Limitations of Quantum Process Tomography

Transcription

1 Invited Paper Limitations of Quantum Process Tomography L.K. Shalm, M. W. Mitchell, and A. M. Steinberg University of Toronto, Physics Department, 6 St. George St., Toronto, ON, Canada ABSTRACT Quantum process tomography is often cited as providing all the information that can be known about a given quantum process. In this paper we have shown that even if two processes appear identical under process tomography, it is possible to distinguish them using an interferometric setup. Using this setup, it is possible to gain more information about a process than just tomography provides. Keywords: Quantum Process Tomography, Decoherence, Completely Positive Maps, Quantum Information. INTRODUCTION Over the past decade, experimental capabilities in a number of fields have evolved to the point that precise preparation, manipulation, and characterisation of quantum states are now possible. Such an unprecedented degree of control of quantum systems promises widespread applications, ranging from quantum information processing to coherent control of reactions to fundamental studies of quantum physics. At the same time, it opens up a broad range of new questions, ranging from technical issues such as how one can most efficiently produce highly-entangled states of many photons; to ones specific to the new science of quantum information, such as what the appropriate measures are for characterizing quantum processes and what the best methods are for obtaining them experimentally; to more fundamental questions about quantum measurement and the best way to describe postselected quantum systems. The issue of characterizing quantum states and processes has become a central theme in quantum information science, and will be crucial to all aspects of the control of quantum systems. 2. PROCESS TOMOGRAPHY For a pure state, a wavefunction ψ is sufficient to completely describe the state. However, in the case of a mixed state a more general representation is required. The state of a system in a d-dimensional Hilbert state can be represented by a dxd density matrix. If a quantum system is in a probabilistic mixture of pure states, ψ i, with probabilities p i, then the density matrix that describes this system is : ρ = i p i ψ i ψ i. () The elements of the density matrix completely describe the state of either a pure or a mixed quantum system, while the latter cannot be described by a wavefunction. A quantum process can be thought of as a black box that takes some initial input quantum state and maps it to some final output state. Mathematically, a quantum process is represented as a completely positive map (CPM) that takes some initial quantum state ρ in and maps it to some final quantum state ρ out. Quantum process tomography (QPT) is one method that allows a given quantum process and the CPM associated with it to be characterized. QPT is analogous to medical imaging methods, 2 except instead of taking a series of two-dimensional images to reconstruct a three-dimensional image, a series of measurements are made on a quantum system that allow an unknown quantum process to be reconstructed. Tomographic techniques have been successfully applied and studied to a number of physical systems, including photons, 3, 4 molecules, 5 trapped ions, 6 and trapped atoms. 7 The reliance of the quantum information community on the power of tomography is underscored by the listing of two-photon QPT 8 as one of the trophies on the Quantum Information Science and Technology Roadmap. 9 Permanent address: Institut de Ciènceies Fotòniques, C. Jordi Girona, 29. Building Nexus II, 834 Barcelona, Spain 6 Quantum Optics and Applications in Computing and Communications II, edited by G.-C. Guo, H.-K. Lo, M. Sasaki, S. Liu, Proc. of SPIE Vol. 563 (SPIE, Bellingham, WA, 25) X/5/$5 doi:.7/

2 U V Figure. A Mach-Zehnder interferometer with two different baths in each arm, U and V, that couple the system to the environment and introduce decoherece. The phase shift φ in the second arm allows the visibility of the interferometer to be measured. A useful way to describe a quantum process ɛ is with the Kraus sum representation. ρ out = ɛ(ρ in )= l K l ρ in K l (2) where K l are the Kraus operators and satisfy the condition that K l K l = I (3) l The Kraus operators that represent a given CPM are not unique. Different sets of Kraus operators can give rise to the same CPM. It is desirable for comparison purposes to rewrite the Kraus representation of a process into a unique form. One particular representation is the Choi Matrix representation. 3 The Choi matrix representation of a CPM is a d 2 xd 2 matrix that describes how a process transforms an input state into an output state. 3. TOMOGRAPHY AND INTERFEROMETRY Does QPT yield all the information it is possible to obtain about a quantum process? The somewhat surprising answer shown by D. L. Oi in 23 4 is that it is possible to extract extra information about a quantum system beyond that yielded solely by tomography through the use of interferometric means. It has been shown than knowing the CPM of two different quantum processes is not sufficient to determine the behavior of a joint quantum system formed by both process. 5 7 In particular, two processes which are indistinguishable under process tomography can behave differently when incorporated into a joint quantum system such as an interferometer. Consider a particle with two internal states which is sent through a Mach-Zehnder interferometer with an independent bath in each arm. Any interaction between the bath and particle that tags the path of the particle through the interferometer causes the paths to become distinguishable and reduces the interferometric visibility (decoherence). If the operations in the upper and lower arms of the interferometer are not unitary, but rather CPMs U and V with the same input and output space, the situation can be modeled by extending the state space of the system by appending two ancillae F and E which are coupled to the two paths by overall unitaries U and Proc. of SPIE Vol

3 V which implement the CPMs as shown in Figure 4 : [ U(ρ) =Tr F U (ρ f f ) U ] (4) [ V (ρ) =Tr E V (ρ e e ) V ], (5) where ρ is the internal state of a particle in the interferometer and { f m } and { e n } form a linear basis describing the ancillae F and E where f and e are the initial states of F and E respectively. By using QPT one can find the CPMs U and V which result from the overall unitaries, but the actual nature of U and V remains unkown as QPT does not interact directly with the bath. Trivially extending U and V to act on the entire interferometric system yields the following: Ũ = U I E (6) Ṽ = V I F. (7) A particle initially in the state Ψ is transformed by the first beamsplitter and then the interaction with the ancilla and is left in state, Ψ = ψ e f (8) ψ Ṽ ψ e f + e iφ ψ Ũ e f [ ) )] ψ (Ṽ + e iφ Ũ + ψ (Ṽ e iφ Ũ e f where ψ m represents the internal state of the particle in mode m (corresponding to either arm or ). After this interaction, when the particle arrives at the second beamsplitter, it will have a probability P (φ) of exiting from the port where φ is a phase shift is applied to the second arm of the interferometer. P (φ) = ) (Ṽ + e iφ Ũ ψ 4 e f 2 (9) = ( { ]}) +Re e iφ 2 Tr[Ũ Ṽρ e f e f () = ( + v cos (φ α)) () 2 where v is the is the amplitude of the of the interference pattern and is called the visibility while α is the shift in the interference fringes. The quantities v and α can also be related to Kraus operators implementing the processes in each arm of the interferometer. 4 ve iα [Ũ [ ] = Tr Ṽρ e f e f ] = Tr K W ρ (2) {K m } = { e m U e } (3) {W n } = { f m V f } where {K m } and {W n } are the Kraus operators that describe the CPMs the two processes obey. If the maximally mixed state is used as the input to the interferometer, then the visibility v of the interferometer reduces to ve iα = ] [K d Tr W (4) where d is the dimension of the Hilbert space of the state. A maximally mixed state is equivalent to averaging over a uniform distribution of pure states. It is important to note that the visibility depends solely on the overlap of the first two Kraus operators K and W which are the operators corresponding to the case when the particle is not tagged by its interaction with the ancilla. Since there is not a unique Kraus representation 62 Proc. of SPIE Vol. 563

4 for any given process, only knowing the CPM for each arm of the interferometer is not enough to predict what the resulting visibility will be. One consequence of this fact is that it is not possible to build a compound hot swappable quantum process made up of independent black box quantum processes where only the CPM is known. Swapping one black box with another that implements the exact same CPM and appears indistinguishable under process tomography can lead to a different behavior in the overall system. In terms of the interferometric setup, if one of the processes is replaced by another process implementing the same CPM, the overall visibility given by Eq. (4) will be different if the new process has a different physical implementation. The reason that the CPM of two individual processes alone is not sufficient to predict their behavior in a joint system like an interferometer is that there is now an extra degree of freedom that has been introduced. A particle in the interferometer is now in a superposition of either being in one arm (process) or the other, and this is not taken into account by tomography of a single process alone EXPERIMENT Any information that can be used to distinguish between the two paths of the interferometer will serve to reduce the resulting visibility. One method to introduce distinguishing information (decoherence) into the interferometer is to alter the momentum of the light passing through one arm of the interferometer. In our setup this was accomplished by placing two quartz prisms adjacent to one another as shown in Figure 2. If the prisms are perfectly aligned, then they will not change the momentum of the light passing through. However, by changing the angle the faces of the prisms make with one another the momentum of the light passing through can be altered. In this case, the bath in both arms corresponds to the momentum of the photons. By changing the momentum shift introduced, the amount of decoherence introduced by the bath can be controlled. In this sense, the momentum bath may be thought of as performing a quantum non-demolition measurement on the presence or absence of a photon in the arm. The external degree of freedom (transverse position or momentum) constitutes the bath in this experiment. (Although they are properties of the photon, they are a disjoint subspace from the one describing polarization.) The momentum shift which occurs for a photon travelling through the prism arm constitutes an interaction which changes the state of the bath depending on the state (presence or absence in that arm) of the system, i.e., a measurement of the presence of the photon. A transverse momentum shift imparted to photons travelling through one arms can of course be equivalently seen as a position-dependent phase shift (due to the dependence of the optical path which must be traversed in the prism on the transverse position of the photon). In this picture, the system-bath interaction is not to modify the state of the bath (a measurement pointer) as a function of a system variable, but rather to phase-shift the system depending on the value of the (uncertain) bath variable. Note that the two viewpoints are equivalent, and lead to the same decoherence. In either case, QPT is insensitive to these effects, which relate not to the polarization of the photon (or the relative phases of the different polarizations), but to its overall global phase. For a fixed number of particles traversing the arm, as when QPT is performed on it, no information about the global phase can be determined; one way to see this is to recall that the phase of a single photon (or any well-defined number of photons) is undefined. On the other hand, in the interferometric setup, the number of particles going through each arm is uncertain, and the phase accumulated in each arm may be measured. Although this is the simplest implementation possible, the results can be extended without loss of generality to more complex processes and decoherence mechanisms RESULTS Process tomography was carried out on each arm independently, and then the visibility of the entire interferometer was measured. This was done for prism angles of,,2,and3. For each arm, the CPM was reconstructed using the data obtained from tomography measurements. Figure 3 shows an example of the calculated Choi matrices for each arm as well as the difference between the elements of the Choi matrices for the two processes. The difference between CPMs of the two arms is negligible and fall within the errors due to incorrect waveplate and prism settings. Proc. of SPIE Vol

5 State Preparation PBS HWP, QWP BS Iris Quartz Wedges Adjustable Phase Shifter QWP HWP PBS Detector BS Translation Stage With PZT Adjustable Phase Shifter Iris State Detection Figure 2. Schematic of the experimental apparatus used to implement the quantum non-demolition measurement model of decoherence. In the first arm of a Mach-Zehnder interferometer, two quartz prisms were used to impart a momentum shift on photons passing through. Adjustable polarization phase shifters in either arm compensated for any unwanted phase shifts picked up by either the mirrors or the prisms. a) Real Part of Choi Matrix in Arm Imaginary Part of Choi Matrix in Arm Real Part of Choi Matrix Imaginary Part of Choi Matrix in Arm 2 in Arm 2 b) c) Difference in Real Part of Choi Matrices Difference in Imaginary Part of Choi Matrices Figure 3. Results when the prisms were aligned with one another (no momentum shift). a) The Choi matrix of the process in arm of the MZ. b) The Choi matrix of the process in arm 2 of the MZ. c) The difference between elements of the Choi matrices in arms and 2. The two processes have an average fidelity of.999 indicating that they are nearly identical under process tomography 64 Proc. of SPIE Vol. 563

6 Measued Visibility as a Function of Deflection Angle 8 Visibility (Percent) Deflection Angle (mrad) Figure 4. Experimentally measured visibility as a function of deflection angle. A gaussian curve was fit to the data, allowing the beam diameter to be calculated. The calculated beam diameter of (.2±.)mm is consistent with the beam used in the experiment. A useful measure for how close two processes are together is given by the fidelity I(ρ). (ρ)ω(ρ) I(ρ) = (ρ) (5) Where is the CPM of the process in the first arm of the interferometer and Ω is the CPM of the process in the second arm. The average fidelities over the bloch sphere between the processes in each arm of the interferometer for prism angles of,,2, and 3 was.999,.999,.998, and.999 respectively. This indicates that the processes in each arm were nearly indistinguishable under process tomography. To measure the interferometric visibility an oscilloscope trace of the intensity pattern for a PZT phase shift was obtained using the pure input states H and V as inputs. The visibilities were observed to be (9±)%, (9±)%, (3±)%, and (2±)% for prism settings of,,2, and 3 respectively. The measured visibilities at different deflection angles is shown in Figure 4. A gaussian was fit to this data, and the diameter of the beam calculated to be (.2±.)mm which is consistent with the beam used in the experiment. Although the different prism settings, corresponding to different black boxes, appear indistinguishable under QPT, their effect on the overall measured visibility is dramatic different. Depending on the prism angles chosen, visibilities between (9±)% and (2±)% were obtained. This is a clear indication that QPT is not able to completely characterize a process and additional information about a process can be obtained. By placing two black boxes in an interferometer that implement the same CPM, it is possible to measure how close the two processes are to one another. If they yield unit visibility, then they can be said to be self-coherent with one another 4 as they both act on the internal state of the photons passing through in an identical manner (for example, both processes apply the same momentum shift). This is extra information about a process that is 4, 5, 7 inaccessible to process tomography. 6. BEYOND TOMOGRAPHY It is also possible to use a similar interferometric technique to gain extra information about the decoherence properties of a map, that is not available through traditional process tomography. Consider the interferometric setup shown in Figure 5. In the top arm of the interferometer, a pair of quartz prisms introduces a Proc. of SPIE Vol

7 U V H Figure 5. A diagram of a MZ interferometer with a set of prisms in the upper arm. The prisms introduce a polarization dependent momentum shift to the photons passing through. Vertically polarized V photons are decohered from horizontally polarized photons H. In the bottom arm, an adjustable unitary is placed. The closest unitary to the process in the upper arm can be found by adjusting the unitary until the maximum interference visibility is found. polarization-dependent momentum shift. The interferometer is aligned such that horizontally polarized H photons experience no momentum shift while vertically polarized photons V photons experience a momentum shift that decoheres them from the horizontally polarized photons. A state such as 2 ( H + V ) iseffectively turned into a mixed state after passing through the prisms. By placing an adjustable unitary in the second arm of the interferometer, the closest unitary to the process in the top arm of the interferometer can be found. This occurs when the visibility given in equation 4 of the interferometer is maximized. A series of waveplates can be used to carry out any arbitrary polarization rotation, and therefore act as an adjustable unitary. For the scenario considered above, the unitary that yields the maximum visibility is the one that rotates the state in second arm to be H polarized. Thus, the Kraus operator K as given in equation 3 which leaves the state unchanged after an interaction with the bath is H H. Now, if a demon sneaks into the lab and changes the orientation of the quartz prisms unbeknownst to the experimenter so that now the vertically polarized photons experience a zero momentum shift while it is the horizontally polarized photons whose momentum is shifted. This new processes appears identical under process tomography to the one previously discussed, but, the interference in the interferometer will be destroyed. However, the experimenter could once again use the waveplates to find the closest unitary to the new process, and discover that the Kraus operator K is now V V. Ordinarily there is not a unique Kraus representation for a process, but by using this interferometric method it is possible to assign physical meaning to the Kraus operator that leaves a state unchanged after an interaction with a bath. This leads to an additional restriction in the Kraus representation that is used to describe a process. This is information that is impossible to obtain using just QPT. Using such an interferometric approach, it may be possible to develop a more robust diagnostic tool for quantum systems. Decoherence that causes errors in complicated quantum systems composed of many interacting processes, such as those needed to construct a quantum computer, could be more accurate analyzed with such a diagnostic technique. 7. CONCLUSIONS Although it is sometimes claimed that process tomography provides all the information that there is to know about a system, this is not true. Extra useful information can be inferred about a process through the use of 66 Proc. of SPIE Vol. 563

8 interferometry. It may be possible to use an interferometric system similar to our setup to build a decoherence analyzer. By combining interferometry with process tomography, extra information about the physical sources of decoherence present can be obtained. Such techniques are vital for catching and correcting errors that arise in complex quantum devices such as a quantum computer or quantum cryptographic communications system. 8. ACKNOWLEDGMENTS We would like to thank Jeff Lundeen and Robert Adamson for their thoughts and comments on this work. This work was supported by the National Science and Engineering Research Council of Canada, Photonics Research Ontario, and the DARPA-QuIST program (managed by AFOSR under Agreement No. F ). REFERENCES. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2) 2. A. M. Cormack, OAMS 4, 325 (98) 3. D. James, P. Kwiat, W. Munro, and A. White, Phys. Rev. A 64, D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Phys. Rev. Lett. 7, 244 (993) 5. T. J. Dunn, I. A. Walmsley and S. Mukamel, Phys Rev. Lett. 74, 884 (995) 6. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano and D. J. Wineland, Phys. Rev. Lett. 77, 428 (996) 7. S. H. Myrskog, J. K. Fox, M. W. Mitchell, and A. M. Steinberg, e-print: quant-ph/ M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, Phys Rev. Lett. 9, 7232 (23) 9. http : //qist.lanl.gov/qcomp map.shtml. K. Kraus, States, Effects, and Operations (Springer-Verlag, Berlin, 983). I.L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455 (997) 2. M.D. Choi, Lin. Alg. Appl., 285 (975) 3. D. W. Leung, e-print: quant-ph/29 4. D. K. L. Oi, Phys. Rev. Lett. 9, 6792 (23) 5. J. Aberg, Phys. Rev. A 7, 23 (24) 6. J. Aberg, e-print: quant-ph/ J. Aberg, e-print: quant-ph/3282 Proc. of SPIE Vol