the quantum states ρ of interest satisfy Tr[ ρ ˆ H ] ≤ U . According to Ref. [22], ifwe choose the cutoff dimension dim( e H ) = N + 1 such that U/h N +1 ≤ γ for some (small) γ > , the reconstructed process output errors are bounded from above as (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( ρ ) − e E ( e ρ )Tr e E ( e ρ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ, (21)where ǫ = 2 √ γ + γ/ (1 − γ ) . (22)Conversely, if we want to achieve a certain upper bound ǫ on the error of approximation(which corresponds to a lower bound on the desired accuracy of the process characterization),we first solve Eq. (22) for γ = γ ( ǫ ) , and then find the minimum N γ ∈ N such that U/h N γ +1 ≤ γ . Any cutoff dimension N + 1 > N γ is then sufficient for our purpose. For γ ≪ , ǫ ≈ √ γ , which yields ǫ = O (1 / √ N ) . (23)This implies that the error of approximation scales as / √ N with the cutoff dimension N + 1 .For example, in order to achieve a 10% error in Eq. (21), we need ǫ = 0 . and thus γ ≈ . . For the input mean energy bound corresponding to one photon ( U = 3 / ω ), therequired cutoff is N ≈ U/γ ≈ . This calculation shows that the above error estimate isextremely conservative.
3. Phase-invariant processes
Many practically relevant processes, including the single-mode processes studied in Sec. 4,exhibit phase invariance. If two input states are identical up to a shift by an optical phase φ , uantum process tomography with coherent states E [ e i ˆ nφ ρe − i ˆ nφ ] = e i ˆ nφ E ( ρ ) e − i ˆ nφ . (24)For such processes, it is convenient to express the probe coherent states in polar coordinates: | α i = (cid:12)(cid:12) re iθ (cid:11) = e i ˆ nθ | r i . Specifically, in these coordinates, we have [9] P mn ( r, θ ) = √ m ! n !( m + n )! e r + iθ ( n − m ) ( − m + n d m + n d r m + n δ ( r ) , (25)and accordingly E mnjk = √ m ! n !( m + n )! d m + n d r m + n (cid:20)Z π d θ π e r + iθ ( n − m ) h j | ̺ E ( r, θ ) | k i (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) r =0 . (26)Hence Eq. (24) can be expressed as h j | E ( | α i h α | ) | k i = e iθ ( j − k ) h j | E ( | r i h r | ) | k i , (27)and the superoperator E [Eq. (26)] has the following explicit representation: E mnjk = √ m ! n !( m + n )! d m + n d r m + n h e r h j | E ( | r i h r | ) | k i i (cid:12)(cid:12)(cid:12)(cid:12) r =0 δ m − j,n − k . (28)In experimental tomography of phase-invariant processes [7, 12], it is sufficient tomeasure the process output for a discrete set of coherent states {| r i i} on the real axis of thephase space. The matrix elements of the output states can then be interpolated as polynomialfunctions h j | E ( | r i h r | ) | k i = Q X l =0 C l ( j, k ) r l , (29)where Q is the degree of the polynomial (which depends on the dimension of the truncatedHilbert space) and C l ( j, k ) are its coefficients. Furthermore, from Eq. (A.3) together withEq. (28), it follows that, for phase-symmetric processes, when j − k is even or odd, h j | E ( | r i h r | ) | k i and its analytic extension to negative values of r are even or odd functionsof r , respectively. By taking into account the symmetric or antisymmetric property of thisfunction, we have additional information to be used in the interpolation procedure; theconstructed polynomial has to contain only even or odd powers of r , respectively. In thisway the precision of process estimation from the experimental data is substantially increased.With the knowledge of the coefficients C l ( j, k ) , Eq. (28) is further simplified to: E mnjk = √ m ! n !( m + n )! d m + n d r m + n " ∞ X s =0 r s s ! Q X l =0 C l ( j, k ) r l r =0 δ m − j,n − k = √ m ! n !( m + n )! ∞ X s =0 Q X l =0 δ m + n, s + l ( m + n )! s ! C l ( j, k ) δ m − j,n − k = √ m ! n ! ⌊ ( m + n ) ⌋ / X s =0 C m + n − s ( j, k ) s ! δ m − j,n − k . (30) uantum process tomography with coherent states d = N + 1 , from Eq. (30) it follows that onlyterms of power l ≤ N of the interpolation polynomial (29) contribute to the process tensor.We have tested this procedure on experimental data [12] and calculated the process tensor in afew microseconds, which is a dramatic improvement in comparison to several hours requiredfor the original procedure [7, 12].
4. Examples: superoperators of important quantum optical processes
In this section, we illustrate our new method by applying it to some fundamental quantumoptical processes, whose effects on coherent states are known. Specifically, using Eqs. (12) or(16), we analytically derive corresponding superoperator tensors E mnjk in the Fock basis. Theresults are summarized in Table 1. For the identity process ( E id ), ̺ E id ( α ) = | α ih α | , the matrix elements of the output states are h j | ̺ E id ( α ) | k i = e −| α | α j ¯ α k √ j ! k ! . (31)Inserting these elements into Eq. (12) yields E mnjk = δ mj δ nk , as expected. For attenuation of light fields ( E att ), the process’s effect on single-mode coherent states is givenby ̺ E att ( α ) = | ηα ih ηα | , where ≤ η < . The matrix elements in the Fock basis are h j | ̺ E att ( α ) | k i = e − η | α | η j + k α j ¯ α k √ j ! k ! . (32)From Eq. (12), we obtain E mnjk = η j + k √ m ! n ! j ! k ! ∂ mα ∂ n ¯ α h e | α | (1 − η ) α j ¯ α k i (cid:12)(cid:12)(cid:12) α, ¯ α =0 = η j + k √ m ! n ! j ! k ! ∂ mα ∂ n ¯ α ∞ X l =0 (1 − η ) l α j + l ¯ α k + l l ! (cid:12)(cid:12)(cid:12) α, ¯ α =0 = s m ! n ! j ! k ! η j + k (1 − η ) m − j ( m − j )! δ m − j,n − k , (33)which depends explicitly on η . uantum process tomography with coherent states a)b) single-photondetector low-reflectivitybeam splitteridler signalparametricdown-conversion pumpbeam † ˆˆ aa aa ˆˆ † Figure 2.
Experimental realizations of (a) photon subtraction and (b) photon addition. Theprocess is heralded by single-photon detection events.
Photon subtraction is defined as a process that removes a single photon from the lightfield, whereas photon addition adds a single photon. Photon subtraction has been used byOurjoumtsev et al. [23] to generate optical Schr¨odinger kittens (coherent superpositions oflow-amplitude coherent states) from squeezed vacuum states for the purpose of quantuminformation processing. Single-photon-added coherent states can be regarded as the resultof the most elementary amplification process of classical light fields by a single quantum ofexcitation; being intermediate between single-photon Fock states (fully quantum-mechanical)and coherent (classical) ones, these states have been demonstrated to be suited for the studyof smooth transition between the particle-like and the wavelike behavior of light [24].Here we discuss idealized single-mode photon subtraction and photon addition. Bothprocesses are non-trace-preserving. For example, photon subtraction can be approximatelyrealized [23] by a highly-transmissive beam splitter, whose reflected mode is directed to adetector and whose transmitted mode constitutes the output, respectively, as illustrated inFig. 2a. Any click in a detector implies extraction of photon(s) from the input mode bythe beam splitter. As the beam splitter has low reflectivity, here single-photon extractionevents are more likely than multi-photon events. An approximate experimental realization ofphoton addition is illustrated in Fig. 2b. The input quantum state ρ enters the signal channelof a parametric down-conversion setup. Provided that detector dark counts are neglected, aphoton detection in the idler mode heralds photon addition to the signal mode, which containsthe output state of the process.The effect of photon subtraction ( E sub ) and addition ( E add ) on coherent states is given by ̺ E sub ( α ) = ˆ a | α ih α | ˆ a † and ̺ E add ( α ) = ˆ a † | α ih α | ˆ a , respectively, where ˆ a and ˆ a † are the photonannihilation and photon creation operators of a single mode, respectively. The matrix elements uantum process tomography with coherent states h j | ̺ E sub ( α ) | k i = e −| α | α j +1 ¯ α k +1 √ j ! k ! , (34) h j | ̺ E add ( α ) | k i = e −| α | p kj α j − ¯ α k − p ( j − k − . (35)The process tensor is found to be E mnjk = ( p ( j + 1)( k + 1) δ m,j +1 δ n,k +1 , for photon subtraction, √ kjδ m,j − δ n,k − , for photon addition, (36)where we have employed Eq. (12). The unitary evolution according to ˆ U Kerr ( χ ) ≡ exp h − iχ (cid:0) ˆ a † ˆ a (cid:1) i for χ = π/ , if applied tocoherent states, generates Schr¨odinger cat states (hereafter denoted as E cat ) [25, 26] ̺ E cat ( α ) = ˆ U Kerr ( π | α ih α | ˆ U † Kerr ( π | α i + i |− α i )( h α | − i h− α | ) , (37)with matrix elements h j | ̺ E cat ( α ) | k i = e −| α | α j ¯ α k √ j ! k ! (cid:2) − j + k + i ( − j − i ( − k (cid:3) . (38)The superoperator tensor for this non-Gaussian unitary process obtained via Eq. (12) is E mnjk = e − i π ( j − k ) δ mj δ nk . (39)Interestingly, this process does not change the total particle number of any input state. Now let us consider the beam splitter as an example of a two-mode process. The unitary beamsplitter transformation is given by [27] ˆ B (Θ) = e Θ2 (ˆ a † ˆ a − ˆ a † ˆ a ) , (40)where Θ is the parameter identifying how the beam splitter transmits or reflects beams.Specifically, its action on coherent state inputs | α i and | α i is given as ̺ E B ( α , α ) = E B ( | α , α ih α , α | )= ˆ B † (Θ)( | α , α ih α , α | ) ˆ B (Θ)= | T α − Rα , Rα + T α ih T α − Rα , Rα + T α | , (41) uantum process tomography with coherent states T ≡ cos(Θ / and R ≡ sin( − Θ / being the transmissivity and reflectivity,respectively. By knowing the effect of the process on two-mode coherent states, we cancalculate the corresponding tensor using Eq. (16), which yields E m m n n j j k k = s m ! m ! n ! n ! j ! j ! k ! k ! j X p =0 k X q =0 (cid:18) j p (cid:19)(cid:18) j m − p (cid:19)(cid:18) k q (cid:19) × (cid:18) k n − q (cid:19) T p +2 q + j + k − m − n × ( − j + k − p − q R j + k + m + n − p − q × δ m + m ,j + j δ n + n ,k + k , (42)as an explicit function of T and R . Another two-mode process of interest is parametric down-conversion (PDC). In PDC, a crystalwith an appreciably large second-order non-linearity is pumped by a laser field. Each ofthe pump photons can spontaneously decay into a pair of identical (degenerate PDC) ornonidentical photons (nondegenerate PDC). Here we consider a nondegenerate PDC process E PDC induced by the transformation [27] ˆ S ( r ) = e r (ˆ a ˆ a − ˆ a † ˆ a † ) . (43)The effect of this unitary process on a two-mode coherent state is given by ̺ E PDC ( α , α ) = E PDC ( | α , α i h α , α | )= ˆ S ( r ) | α , α i h α , α | ˆ S † ( r ) . (44)In Appendix B, we derive the process tensor in the Fock basis. The result can be expressedas: E m m n n j j k k = s n ! m ! m ! n ! j ! k ! k ! j ! (tanh r ) m + n − j − k ( m − j )! ( n − k )! (cosh r ) j + k − j − k +2 × F (cid:0) − j , m + 1; m − j + 1; tanh r (cid:1) × F (cid:0) − k , n + 1; n − k + 1; tanh r (cid:1) × δ m − m ,j − j δ n − n ,k − k , (45)with F ( α, β ; γ ; z ) := 1 + ∞ X n =1 ( α ) n ( β ) n ( γ ) n z n n ! , (46)the hypergeometric function, ( x ) n := Γ( x + n ) / Γ( x ) the Pochhammer symbol and Γ( · ) theGamma function [28]. uantum process tomography with coherent states Table 1.
Process tensor E mnjk for some quantum optical processes.Operation E ̺ E ( α ) Process tensor E mnjk Identity ( E id ) | α ih α | δ mj δ nk Attenuation ( E att ) | ηα ih ηα | q m ! n ! j ! k ! η j + k (1 − η ) m − j ( m − j )! δ m − j,n − k Photon addition ( E add ) ˆ a † | α ih α | ˆ a √ kjδ m,j − δ n,k − Photon subtraction ( E sub ) ˆ a | α ih α | ˆ a † p ( j + 1)( k + 1) δ m,j +1 δ n,k +1 Cat generation ( E cat ) ( | α i + i |− α i ) e − i π ( j − k ) δ mj δ nk × ( h α | − i h− α | ) Beam splitter ( E B ) | T α − Rα , Rα + T α i q m ! m ! n ! n ! j ! j ! k ! k ! P j p =0 P k q =0 ( − j + k − p − q ×h T α − Rα , Rα + T α | × (cid:0) j p (cid:1)(cid:0) j m − p (cid:1)(cid:0) k q (cid:1)(cid:0) k n − q (cid:1) × T p +2 q + j + k − m − n × R j + k + m + n − p − q × δ m + m ,j + j δ n + n ,k + k Parametric down- e r (ˆ a ˆ a − ˆ a † ˆ a † ) | α , α i q m ! m ! n ! n ! j ! j ! k ! k ! conversion ( E PDC ) × h α , α | e r (ˆ a † ˆ a † − ˆ a ˆ a ) × (tanh r ) m n − j − k ( m − j )! ( n − k )! (cosh r ) j k − j − k × F (cid:0) − j , m + 1; m − j + 1; tanh r (cid:1) × F (cid:0) − k , n + 1; n − k + 1; tanh r (cid:1) × δ m − m ,j − j δ n − n ,k − k
5. Conclusions
Coherent states are easily generated probe states for tomography of unknown quantum-optical processes. Here, we have presented a new, more efficient data processing techniquefor estimating a quantum process from similar experimental procedure of Ref. [7]. Theoriginal formulation was based on regularization and filtering of the Glauber-Sudarshanrepresentations for quantum states, which are cumbersome to implement numerically.Furthermore, Ref. [7] introduces additional errors associated with regularization of the P function. In contrast, our new method to determine the process superoperator [Eq. (12) orEq. (16)] is mathematically simpler, computationally faster and unique up to the choice ofthe energy cutoff. Moreover, we presented straightforward generalizations of coherent statequantum process tomography to multi-mode and non-trace-preserving conditional processes.We have illustrated the new framework through several examples (summarized inTable 1). We have shown that it is straightforward to derive analytically exact and uniqueclosed-form expressions for the superoperators for quantum optical processes whose effect uantum process tomography with coherent states | α i in the immediate vicinity of the vacuum state.This is due to the entireness property of the image of processes on coherent states. It thusappears sufficient to perform tomography experiments only for a range of coherent stateswhose mean photon number is much smaller than that required for the method of Ref. [7] (seethe suppl. material therein). However, coherent state quantum process tomography relies onthe ability to approximately determine all the derivatives of a function which is obtained byinterpolation from measured experimental data. Minimization of errors associated with thiscalculation imposes a lower bound on the phase space region over which the measurementsneed to be performed. For the time being, we have provided an evaluation of the error in theprocess estimation by introducing a truncation of the Fock space. For the class of processesrespecting a certain energy constraint (which includes all processes that do not amplify theenergy), we have determined (i) the cutoff dimension that is sufficient in order to achievea certain degree of approximation accuracy, as well as (ii) the upper bound on the error ofestimation for a given cutoff dimension. Acknowledgments:
We acknowledge financial support by NSERC, i CORE, MITACS, QuantumWorks andGeneral Dynamics Canada. AIL is a CIFAR Scholar, and BCS is a CIFAR Fellow. We wouldalso like to thank Connor Kupchak for helpful discussions.
Appendix A. Proof that h j | ̺ E ( α ) | k i is an entire function According to Eq. (12), by knowing the complex-valued function h j | ̺ E ( α ) | k i (of the variable α ) for any j and k , one can determine the process tensor E mnjk . Here we show that this functionis an entire function so it can be represented as a power series that converges uniformly onany compact domain.As a completely-positive quantum operation, E possesses a Kraus decomposition E ( ρ ) = P Li =1 ˆ K i ρ ˆ K † i , where L ≤ dim( H ) and ˆ K i are some Kraus operators on H (whose explicitform is not needed for our purpose). Hence we can rewrite the matrix elements of the outputstate as h j | ̺ E ( α ) | k i = L X i =1 h j | ˆ K i | α i h α | ˆ K † i | k i = L X i =1 h α | ˆ K † i | k i h j | ˆ K i | α i = h α | E ∗ ( | k i h j | ) | α i , (A.1) uantum process tomography with coherent states E ∗ : B ( H ) → B ( H ) , ˆ B L X i =1 ˆ K † i ˆ B ˆ K i , (A.2)is the dual or adjoint map [29]. The complex-valued function h α | ˆ A | α i (referred to as Husimifunction if ˆ A is a density operator)—where ˆ A is any bounded operator on H —is an entirefunction of the two variables α and ¯ α [13, 30]. Hence the right hand side of Eq. (A.1) impliesthat the function h j | ̺ E ( α ) | k i is an entire function. By representing the coherent states inEq. (A.1) in the Fock basis and using Eq. (6), we obtain h j | ̺ E ( α ) | k i = e −| α | ∞ X n =0 ∞ X m =0 α n ¯ α m √ n ! m ! E nmjk , (A.3)which is a power series of the complex variables α and ¯ α , hence convergent everywhere[13, 30]. Appendix B. Process tensor for parametric down-conversion