PHYSICAL REVIEW A 75, 062331 2007 Direct characterization of quantu dynaics: General theory M. Mohseni 1,2 and D. A. Lidar 2,3 1 Departent of Cheistry and Cheical Biology, Harvard University, 12 Oxford St., Cabridge, Massachusetts 012138, USA 2 Departent of Cheistry, University of Southern California, Los Angeles, California 90089, USA 3 Departents of Electrical Engineering and Physics, University of Southern California, Los Angeles, California 90089, USA Received 28 March 2007; published 26 June 2007 The characterization of the dynaics of quantu systes is a task of both fundaental and practical iportance. A general class of ethods which have been developed in quantu inforation theory to accoplish this task is known as quantu process toography QPT. In an earlier paper M. Mohseni and D. A. Lidar Phys. Rev. Lett. 97, 170501 2006 we presented an algorith for direct characterization of quantu dynaics DCQD of two-level quantu systes. Here we provide a generalization by developing a theory for direct and coplete characterization of the dynaics of arbitrary quantu systes. In contrast to other QPT schees, DCQD relies on quantu error-detection techniques and does not require any quantu state toography. We deonstrate that for the full characterization of the dynaics of nd-level quantu systes with d prie, the inial nuber of required experiental configurations is reduced quadratically fro d 4n in separable QPT schees to d 2n in DCQD. DOI: 10.1103/PhysRevA.75.062331 PACS nubers: 03.67.Pp, 03.65.Wj I. INTRODUCTION The characterization of quantu dynaical systes is a fundaental proble in quantu physics and quantu cheistry. Its ubiquity is due to the fact that knowledge of quantu dynaics of open or closed quantu systes is indispensable in prediction of experiental outcoes. In particular, accurate estiation of an unknown quantu dynaical process acting on a quantu syste is a pivotal task in coherent control of the dynaics, especially in verifying and/or onitoring the perforance of a quantu device in the presence of decoherence. The procedures for characterization of quantu dynaical aps are traditionally known as quantu process toography QPT 1 3. In ost QPT schees the inforation about the quantu dynaical process is obtained indirectly. The quantu dynaics is first apped onto the states of an enseble of probe quantu systes, and then the process is reconstructed via quantu state toography of the output states. Quantu state toography is itself a procedure for identifying a quantu syste by easuring the expectation values of a set of noncouting observables on identical copies of the syste. There are two general types of QPT schees. The first is standard quantu process toography SQPT 1,4,5. In SQPT all quantu operations, including preparation and state toography easureents, are perfored on the syste whose dynaics is to be identified the principal syste, without the use of any ancillas. The SQPT schee has already been experientally deonstrated in a variety of systes including liquid-state nuclear agnetic resonance NMR 6 8, optical 9,10, atoic 11, and solid-state systes 12. The second type of QPT schee is known as ancilla-assisted process toography AAPT 13 16. In AAPT one akes use of an ancilla auxilliary syste. First, the cobined principal syste and ancilla are prepared in a faithful state, with the property that all inforation about the dynaics can be iprinted on the final state 13,15,16. The relevant inforation is then extracted by perforing quantu state toography in the joint Hilbert space of syste and ancilla. The AAPT schee has also been deonstrated experientally 15,17. The total nuber of experiental configurations required for easuring the quantu dynaics of nd-level quantu systes qudits is d 4n for both SQPT and separable AAPT, where separable refers to the easureents perfored at the end. This nuber can in principle be reduced by utilizing nonseparable easureents, e.g., a generalized easureent 1. However, the nonseparable QPT schees are rather ipractical in physical applications because they require any-body interactions, which are not experientally available or ust be siulated at high resource cost 3. Both SQPT and AAPT ake use of a apping of the dynaics onto a state. This raises the natural question of whether it is possible to avoid such a apping and instead perfor a direct easureent of quantu dynaics, which does not require any state toography. Moreover, it sees reasonable that by avoiding the indirect apping one should be able to attain a reduction in resource use e.g., the total nuber of easureents required, by eliinating redundancies. Indeed, there has been a growing interest in the developent of direct ethods for obtaining specific inforation about the states or dynaics of quantu systes. Exaples include the estiation of general functions of a quantu state 18, detection of quantu entangleent 19, easureent of nonlinear properties of bipartite quantu states 20, reconstruction of quantu states or dynaics fro incoplete easureents 21, estiation of the average fidelity of a quantu gate or process 22,23, and universal source coding and data copression 24. However, these schees cannot be used directly for a coplete characterization of quantu dynaics. In Ref. 25 we presented such a schee, which we called direct characterization of quantu dynaics DCQD. In trying to address the proble of direct and coplete characterization of quantu dynaics, we were inspired by the observation that quantu error detection QED 1 provides a eans to directly obtain partial inforation about the 1050-2947/2007/756/06233115 062331-1 2007 The Aerican Physical Society
M. MOHSENI AND D. A. LIDAR PHYSICAL REVIEW A 75, 062331 2007 A B Bell state preparation BSM FIG. 1. Color online Scheatic of DCQD for a single qubit, consisting of Bell-state-type preparations, application of the unknown quantu ap, E, and Bell-state easureent BSM. nature of a quantu process, without ever revealing the state of the syste. In general, however, it is unclear if there is a fundaental relationship between QED and QPT, naely whether it is possible to copletely characterize the quantu dynaics of arbitrary quantu systes using QED. And, providing the answer is affirative, how the physical resources scale with syste size. Moreover, one would like to understand whether entangleent plays a fundaental role, and what potential applications eerge fro such a theory linking QPT and QED. Finally, one would hope that this approach ay lead to new ways of understanding and/or controlling quantu dynaical systes. We addressed these questions for the first tie in Ref. 25 by developing the DCQD algorith in the context of two-level quantu systes. In DCQD see Fig. 1 the state space of an ancilla is utilized such that experiental outcoes fro a Bell-state easureent provide direct inforation about specific properties of the underlying dynaics. A coplete set of probe states is then used to fully characterize the unknown quantu dynaics via application of a single Bell-state easureent device 3,25. Here we generalize the theory of Ref. 25 to arbitrary open quantu systes undergoing an unknown, copletely positive CP quantu dynaical ap. In the generalized DCQD schee, each probe qudit with d prie is initially entangled with an ancillary qudit syste of the sae diension, before being subjected to the unknown quantu process. To extract the relevant inforation, the corresponding easureents are devised in such a way that the final joint probability distributions of the outcoes are directly related to specific sets of the dynaical superoperator s eleents. A coplete set of probe states can then be utilized to fully characterize the unknown quantu dynaical ap. The preparation of the probe systes and the easureent schees are based on QED techniques, however, the objective and the details of the error-detection schees are different fro those appearing in the protection of quantu systes against decoherence the original context of QED. More specifically, we develop error-detection schees to directly easure the coherence in a quantu dynaical process, represented by off-diagonal eleents of the corresponding superoperator. We explicitly deonstrate that for characterizing a dynaical ap on n qudits, the nuber of required experiental configurations is reduced fro d 4n,in SQPT and separable AAPT, to d 2n in DCQD. A useful feature of DCQD is that it can be efficiently applied to partial characterization of quantu dynaics 25,26. For exaple, it can be used for the task of Hailtonian identification, and also for siultaneous deterination of the relaxation tie T 1 and the dephasing tie T 2. This paper is organized as follows. In Sec. II, we provide a brief review of copletely positive quantu dynaical aps, and the relevant QED concepts such as stabilizer codes and noralizers. In Sec. III, we deonstrate how to deterine the quantu dynaical populations, or diagonal eleents of a superoperator, through a single enseble easureent. In order to further develop the DCQD algorith and build the required notations, we introduce soe leas and definitions in Sec. IV, and then we address the characterization of quantu dynaical coherences, or offdiagonal eleents of a superoperator, in Sec. V. In Sec. VI, we show that easureent outcoes obtained in Sec. V provide d 2 linearly independent equations for estiating the coherences in a process, which is in fact the axiu aount of inforation that can be extracted in a single easureent. A coplete characterization of the quantu dynaics, however, requires obtaining d 4 independent real paraeters of the superoperator for nontrace preserving aps. In Sec. VII, we deonstrate how one can obtain coplete inforation by appropriately rotating the input state and repeating the above algorith for a coplete set of rotations. In Secs. VIII and IX, we address the general constraints on input stabilizer codes and the iniu nuber of physical qudits required for the encoding. In Sec. X and Sec. XI, we define a standard notation for stabilizer and noralizer easureents and then provide an outline of the DCQD algorith for the case of a single qudit. For convenience, we provide a brief suary of the entire DCQD algorith in Sec. XII. We conclude with an outlook in Sec. XIII. In the Appendix, we generalize the schee for arbitrary open quantu systes. For a discussion of the experiental feasibility of DCQD see Ref. 25, and for a detailed and coprehensive coparison of the required physical resources in different QPT schees see Ref. 3. II. PRELIMINARIES In this section we introduce the basic concepts and notation fro the theory of open quantu syste dynaics and quantu error detection, required for the generalization of the DCQD algorith to qudits. A. Quantu dynaics The evolution of a quantu syste open or closed can, under natural assuptions, be expressed in ters of a copletely positive quantu dynaical ap E, which can be represented as 1 E =,n=0 n E E n. Here is the initial state of the syste, and the E are a set of error operator basis eleents in the Hilbert-Schidt space of the linear operators acting on the syste. That is, any arbitrary operator acting on a d-diensional quantu syste can be expanded over an orthonoral and unitary error operator basis E 0,E 1,...,E, where E 0 =I and tre i E j =d ij 27. The n are the atrix eleents of the superoperator, or process atrix, which encodes all the inforation about the dynaics, relative to the basis set E 1 062331-2
DIRECT CHARACTERIZATION OF QUANTUM DYNAMICS: 1. For an n-qudit syste, the nuber of independent atrix eleents in is d 4n for a non-trace-preserving ap and d 4n d 2n for a trace-preserving ap. The process atrix is positive and Tr 1. Thus can be thought of as a density atrix in the Hilbert-Schidt space, when we often refer to its diagonal and off-diagonal eleents as quantu dynaical population and quantu dynaical coherence, respectively. In general, any successive operation of the error operator basis can be expressed as E i E j = k i,j,k E k, where i, j,k =0,1,...,. However, we use the very nice error operator basis in which E i E j = i,j E i*j, det E i =1, i,j is a dth root of unity, and the operation * induces a group on the indices 27. This provides a natural generalization of the Pauli group to higher diensions. Any eleent E i can be generated fro appropriate products of X d and Z d, where X d k=k+1, Z d k= k k, and X d Z d = 1 Z d X d 27,28. Therefore, for any two eleents E i=a,q,p = a X q p d Z d and E j=a,q,p= a X q d Zd p where 0q, pd of the singlequdit Pauli group, we always have E i E j = pq qp E j E i, 2 where pq qp kod d. 3 The operators E i and E j coute if and only if k=0. Henceforth, all algebraic operations are perfored in odd arithetic, and all quantu states and operators, respectively, belong to and act on a d-diensional Hilbert space. For siplicity, fro now on we drop the subscript d fro the operators. B. Quantu error detection In the last decade the theory of quantu error correction QEC has been developed as a general ethod for detecting and correcting quantu dynaical errors acting on ultiqubit systes such as a quantu coputer 1. QEC consists of three steps: preparation, quantu error detection QED or syndroe easureents, and recovery. In the preparation step, the state of a quantu syste is encoded into a subspace of a larger Hilbert space by entangling the principal syste with soe other quantu systes using unitary operations. This encoding is designed to allow detection of arbitrary errors on one or ore physical qubits of a code by perforing a set of QED easureents. The easureent strategy is to ap different possible sets of errors only to orthogonal and undefored subspaces of the total Hilbert space, such that the errors can be unabiguously discriinated. Finally the detected errors can be corrected by applying the required unitary operations on the physical qubits during the recovery step. A key observation relevant for our purposes is that by perforing QED one can actually obtain partial inforation about the dynaics of an open quantu syste. For a qudit in a general state c in the code space, and for arbitrary error basis eleents E and E n, the Knill- Laflae QEC condition for degenerate codes is PHYSICAL REVIEW A 75, 062331 2007 FIG. 2. Color online A scheatic diagra of quantu error detection QED. The projective easureents corresponding to eigenvalues of stabilizer generators are represented by arrows. For a nondegenerate QEC code, after the QED, the wave function of the ultiqubit syste collapses into one of the orthogonal subspaces each of which is associated with a single error operator. Therefore, all errors can be unabiguously discriinated. For degenerate codes, by perforing QED the code space also collapses into a set orthogonal subspaces. However, each subspace has ultiple degeneracies aong k error operators in a subset of the operator basis, k d i.e., E =1 E i 2 1 i=0. In this case, one cannot distinguish between k different operators within a particular subset E 0 =1. c E n E c = n, where n is a Heritian atrix of coplex nubers 1. For nondegenerate codes, the QEC condition reduces to c E n E c = n ; i.e., in this case the errors always take the code space to orthogonal subspaces. The difference between nondegenerate and degenerate codes is illustrated in Fig. 2. In this work, we concentrate on a large class of error-correcting codes known as stabilizer codes 29; however, in contrast to QEC, we restrict our attention alost entirely to degenerate stabilizer codes as the initial states. Moreover, by definition of our proble, the recovery and/or correction step is not needed or used in our analysis. A stabilizer code is a subspace H C of the Hilbert space of n qubits that is an eigenspace of a given Abelian subgroup S of the n-qubit Pauli group with the eigenvalue +1 1,29. In other words, for c H C and S i S, we have S i c = c, where S i s are the stabilizer generators and S i,s j =0. Consider the action of an arbitrary error operator E on the stabilizer code c, E c. The detection of such an error will be possible if the error operator anticoutes with at least one of the stabilizer generators, S i,e=0. That is, by easuring all generators of the stabilizer and obtaining one or ore negative eigenvalues we can deterine the nature of the error unabiguously as S i E c = ES i c = E c. A stabilizer code n,k,d c represents an encoding of k logical qudits into n physical qudits with code distance d c, such that an arbitrary error on any subset of t=d c 1/2 or fewer qudits can be detected by QED easureents. A stabilizer group with n k generators has d n k eleents and the code space is d k diensional. Note that this is valid when d is a power of a prie 28. The unitary operators that preserve the stabilizer group by conjugation, i.e., USU =S, are called the noralizer of the stabilizer group, NS. Since the noralizer eleents preserve the code space they can be used to perfor certain logical operations in the code space. 062331-3
M. MOHSENI AND D. A. LIDAR PHYSICAL REVIEW A 75, 062331 2007 However, they are insufficient for perforing arbitrary quantu operations 1. Siilarly to the case of a qubit 25, the DCQD algorith for the case of a qudit syste consists of two procedures: i a single experiental configuration for characterization of the quantu dynaical populations, and ii experiental configurations for characterization of the quantu dynaical coherences. In both procedures we always use two physical qudits for the encoding, the principal syste A and the ancilla B, i.e., n=2. In procedure i characterizing the diagonal eleents of the superoperator the stabilizer group has two generators. Therefore it has d 2 eleents and the code space consists of a single quantu state i.e., k=0. In procedure ii characterizing the off-diagonal eleents of the superoperator the stabilizer group has a single generator, thus it has d eleents, and the code space is two diensional. That is, we effectively encode a logical qudit i.e., k =1 into two physical qudits. In the next sections, we develop the procedures i and ii in detail for a single qudit with d being a prie, and in the Appendix we address the generalization to systes with d being an arbitrary power of a prie. III. CHARACTERIZATION OF QUANTUM DYNAMICAL POPULATION To characterize the diagonal eleents of the superoperator, or the population of the unitary error basis, we use a nondegenerate stabilizer code. We prepare the principal qudit, A, and an ancilla qudit, B, in a coon +1 eigenstate c of the two unitary operators E A i E B j and E A i E B j, such that E A i E B j,e A i E B j =0 e.g., X A X B and Z A Z B d 1. Therefore, siultaneous easureent of these stabilizer generators at the end of the dynaical process reveals arbitrary single qudit errors on the syste A. The possible outcoes depend on whether a specific operator in the operator-su representation of the quantu dynaics coutes with E A i E B j and E A i E B j, with the eigenvalue +1, or with one of the eigenvalues, 2,..., d 1. The projection operators corresponding to outcoes k and k, where k,k=0,1,...,d 1, have the for P k = 1 d d 1 lk E A i E B j l and P k = 1 d d 1 lk E A i E B j l. The joint probability distribution of the couting Heritian operators P k and P k on the output state E =,n n E E n, where = c c,is TrP k P k E = 1 d 2,n=0 d 1 d 1 n lk lk TrE n E A i l E A i l E E B j l E B j l. Using E i E = i E E i and the relation E A i E B j l E A i E B j l =, we obtain TrP k P k E = 1 d 2,n=0 d 1 d 1 n i kl i kl n, where we have used the QED condition for nondegenerate codes FIG. 3. Color online A diagra of the error-detection easureent for estiating quantu dynaical population. The arrows represent the projection operators P k P k corresponding to different eigenvalues of the two stabilizer generators S and S. These projective easureents result in a projection of the wave function of the two-qudit systes, after experiencing the dynaical ap, into one of the orthogonal subspaces each of which is associated to a specific error operator basis. By calculating the joint probability distribution of all possible outcoes, P k P k, for k,k=0,...,d, we obtain all d 2 diagonal eleents of the superoperator in a single enseble easureent. TrE n E = c E n E c = n i.e., the fact that different errors should take the code space to orthogonal subspaces, in order for errors to be unabiguously detectable, see Fig. 3. Now, using the discrete Fourier transfor identities d 1 i kl =d i,k and d 1 l =0 i kl =d i,k, we obtain TrP k P k E = i,k i,k = 0 0. 4 =0 Here, 0 is defined through the relations i 0 =k and i 0 =k, i.e., E 0 is the unique error operator that anticoutes with the stabilizer operators with a fixed pair of eigenvalues k and k corresponding to the experiental outcoes k and k. Since each P k and P k operator has d eigenvalues, we have d 2 possible outcoes, which gives us d 2 linearly independent equations. Therefore, we can characterize all the diagonal eleents of the superoperator with a single enseble easureent and 2d detectors. In order to investigate the properties of the pure state c, we note that the code space is one diensional i.e., it has only one vector and can be Schidt decoposed as c = d 1 k=0 k k A k B, where k are non-negative real nubers. Suppose Zk= k k; without loss of generality the two stabilizer generators of c can be chosen to be X A X B q and Z A Z B d 1 p. We then have c X A X B q c =1 and c Z A Z B d 1 p c =1 for any q and p, where 0q, pd. This results in the set of equations d 1 k=0 k k+q =1 for all q, which have only one positive real solution, 0 = 1 = = k =1/ d; i.e., the stabilizer state, c, is a axially entangled state in the Hilbert space of the two qudits. In the reaining parts of this paper, we first develop an algorith for extracting optial inforation about the dynaical coherence of a d-level quantu syste with d being a prie, through a single experiental configuration, in Secs. IV VI. Then, we further develop the algorith to obtain coplete inforation about the off-diagonal eleents of the superoperator by repeating the sae schee for different input states, Sec. VII. In the Appendix, we address the generalization of the DCQD algorith for qudit systes with d 062331-4
DIRECT CHARACTERIZATION OF QUANTUM DYNAMICS: being a power of a prie. In the first step, in the next section, we establish the required notation by introducing soe leas and definitions. IV. BASIC LEMMAS AND DEFINITIONS Lea 1. Let 0q, p,q, pd, where d is prie. Then, for given q, p, q and kod d, there is a unique p that solves pq qp=kod d. Proof. We have pq qp=kod d=k+td, where t is an integer. The possible solutions for p are indexed by t as pt=pq k td/q. We now show that if pt 1 is a solution for a specific value t 1, there exists no other integer t 2 t 1 such that pt 2 is another independent solution to this equation, i.e., pt 2 pt 1 od d. First, note that if pt 2 is another solution then we have pt 1 = pt 2 +t 2 t 1 d/q. Since d is prie, there are two possibilities: a q divides t 2 t 1, then t 2 t 1 d/q=±nd, where n is a positive integer; therefore we have pt 2 = pt 1 od d, which contradicts our assuption that pt 2 is an independent solution fro pt 1. b q does not divide t 2 t 1, then t 2 t 1 d/q is not a integer, which is unacceptable. Thus, we have t 2 =t 1, i.e., the solution pt is unique. Note that the above arguent does not hold if d is not prie, and therefore, for soe q there could be ore than one p that satisfies pq qpkod d. In general, the validity of this lea relies on the fact that Z d is a field only for prie d. Lea 2. For any unitary error operator basis E i acting on a Hilbert space of diension d, where d is a prie and i=0,1,...,, there are d unitary error operator basis eleents, E j, that anticoute with E i with a specific eigenvalue k, i.e., E i E j = k E j E i, where k=0,...,d 1. Proof. We have E i E j = pq qp E j E i, where 0q, p,q, p d, and pq qpkod d. Therefore, for fixed q, p, and k od d we need to show that there are d solutions q, p. According to Lea 1, for any q there is only one p that satisfies pq qp=kod d; but qcan have d possible values, therefore there are d possible pairs of q, p. Definition 1. We introduce d different subsets, W k i, k =0,1,...,d 1, of a unitary error operator basis E j i.e., W k i E j. Each subset contains d ebers which all anticoute with a particular basis eleent E i, where i =0,1,...,, with fixed eigenvalue k. The subset W 0 i which includes E 0 and E i is in fact an Abelian subgroup of the single-qudit Pauli group, G 1. V. CHARACTERIZATION OF QUANTUM DYNAMICAL COHERENCE For characterization of the coherence in a quantu dynaical process acting on a qudit syste, we prepare a twoqudit quantu syste in a nonseparable eigenstate ij of a unitary operator S ij =E i A E j B. We then subject the qudit A to the unknown dynaical ap, and easure the sole stabilizer operator S ij at the output state. Here, the state ij is in fact a degenerate code space, since all the operators E A that anticoute with E i A, with a particular eigenvalue k, perfor the sae transforation on the code space and cannot be distinguished by the stabilizer easureent. If we express the spectral decoposition of S ij =E A i E B j as S ij = k k P k, the projection operator corresponding to the outcoe k can be written as P k = 1 d d 1 lk E A i E B j l. The post-easureent state of the syste, up to a noralization factor, will be P k EP k = 1 d 2 PHYSICAL REVIEW A 75, 062331 2007,n=0 d 1 d 1 n lk lk E i A E j B l E E n E i A E j B l. Using the relations E i E = i E E i, E n E i = i ne i E n E A i E B j l E A i E B j l = we have P k EP k = 1 d 1 d 1 d 2 i kl k i n l n E E n.,n=0 and Now, using the discrete Fourier transfor properties d 1 i kl =d i,k and d 1 l =0 k inl =d in,k, we obtain P k EP k = E A E A + n E A E A n + * n E A n E A. Here, the suation runs over all E A and E n B that belong to the sae W k i ; see Lea 2. That is, the suation is over all unitary operator basis eleents E A and E n B that anticoute with E i A with a particular eigenvalue k. Since the nuber of eleents in each W k is d, the state of the two-qudit syste after the projective easureent coprises d +2dd 1/2=d 2 ters. The probability of getting the outcoe k is TrP k E = +2 Re n TrE A n E A. Therefore, the noralized post-easureent states are k = P k EP k /TrP k E. These d equations provide us with inforation about off-diagonal eleents of the superoperator if and only if TrE A n E A 0. Later we will derive soe general properties of the state such that this condition can be satisfied. Next we easure the expectation value of any other unitary operator basis eleent T rs =E A r E B s on the output state, such that E A r I, E B s I, T rs NS, and T rs S ij a, where 0ad. Let us write the spectral decoposition of T rs as T rs = k k P k. The joint probability distribution of the couting Heritian operators P k and P k on the output state E is TrP k P k E. The average of these joint probability distributions of P k and P k over different values of k becoes k k TrP k P k E=TrT rs P k E=TrT rs k, which can be explicitly written as 5 6 062331-5
M. MOHSENI AND D. A. LIDAR PHYSICAL REVIEW A 75, 062331 2007 FIG. 4. Color online A diagra of the error-detection easureent for estiating quantu dynaical coherence: we easure the sole stabilizer generator at the output state, by applying projection operators corresponding to its different eigenvalues P k. We also easure d 1 couting operators that belong to the noralizer group. Finally, we calculate the probability of each stabilizer outcoe, and joint probability distributions of the noralizers and the stabilizer outcoes. Optially, we can obtain d 2 linearly independent equations by appropriate selection of the noralizer operators as it is shown in the next section. TrT rs k = TrE A E A r E B s E A + n TrE A n E A r E B s E A * + n TrE A E A r E B s E A n. Using E A r E A = r E A E A r and E A r E A n = r ne A n E A r this becoes 1 TrT rs k = TrP k E r TrT rs + r n TrE A n E A T rs + r * n n TrE A E A n T rs. Therefore, we have an additional set of d equations to identify the off-diagonal eleents of the superoperator, provided that TrE A n E A T rs 0. Suppose we now easure another unitary operator T r s =E r A E B s that coutes with S ij, i.e., T r s NS, and also coutes with T rs, and satisfies the relations T r s T rs b S a ij where 0a,bd, E A B r I and E s I. Such a easureent results in d equations for TrT r s k, siilar to those for TrT rs k. However, for these equations to be useful for characterization of the dynaics, one ust show that they are all linearly independent. In the next section, we find the axiu nuber of independent and coutating unitary operators T rs such that their expectation values on the output state, TrT rs k, result in linearly independent equations to be d 1, see Fig. 4. That is, we find an optial Abelian set of unitary operators such that the joint probability distribution functions of their eigenvalues and stabilizer eigenvalues at the output state are linearly independent. VI. LINEAR INDEPENDENCE AND OPTIMALITY OF MEASUREMENTS Before presenting the proof of linear independence of the functions TrT rs k and of the optiality of the DCQD algo- 7 rith, we need to introduce the following leas and definitions. Lea 3. If a stabilizer group, S, has a single generator, the order of its noralizer group, NS, isd 3. Proof. Let us consider the sole stabilizer generator S 12 =E A 1 E B 2, and a typical noralizer eleent T 1 2 =E 1 A E B 2, where E A 1 =X q 1Z p 1, E B 2 =X q 2Z p 2, E A 1 =X q 1Z p B 1, and E 2 =X q 2Z p 2. Since S 12 and T 1 2 coute, we have S 12T 1 2 = 2 i=1 pi q i qi p i T1 2 S 12, where 2 i=1 p i q i q i p i 0od d. We note that for any particular code with a single stabilizer generator, all q 1, p 1, q 2, and p 2 are fixed. Now, by Lea 1, for given values of q 1, p 1, and q 2 there is only one value for p 2 that satisfies the above equation. However, each q 1, p 1, and q 2 can have d different values. Therefore, there are d 3 different noralizer eleents T 1 2. Lea 4. Each Abelian subgroup of a noralizer, which includes the stabilizer group S a ij as a proper subgroup, has order d 2. Proof. Suppose T rs is an eleent of NS, i.e., it coutes with S ij. Moreover, all unitary operators of the for T b rs S a ij, where 0a,bd, also coute. Therefore, any Abelian subgroup of the noralizer, ANS, which includes S a ij as a proper subgroup, is at least an order of d 2. Now let T r s be any other noralizer eleent, i.e., T rs T rs b S a ij with 0a, bd, which belongs to the sae Abelian subgroup A. In this case, any operator of the for T b r Trs s b S a ij would also belong to A. Then all eleents of the noralizer should coute or A=NS, which is unacceptable. Thus, either T r s =T b rs S a ij or T r sa, i.e., the order of the Abelian subgroup A is at ost d 2. Lea 5. There are d+1 Abelian subgroups, A, inthe noralizer NS. Proof. Suppose that the nuber of Abelian subgroups which includes the stabilizer group as a proper subgroup is n. Using Leas 3 and 4, we have d 3 =nd 2 n 1d, where the ter n 1d has been subtracted fro the total nuber of eleents of the noralizer due to the fact that the eleents of the stabilizer group are coon to all Abelian subgroups. Solving this equation for n, we find that n= d2 1 d 1 =d+1. Lea 6. The basis of eigenvectors defined by d+1 Abelian subgroups of NS are utually unbiased. Proof. It has been shown 30 that if a set of traceless and utually orthogonal dd unitary atrices can be partitioned into d+1 subsets of equal size, such that the d 1 unitary operators in each subset coute, then the basis of eigenvectors corresponding to these subsets are utually unbiased. We note that, based on Leas 3, 4, and 5, and in the code space i.e., up to ultiplication by the stabilizer eleents S a ij, the noralizer NS has nontrivial eleents, and each Abelian subgroup A, has d 1 nontrivial couting operators. Thus, the bases of eigenvectors defined by d+1 Abelian subgroups of NS are utually unbiased. Lea 7. Let C be a cyclic subgroup of A, i.e., CANS. Then, for any fixed TA, the nuber of distinct left right cosets, TCCT, in each A is d. Proof. We note that the order of any cyclic subgroup 062331-6
DIRECT CHARACTERIZATION OF QUANTUM DYNAMICS: CA, such as T b rs with 0bd, isd. Therefore, by Lea 4, the nuber of distinct cosets in each A is d2 d =d. Definition 2. We denote the cosets of an invariant cyclic subgroup, C a, of an Abelian subgroup of the noralizer, A v, by A v /C a, where v=1,2,...,d+1. We also represent generic ebers of A v /C a as T b rs S a ij, where 0a,bd. The ebers of a specific coset A v /C a0 are denoted as T b a rs S 0 ij, where a 0 represents a fixed power of stabilizer generator S ij, that labels a particular coset A v /C a0, and b 0bd labels different ebers of that particular coset. b a S 0 ij Lea 8. The eleents of a coset, T rs where T rs =E A r E B s, S ij =E A i E B j, and 0bd anticoute with E A i with different eigenvalues k. That is, there are no two different ebers of a coset, A v /C a0, that anticoute with E A i with the sae eigenvalue. Proof. First we note that for each T b rs =E A r b E B s b, the unitary operators acting only on the principal subsyste, E A r b, ust satisfy either a E A r b =E A i or b E A r b E A i.inthe case a, and due to T rs,s ij =0, we should also have E B s b =E B j, which results in T b rs =S ij ; i.e., T b rs is a stabilizer and not a noralizer. This is unacceptable. In the case b, in particular for b=1, we have E A r E A i = r ie A i E A r. Therefore, for arbitrary b we have E A r b E A i = br ie A i E A r b. Since 0bd, we conclude that br i br i for any two different values of b and b. As a consequence of this lea, different E A r b, for 0 bd, belong to different W i k s. Lea 9. For any fixed unitary operator E A r W i k, where k0, and any other two independent operators E A A and E n that belong to the sae W i k, we always have r r n, where E A r E A = r E A E A r and E A r E A n = r ne A n E A r. Proof. We need to prove for operators E A r,e A,E A i n W k where k0, that we always have E A E A n r r n. Let us prove the converse, r = r n E A =E A A n. We define E i =X q iz p i, E A r =X q rz p r, E A =X q Z p, E A n =X q nz p n. Based on the i definition of subsets W k with k0, we have p i q q i p p i q n q i p n =kod d=k+td I, where t is an integer nuber. We need to show if p r q q r p p r q n q r p n =kod d=k+td II, then E A =E n A. We divide the equations I by q i q or q i q n to get = k+td q i q + p q = k+td q i q n + p n q n q r q or q r q n to get p r p i q i I. We also divide the equations II by q r = k+td q r q + p q = k+td q r q n + p n q n II. By subtracting the equation II fro I we get q n k + td q i k + td q r = q k + td q i Siilarly, we can obtain the equation k + td p nk + tdp i p r = p k + td p i k + td q r. 8 k + td p r. 9 Note that the expressions within the parentheses in both equations 8 or 9 cannot be siultaneously zero, because it will result in p i q r q i p r =0, which is unacceptable for k 0. Therefore, the expression within the parentheses in at least one of the equations 8 or 9 is nonzero. This results in q n =q and/or p n = p. Consequently, considering the equation 8, we have E A =E n A. A. Linear independence of the joint distribution functions Theore 1. The expectation values of noralizer eleents on a post-easureent state, k, are linearly independent if these eleents are the d 1 nontrivial ebers of a coset A v /C a0. That is, for two independent operators T rs, T r s A v/c a0, we have TrT rs k c TrT r s k, where c is an arbitrary coplex nuber. Proof. We know that the eleents of a coset can be written as T b a rs S 0 ij =E A r E B s b a S 0 ij, where b=1,2,...,d 1. We also proved that E A r b i belongs to different W k k0 for different values of b see Lea 8. Therefore, according to Lea 9 and regardless of the outcoe of k after easuring the stabilizer S ij, there exists one eber in the coset A v /C a0 that has different eigenvalues r with all independent ebers E A W i a k. The expectation value of T 0 rs is TrT b rs S ij PHYSICAL REVIEW A 75, 062331 2007 a 0 k = b S ij TrE A T b a rs S 0 ij E A, + n TrE A n T b a rs S 0 ij E A * + n TrE A T b a rs S 0 ij E A n, TrT b rs k = br TrT b rs 10 + br n TrE A n E A T b rs + br * n n TrE A E A n T b rs, 11 where r r n for all eleents E A,E A n, that belong to a specific W i k. Therefore, for two independent ebers of a coset denoted by b and b i.e., b b, we have br, br n,...c br, br n,... for all values of 0 b,bd, and any coplex nuber c. We also note that we have TrE A n E A T b rs c TrE A n E A T b rs, since b b Trs is a noralizer, not a stabilizer eleent, and its action on the state cannot be expressed as a global phase. Thus, for any two independent ebers of a coset A v /C a0, we always have TrT b rsk c TrT b rs k. In suary, after the action of the unknown dynaical process, we easure the eigenvalues of the stabilizer generator, E A i E B j, that has d eigenvalues for k=0,1,...,d 1 and provides d linearly independent equations for the real and iaginary parts of n. This is due to the fact that the outcoes corresponding to different eigenvalues of a unitary operator are independent. We also easure expectation values of all the d 1 independent and couting noralizer operators T b a rs S 0 ij A v /C a0, on the post-easureent state k, which provides d 1 linearly independent equations for each outcoe k of the stabilizer easureents. Overall, we obtain d+dd 1=d 2 linearly independent equations for characterization of the real and iaginary parts of n by a single enseble easureent. In the following, we show 062331-7
M. MOHSENI AND D. A. LIDAR PHYSICAL REVIEW A 75, 062331 2007 that the above algorith is optial. That is, within the d 2 Hilbert space of principal syste and ancilla, there does not exist any other possible strategy that can provide ore than d 2 linearly independent equations by a single easureent on the output state E. B. Optiality Theore 2. The axiu nuber of couting noralizer eleents that can be easured siultaneously to provide linear independent equations for the joint distribution functions TrT b rs S a ij k is d 1. Proof. Any Abelian subgroup of the noralizer has order d 2 see Lea 4. Therefore, the desired noralizer operators should all belong to a particular A v and are liited to d 2 ebers. We already showed that the outcoes of easureents for d 1 eleents of a coset A v /C a, represented by T b rs S a ij with b0, are independent see Theore 1. Now we show that easuring any other operator, T b rs S a ij, fro any other coset A v /C a, results in linearly dependent equations for the functions w=trt b rs S a ij k and w=trt b rs S a ij k as the following: w =TrT b rs S a ij k = TrE A T b rs S a ij E A + n TrE A n T b rs S a ij E A * + n TrE A T b rs S a ij E A n, w =TrT b rs S a ij k = TrE A T b rs S ij a E A + n TrE A n T b rs S a A ij E * + n tre A T b rs S a A ij En. Using the coutation relations T b rs S a ij E A = br +ai E A T b rs S a ij, we obtain w = br +ai TrT b rs + br +ai n TrE A n E A T b rs + br n +ai * n n TrE A E A n T b rs w = br +ai trt b rs + br +ai n TrE A n E A T b rs + br n +ai * n n TrE A E A n T b rs, where we also used the fact that both S a ij and S a ij are stabilizer eleents. Since all of the operators E A belong to the sae W i k, we have i =i n =k, and obtain w = ak br TrT b rs + br n TrE A n E A T b rs + br * n n TrE A E A n T b rs, w = ak br TrT b rs + br n TrE A n E A T b rs + br * n n TrE A E A n T b rs. Thus, we have w= a ak w, and consequently the easureents of operators fro other cosets A v /C a do not provide any new inforation about n beyond the corresponding easureents fro the coset A v /C a. For another proof of the optiality, based on fundaental liitation of transferring inforation between two parties given by the Holevo bound see Ref. 26. In principle, one can construct a set of non-abelian noralizer easureents, fro different A v, where v=1,2,...,d+1, to obtain inforation about the off-diagonal eleents n. However, deterining the eigenvalues of a set of noncouting operators cannot be done via a single easureent. Moreover, as entioned above, by easuring the stabilizer and d 1 couting noralizer eleents, one can in principle transfer log 2 d 2 bits of classical inforation between two parties, which is the axiu allowed by the Holevo bound 31. Therefore, other strategies involving non-abelian, or a ixture of Abelian and non-abelian noralizer easureents, cannot iprove the above schee. It should be noted that there are several possible alternative sets of Abelian noralizers that are equivalent for this task. We address this issue in the next lea. Lea 10. The nuber of alternative sets of Abelian noralizer easureents that can provide optial inforation about quantu dynaics, in one enseble easureent, is d 2. Proof. We have d+1 Abelian noralizers A v see Lea 5. However, there are d of the that contain unitary operators that act nontrivially on both qudit systes A and B, i.e., T b rs =E A r E B s b, where E A r I, E B s I. Moreover, in each A v we have d cosets see Lea 5 that can be used for optial characterization of n. Overall, we have d 2 possible sets of Abelian noralizers that are equivalent for our purpose. In the next section, we develop the algorith further to obtain coplete inforation about the off-diagonal eleents of the superoperator by repeating the above schee for different input states. VII. REPEATING THE ALGORITHM FOR OTHER STABILIZER STATES We have shown that by perforing one enseble easureent one can obtain d 2 linearly independent equations for n. However, a coplete characterization of quantu dynaics requires obtaining d 4 d 2 independent real paraeters of the superoperator or d 4 for nontrace preserving aps. We next show how one can obtain coplete inforation by appropriately rotating the input state and repeating the above algorith for a coplete set of rotations. Lea 11. The nuber of independent eigenkets for the error operator basis E j, where j=1,2,...,, is d+1. These eigenkets are utually unbiased. 062331-8
DIRECT CHARACTERIZATION OF QUANTUM DYNAMICS: Proof. We have unitary operators, E i. We note that the operators E a i for all values of 1ad 1 coute and have a coon eigenket. Therefore, overall we have d 2 1/d 1=d+1 independent eigenkets. Moreover, it has been shown 30 that if a set of traceless and utually orthogonal dd unitary atrices can be partitioned into d +1 subsets of equal size, such that the d 1 unitary operators in each subset coute, then the basis of eigenvectors defined by these subsets are utually unbiased. Let us construct a set of d+1 stabilizer operators E A i E B j, such that the following conditions hold: a E A i,e B j I, b E A i a E A i for ii and 1ad 1. Then, by preparing the eigenstates of these d + 1 independent stabilizer operators, one at a tie, and easuring the eigenvalues of S ij and its corresponding d 1 noralizer operators T b rs S a ij A v /C a, one can obtain d+1d 2 linearly independent equations to characterize the superoperator s off-diagonal eleents. The linear independence of these equations can be understood by noting that the eigenstates of all operators E A i of the d+1 stabilizer operator S ij are utually unbiased i.e., the easureents in these utual unbiased bases are axially noncouting. For exaple, the bases 0,1, + X, X and + Y, Y the eigenstates of the Pauli operators Z, X, and Y are utually unbiased, i.e., the inner products of each pair of eleents in these bases have the sae agnitude. Then easureents in these bases are axially noncouting 32. To obtain coplete inforation about the quantu dynaical coherence, we again prepare the eigenkets of the above d+1 stabilizer operators E A i E B j, but after the stabilizer easureent we calculate the expectation values of the operators T r S s a ij belonging to other Abelian subgroups A v /C a b of the noralizer, i.e., A v A v. According to Lea 6 the bases of different Abelian subgroups of the noralizer are utually unbiased, therefore, the expectation values of b T r S s a ij and T b rs S a ij fro different Abelian subgroups A v and A v are independent. In order to ake the stabilizer easureents also independent we choose a different superposition of logical basis in the preparation of d+1 possible stabilizer state in each run. Therefore in each of these easureents we can obtain at ost d 2 linearly independent equations. By repeating these easureents for d 1 different A v over all d+1 possible input stabilizer states, we obtain d+1d 1d 2 =d 4 d 2 linearly independent equations, which suffice to fully characterize all independent paraeters of the superoperator s off-diagonal eleents. In the next section, we address the general properties of these d + 1 stabilizer states. VIII. GENERAL CONSTRAINTS ON THE STABILIZER STATES PHYSICAL REVIEW A 75, 062331 2007 The restrictions on the stabilizer states can be expressed as follows: Condition 1. The state = ij ij is a nonseparable pure state in the Hilbert space of the two-qudit syste H. That is, ij AB A B. Condition 2. The state ij is a stabilizer state with a sole stabilizer generator S ij =E A i E B j. That is, it satisfies S a ij ij = ak ij, where k0,1,...,d 1 denotes a fixed eigenvalue of S ij, and a=1,...,d 1 labels d 1 nontrivial ebers of the stabilizer group. The second condition specifies the stabilizer subspace, V S, that the state lives in, which is the subspace fixed by all the eleents of the stabilizer group with fixed eigenvalues k. More specifically, an arbitrary state in the entire Hilbert d 1 space H can be written as = u,u =0 uu u A u B where u and u are bases for the Hilbert spaces of the qudits A and B, such that X q u=u+q and Z p u= pu u. However, we can expand in another basis as d 1 = v,v =0 vv v A v B, such that X q v= qv v and Z p v =v+ p. Let us consider a stabilizer state fixed under the action of a unitary operator E A i E B j =X A q X B q Z A p Z B p with eigenvalue k. Regardless of the basis chosen to expand ij, we should always have S ij ij = k ij. Consequently, we have the constraints pu pu=k, for the stabilizer subspace V S spanned by the u ubasis, and qv p qv p=k, ifv S is spanned by v v basis, where is addition odd. Fro these relations, and also using the fact that the bases v and u are related by a unitary transforation, one can find the general properties of V S for a given stabilizer generator E A i E B j and a given k. We have already shown that the stabilizer states should also satisfy the set of conditions TrE A n E A 0 and TrE A n E A T b rs 0 for all operators E A belonging to the sae W i k, where T b rs 0bd 1 are the ebers of a particular coset A v /C a of an Abelian subgroup, A v, of the noralizer NS. These relations can be expressed ore copactly as follows. Condition 3. For stabilizer state = ij ij c c and for all E A W i k, we have c E A n E A T b rs c 0, 12 where here 0bd 1. Before developing the iplications of the above forula for the stabilizer states we give the following definition and lea. Definition 3. Let l L be the logical basis of the code space that is fixed by the stabilizer generator E A i E B j. The stabilizer state in that basis can be written as c = d 1 l l L, and all the noralizer operators, T rs, can be generated fro tensor products of logical operations X and Z defined as Zl L = l l L and Xl L =l+1. For exaple, l L =kk, Z =Z I, and X=X X, where Xk=k+1 and Zk= k k. Lea 12. For a stabilizer generator E A i E B j and all unitary operators E A W i k, we always have E A n E A = c Z a, where Z is the logical Z operation acting on the code space and a and c are integers. Proof. Let us consider E A i =X q iz p i, and two generic operators E A n and E A that belong to W i k, E A =X q Z p A and E n =X q nz p i n. Fro the definition of W k see Definition 1 we have p i q q i p = p i q n q i p n =kod d=k+td. We can solve these two equations to get q q n =q i p q n q p n /k+td and p p n = p i p q n q p n /k+td. We also define p q n q p n =k+td. Therefore, we obtain q q n =q i a and p p n = p i a, where we have introduced 062331-9
M. MOHSENI AND D. A. LIDAR PHYSICAL REVIEW A 75, 062331 2007 a = k + td/k + td. 13 Moreover, we have E n A =X td q n Z td p n for soe other integer t. Then we get E n A E A = c X td+q q n Z td+p p n = c X q q n Z p q n = c X q iz p i a, where c=td p n td+q q n. However, X q iz p i I acts as logical Z on the code subspace, which is the eigenstate of E A i E B j. Thus, we obtain E A n E A = c Z a. Based on the above lea, for the case of b=0 we obtain d 1 c E A n E A c = c c Z a c = c al l 2. Therefore, our constraint in this case becoes d 1 k=0 al l 2 0, which is not satisfied if the stabilizer state is axially entangled. For b0, we note that T b rs are in fact the noralizers. By considering the general for of the noralizer eleents as T b rs =X q Z p b, where q, p0,1,...,d 1, weobtain c E A n E A T b rs c = c c Z a X q Z p b c d 1 = c al+bq bpl * l l+bq k=0 d 1 = c+abq a+bpl * l l+bq. Overall, the constraints on the stabilizer state, due to condition iii, can be suarized as d 1 a+bpl * l l+bq 0. 14 This inequality should hold for all b0,1,...,d 1, and all a defined by Eq. 13, however, for a particular coset A v /C a the values of q and p are fixed. One iportant property of the stabilizer code, iplied by the above forula with b=0, is that it should always be a nonaxially entangled state. In the next section, by utilizing the quantu Haing bound, we show that the iniu nuber of physical qudits, n, needed for encoding the required stabilizer state is in fact two. IX. MINIMUM NUMBER OF REQUIRED PHYSICAL QUDITS In order to characterize off-diagonal eleents of a superoperator we ust use degenerate stabilizer codes, in order to preserve the coherence between operator basis eleents. Degenerate stabilizer codes do not have a classical analog 1. Due to this fact, the classical techniques used to prove bounds for nondegenerate error-correcting codes cannot be applied to degenerate codes. In general, it is yet unknown if there are degenerate codes that exceed the quantu Haing bound 1. However, due to the siplicity of the stabilizer codes used in the DCQD algorith and their syetry, it is possible to generalize the quantu Haing bound for the. Let us consider a stabilizer code that is used for encoding k logical qudits into n physical qudits such that we can correct any subset of t or fewer errors on any n e n of the physical qudits. Suppose that 0 jt errors occur. Therefore, there are n e j possible locations, and in each location there are different operator basis eleents that can act as errors. The total possible nuber of errors is t j=0 n e j j. If the stabilizer code is nondegenerate each of these errors should correspond to an orthogonal d k -diensional subspace; but if the code is uniforly g-fold degenerate i.e., with respect to all possible errors, then each set of g errors can be fit into an orthogonal d k -diensional subspace. All these subspaces ust be fit into the entire d n -diensional Hilbert space. This leads to the following inequality: t j=0 n e d2 1 j d k d n. j g 15 We are always interested in finding the errors on one physical qudit. Therefore, we have n e =1, j0,1 and cn e j =1, d 1 2 1 j d k and Eq. 15 becoes j=0 g d n. In order to characterize diagonal eleents, we use a nondegenerate stabilizer code with n=2, k=0, and g=1, and we have 1 j=0 j =d 2. For off-diagonal eleents, we use a degenerate stabilizer code with n = 2, k = 1, and g = d, and we have 1 j=0 j d d =d 2. Therefore, in both cases the upper-bound of the quantu Haing bound is satisfied by our codes. Note that if instead we use n=k, i.e., if we encode n logical qudits j 1 into n separable physical qubits, we get j=0 g 1. This can only be satisfied if g=d 2, in which case we cannot obtain any inforation about the errors. The above arguent justifies condition i of the stabilizer state being nonseparable. Specifically, it explains why alternative encodings such as n=k=2 and n=k=1 are excluded fro our discussions. However, if we encode zero logical qubits into one physical qubit, i.e., n = 1,k = 0, then, by using a d-fold degenerate d 1 2 1 j code, we can obtain j=0 d =d which satisfies the quantu Haing bound and could be useful for characterizing off-diagonal eleents. For this to be true, the code c should also satisfy the set of conditions c E A n E A c 0 and c E A n E A T b rs c 0. Due to the d-fold degeneracy of the code, the condition c E A n E A c 0 is autoatically satisfied. However, the condition c E A n E A T b rs c 0 can never be satisfied, since the code space is one-diensional, i.e., d k =1, and the noralizer operators cannot be defined. That is, there does not exist any nontrivial unitary operator b T rs that can perfor logical operations on the onediensional code space. We have deonstrated how we can characterize quantu dynaics using the ost general for of the relevant stabilizer states and generators. In the next section, we choose a 062331-10