Quantum state tomography via mutually unbiased measurements in driven cavity QED systems

We present a feasible proposal for quantum tomography of qubit and qutrit states via mutually unbiased measurements in dispersively coupled driven cavity QED systems. We first show that measurements in the mutually unbiased bases (MUBs) are practically implemented by projecting the detected states onto the computational basis after performing appropriate unitary transformations. The measurement outcomes can then be determined by detecting the steady-state transmission spectra (SSTS) of the driven cavity. It is found that all the measurement outcomes for each MUB (i.e., all the diagonal elements of the density matrix of each detected state) can be read out directly from only one kind of SSTS. In this way, we numerically demonstrate that the exemplified qubit and qutrit states can be reconstructed with the fidelities 0.952 and 0.961, respectively. Our proposal could be straightforwardly extended to other high-dimensional quantum systems provided that their MUBs exist.


Introduction
One of the essential tasks in quantum information processing is how to extract the complete information of an unknown quantum state. A powerful method to achieve this aim is known as quantum state tomography (QST), i.e., reconstructing the density matrix of this unknown quantum state [1]. Due to its particular importance in quantum information processing, QST has received considerable attention in recent decades, and a great deal of significant advancements have been achieved in theoretical [2][3][4][5][6][7] and experimental aspects [8][9][10][11][12].
In order to realize QST, one first needs to perform a series of projective measurements on a large enough number of identically prepared copies of the quantum state, and then reconstructs its density matrix from these measurement outcomes. Prior to measurements, a crucial issue is the choice of measurement sets [13]. To increase the accuracy and efficiency in QST, several kinds of measurement sets have been explored and employed, including the set of the measurement bases, e.g., standard projective measurement bases [2,3], equidistant states [14], symmetric informationally complete positive operator-valued measures [15,16], mutually unbiased bases (MUBs) [17], and so on.
MUBs are a typical kind of measurement bases, which are defined by the property that the squared overlaps between a basis state in one basis and all basis states in the other bases are the same [17]. Physically, this means that the measurement of a particular basis state does not reveal any information about the state if it was prepared in another basis. Due to this distinct property, MUBs have extensive applications in the field of quantum information. Therein, a typical application is QST [18][19][20][21][22][23]. In the context of QST, Wootters and Fields [18] proved that the measurements in the MUBs provide a minimal and optimal way to realize QST (called MUBs-QST hereafter) in the sense of maximizing information extraction from each measurement and minimizing the effects of statistical errors in the measurements. By far, several experiments have been demonstrated to

MUBs and MUBs-QST
In a d-dimensional quantum system, two orthogonal bases {| } ( ) where ( ) ¢ l l labels one of the basis states in the ( ) ¢ k k th orthogonal basis. It is known that the number of MUBs for any dimension d is at most + d 1 [18], and exactly + d 1 if the dimension d is a prime [19] or a power of a prime [18]. Such + d 1 MUBs constitute a complete set if each pair of MUBs in this set is mutually unbiased [18]. Nevertheless, whether a complete set of MUBs exists or not in other finite dimensions is still unknown, for instance, d=6 [30][31][32].
It is well known that the measurements in the MUBs provide a minimal and optimal way to completely determine the density matrix of an unknown quantum state [18]. In terms of the MUBs, the density matrix of an arbitrary d-dimensional quantum state can be represented as [19] ( ) , defines a complete set of projective measurements, is the probability of projecting r d onto the basis state | ( ) y ñ d k l , of the kth MUB, and I is an identity operator. The measured probability can be equivalently written as where U d k is the unitary transformation that transforms the standard computational basis is the phase operation and -R 1 is its inverse operator.

Implementation of single-qubit unitary transformations
We consider a cavity QED system sketched in figure 1(a), wherein a qubit (two-level atom) shown in figure 1(b) is dispersively coupled to a cavity. In order to implement single-qubit unitary transformations, we only need to add a classical driving field on the qubit, see figure 1(b). Under the rotating-wave approximation, the whole system can be described by the Hamiltonian ( = 1; hereafter the same) In a frame rotating at the frequency ν of the classical driving field for both the qubit and the cavity, the Hamiltonian (4) is changed to  Figure 1. (a) Schematic diagram of a cavity QED system, wherein a qubit (two-level atom) or a qutrit (three-level atom) is dispersively coupled to a cavity. The qubit or qutrit states can be nondestructively read out by applying a classical driving field with the frequency w d on one side of the cavity and then detecting its SSTS T ss on the other side. κ denotes the photon decay rate of the cavity, g 1 the qubit decay rate, g j 1 ( ) = j 1, 2 the qutrit decay rates. (b) A classical driving field with the amplitude Ω and the frequency ν is applied on the qubit to implement the required single-qubit unitary transformations. (c) Two classical driving fields with the amplitude W 1 and the frequency n 1 , and the amplitude W 2 and the frequency n 2 , are applied between the energy levels | ñ 0 and | ñ 1 , and between | ñ 1 and | ñ 2 , respectively, to realize the required single-qutrit unitary transformations.
being the frequency detuning of the cavity from the classical driving field, and w n D = a a being the frequency detuning of the qubit from the classical driving field. Similar to [33], in the dispersive regime (i.e., on the Hamiltonian(5) to eliminate the direct qubit-cavity coupling. Using the Hausdorff expansion to second order in the small parameter D g , the effective Hamiltonian reads as˜˜( As the average photon number of the cavity † á ñã a 0, this Hamiltonian(6) can generate rotations of the qubit about any axis on the Bloch sphere by adjusting the frequency ν of the classical driving field. First, if we adjust n w = + L a so thatD = 0 a , the rotations of the qubit around x axis on the Bloch sphere can be generated, that is, the unitary transformation ( ) x . Secondly, if the driving frequency is adjusted as n w = + -W L a , we can realize the required Hadamard gate  with the duration time ( ) can generate rotations of the qubit about z axis on the Bloch sphere since ( )

SSTS of a driven cavity with a qubit
For the readout of the qubit states, we need to apply another classical driving field on one side of the cavity, and then detect its SSTS on the other side, see figure 1(a). The interaction Hamiltonian between the applied driving and the cavity reads as where ò is the time-independent real amplitude and w d is the frequency of the applied driving. The total Hamiltonian H T of such a driven cavity-qubit system includes H d plus the first three terms of equation (4). In the dispersive regime and in a frame rotating at the driving frequency w d for the cavity, the effective Hamiltonian of the whole system is derived as r is the detuning between the driving frequency and the cavity frequency. Under the Born-Markov approximation, the dynamics of the whole system is governed by the master equation [29] [ where ρ is the density matrix of the system and [ ] † † † Here, g = T 1 1 1 is the qubit energy decay rate, g f the qubit dephasing rate, and κ the photon decay rate of the cavity. From the master equation (9), the coupled equations of motion related to the desired quantity † á ñ a a are given by of the driven cavity is analytically derived from equations (10a)-(10c) as where the subscript ss denotes steady state.
From equation (8), it can be seen that the interaction Hamiltonian between the qubit and the cavity is † . Therefore, the transmission spectra readout method proposed above is of the nondestructive property [29]. The interaction Hamiltonian H int also indicates that the cavity frequency is shifted by -L (or L), if the qubit is in the computational basis state | ñ 0 (or | ñ 1 ). Hence, the shifts of the cavity frequency can mark the computational basis states of the qubit. On the other hand, only the incident photon whose frequency is equivalent to one of the state-dependent frequencies w  L r of the cavity can transmit the cavity and then be detected. Physically, the detected probability is exactly the superposed probability of the computational basis state in the qubit state. Therefore, the SSTS T ss (11) has the feature: the relative height of each transmitted peak marked one of the computational basis states corresponds to its superposed probability in the detected qubit state. This can also be verified through numerical experiments, see section 3.3 as well. This manifest advantage allows us to directly read out all the diagonal elements of the density matrix of the detected qubit state with only one kind of such SSTS.

Numerical demonstration of MUBs-QST of qubit states
We now numerically demonstrate the MUBs-QST of qubit states in detail with the presented single-qubit unitary transformations in section 3.1 and the SSTS in section 3.2.
The density matrix of an arbitrary qubit state to be determined can be represented as can be directly read out from figure 2(b) and (c), respectively. Finally, inserting these projective measurement outcomes and the MUBs for d=2 [18] into equation (2), we can obtain the reconstructed state normalized as  However, it is noted that the reconstructed density matrix (14) is unphysical because it has a negative eigenvalue violating the property of positive semidefiniteness of all physical density matrices. To avoid this problem, we employ the commonly used maximum likelihood estimation (MLE) technique [2] to derive a physical density matrix most likely to have returned the numerically simulated results (14). In this manner, a physical density matrix is obtained as˜(

. Implementation of single-qutrit unitary transformations
We consider a setup shown in figure 1(a), wherein a qutrit (three-level atom with the cascade configuration, without loss of generality) displayed in figure 1(c) is dispersively coupled to a cavity. The Hamiltonian of the qutrit-cavity system reads as where w j is the frequency of the energy level | ñ j , g j is the coupling strength between the cavity and the transition | | ñ « + ñ j j 1 , and | | P = ñá ¢ On the other hand, similar to section 3.1, a classical driving field with the amplitude W 1 and the frequency n 1 is applied between | ñ 0 and | ñ 1 . By adjusting the frequency n 1 , an arbitrary unitary transformation between | ñ 0 and | ñ 1 can be produced, i.e., 1 . Likewise, we add another classical driving field with the amplitude W 2 and the frequency n 2 between | ñ 1 and | ñ 2 . We can generate an arbitrary unitary transformation between | ñ 1 and | ñ 2 , 2 , e.g., 2 . Finally, the required inverse operator of the Fourier transformation can be constructed with the combination of the above unitary transformations, e.g.,

SSTS of a driven cavity with a qutrit
Similar to section 3.2, a classical driving field with the same interaction Hamiltonian (7) is employed to achieve the readout of the qutrit states. The total Hamiltonian of such a driven cavity-qutrit system is   = + H T d . In the dispersive regime and in a frame rotating at the driving frequency w d for the cavity, the efficient Hamiltonian of the whole system is derived as are Lamb shift and Stark shift of energy level | ñ j , respectively.
The master equation to describe the dynamics of the whole system is given by [29] [ where g = T 1 j j 1 1 and g f j are the energy decay rate and the dephasing rate for energy level | ñ j , respectively. From the master equation (20), we can derive a set of coupled equations of emotion associated with the desired quantity † á ñ a a , which includes equation (10a) and Similar to the qubit case, from equations (21e)-(21g), we can obtain ( )  áP ñ áP ñ 0 00 00 , ( )  áP ñ áP ñ 0 11 11 , and ( )  áP ñ áP ñ 0 22 22 . Further, the normalized SSTS of the driven cavity can be analytically derived from equations (10a) and(21a)-(21d) as , and l g = -+ C S i 2 1 2 . The physical mechanism of the SSTS readout method is similar to the qubit case explained in section 3.2. This is also a kind of nondestructive measurement due to the fact that the interaction Hamiltonian between the qutrit and the cavity (i.e., Stark shift term in equation (19) 100 MHz [33]. Also, the errors of the amplitude and the frequency of the classical driving field are both assumed to be e = 5%, that is, the amplitude and the frequency are taken as ( ) e + W 1 and ( ) e n + 1 , respectively. The fidelity for quantum gates is defined as Nevertheless, imperfect unitary transformations would lead to imperfect projector of MUBs and further the errors of the measurement outcomes. As discussed in [28], the imperfect projector is assumed as where F is the fidelity of the mixed quantum state ¢ P with respect to the ideal pure state P. The projector is completely mixed state ¢ = P I 1 2 for = F 1 2 , and is the ideal one ¢ = P P for F=1. With the imperfect projector, the measurement outcomes is acquired as Tr . 30 2 2 If the statistical error of the measurement outcome ( ) r = p P Tr is desired to be below δ, the statistical error of the experimental values of ( ) r ¢ P Tr should be smaller than This indicates that increasing the number of the experimental repetitions can compensate for the reduced fidelity in the projection states to a certain extent.

Conclusion
We have presented an experimentally feasible proposal for MUBs-QST of qubit and qutrit states in dispersively coupled driven cavity QED systems. Due to the property of the MUBs, our proposal requires projections from optimal and minimal number of measurement bases to be performed [18]. It has been shown that the measurements in the MUBs are practically realized by projecting the detected states onto the computational basis after performing proper unitary transformations, which can be readily implemented by adjusting the classical driving field applied on the qubit/qutrit. The projective measurement outcomes are then read out directly from the SSTS of the driven cavity. We have shown that only one kind of SSTS is sufficient to determine all the projective measurement outcomes for each MUB, i.e., all the diagonal elements of the density matrix of the detected state. This is essentially less than the number of the usual projective measurements [2,3], wherein only one diagonal element of the density matrix can be determined each time. It has been numerically shown that MUBs-QST of the exemplified qubit and qutrit states can be realized with the fidelities 0.952 and 0.961, respectively. We believe that our proposal can be extended to other high-dimensional quantum systems in a straightforward way if their MUBs exist.