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Quantum State and Process Tomography: measuring mixed states. Now for something different: Instead of "weak" measurements, let's try to measure everything. Instead of a state overdetermined by preparation and postselection, let's consider states which may be incompletely defined.
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Quantum State and Process Tomography: measuring mixed states • Now for something different: • Instead of "weak" measurements, let's try to measure everything. • Instead of a state overdetermined by preparation and postselection, let's consider states which may be incompletely defined. • Description of mixed states • Density matrices • Wigner functions • Superoperators • Brief nod to "traditional" (!) quantum state tomography • Two-photon state & process tomography • Optical-latice state & process tomography 2 Dec 2003
Diagonal elements = probabilities Off-diagonal elements = "coherences" (provide info. about relative phase) Recall: mixed states are described by density matrices, not by wave functions. Can't determine the coefficients by measuring a single system - to extract all this information, we must study a large ensemble of identical particles.
How to extract coefficients? Measure the expectation value of various operators A ; each one provides a given linear combination of the matrix elements. To measure all n2 elements for an n-dimensional system, should make n2 different (linearly independent) measurements.
The Wigner function is one of several phase-space distributions which can play the same role as the density matrix: It is not a probability (since what is the probability of an event which can never be tested?), but it is the unique "quasi-probability distribution" satisfying: Like the density matrix, it can be used to extract any expectation value: What about continuous variables? It may be negative, but P(x) and P(p) never are... [in fact, W is almost always negative somewhere.] (In principle, an infinite number of observables must be measured to extract all the infinite number of points in W(x,p).)
Signal (weak) Local oscillator (strong) |Es + |ELO| eif|2 |ELO|2 + 2 |ELO| Re Es cos f - 2 |ELO| Im Es sin f + ... How does one measure these things? Heterodyning allows one to measure Re (Es eiffor various f thus we can extract integrals along every possible angle... Much work over the last 10-20 years applying algorithms from medical imaging to extract Wigner functions, e.g., of light...
E.g.: Wigner function of a photon Lvovsky et al., Physical Review Letters 87 , 050402 (2001) Circularly symmetric: no phase-dependence when you homodyne. Dip in each marginal at 0 -- the only way this can be is: negative quasiprobability at E1=E2=0. Dip at middle is related to the Hong-Ou-Mandel dip, and its high- photon-number analog: put another way, it's our old discussion of interference of number states, and how the photons tend to bunch. But with quantum information in mind, let's think about something different: polarisation states of photon pairs, and how they evolve...
Two-photon Process Tomography "Black Box" 50/50 Beamsplitter Two waveplates per photon for state preparation Detector A HWP HWP PBS QWP QWP SPDC source QWP QWP PBS HWP HWP Detector B Argon Ion Laser Two waveplates per photon for state analysis
Hong-Ou-Mandel Interference r t + t r How often will both detectors fire together? r2+t2 = 0; total destructive interference. ...iff the processes (& thus photons) indistinguishable. If the photons have same polarisation, no coincidences. Only in the singlet state |HV> – |VH> are the two photons guaranteed to be orthogonal. This interferometer is a "Bell-state filter," needed for quantum teleportation and other applications. Our Goal: use process tomography to test this filter.
“Measuring” the superoperator Coincidencences Output DM Input } HH } } 16 input states } HV etc. VV 16 analyzer settings VH
“Measuring” the superoperator Superoperator Input Output DM HH HV VV VH Output Input etc.
Testing the superoperator LL= input state Predicted Nphotons = 297 ± 14
Testing the superoperator LL= input state Predicted Nphotons = 297 ± 14 Observed Nphotons = 314
So, How's Our Singlet State Filter? 1/2 -1/2 -1/2 1/2 Bell singlet state: = (HV-VH)/√2 Observed , but a different maximally entangled state:
Model of real-world beamsplitter f f Singlet filter multi-layer dielectric AR coating 45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser) birefringent element + singlet-state filter + birefringent element Necessary correction determined from leading Kraus operator...
Superoperator provides informationneeded to correct & diagnose operation Measured superoperator, in Bell-state basis: Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors. The ideal filter would have a single peak. Leading Kraus operator allows us to determine unitary error. FUTURE: more efficient extraction of information for better correction of errors iterative search for optimal encodings in presence of collective noise;...
Tomography in Optical Lattices Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of mK depth) Complete characterisation of process on arbitrary inputs?
Lattice experimental setup Setup for lattice with adjustable position & velocity
Quantum state reconstruction D x Wait… Shift… Measure ground state population (former for HO only; latter requires only symmetry) [Now, we can also perform translation directly in both x and p]
Oscillations in lattice wells [essentially a measure of Q(r,) at fixed r-- recall, r is set by size of shift and by length of delay]
Extracted phase-space distributions(Q rather than W in this case)
Data:"W-like" [Pg-Pe](x,p) for a mostly-excited incoherent mixture (For 2-level subspace, can also choose 4 particular measurements and directly extract density matrix)
Atomic state measurement(for a 2-state lattice, with c0|0> + c1|1>) initial state displaced delayed & displaced left in ground band tunnels out during adiabatic lowering (escaped during preparation) |c0 + i c1 |2 |c0|2 |c0 + c1 |2 |c1|2
Extracting a superoperator:prepare a complete set of input states and measure each output
Superoperator for resonant drive Bloch sphere predicted from truncated harmonic-oscillator plus decoherence as measured previously. Operation: x (resonantly couple 0 and 1 by modulating lattice periodically) Measure superoperator to diagnose single-qubit operation (and in future, to correct for errors and decoherence) Observed Bloch sphere Upcoming goals: generate tailored pulse sequences to preserve coherence; determine whether decoherence is Markovian; et cetera.
SUMMARY Any pure or mixed state may be represented by a density matrix or phase-space distribution (e.g., Wigner function). These can be reconstructed by making repeated measurements in various bases (n2 measurements for a density matrix). A superoperator determines the time-evolution of a density matrix (including decoherence), and requires n4 measurements. Elements in quantum-information systems can be characterized by performing such measurements. More work needs to be done on (a) optimizing the extraction of useful information (b) determining how to use the resulting superoperators.
References Your favorite quantum optics text -- Loudon, Walls/Milburn, Meystre/Sargent, Milonni, Scully/Zubairy, etc. -- for introduction to quantum optics & phase-space methods. Schleich's Quantum Optics in Phase Space. Leonhardt's Measuring the Quantum State of Light. Theory: Wigner, Phys. Rev. 40, 749 (1932) Hillery et al., Phys. Rep. 106, 121 (1984) Early tomography experiments: Smithey et al, PRL 70, 1244 (1993) (light modes) Dunn et al., Phys. Rev. Lett. 74, 884 (1995) (molecules) Measurement of negative Wigner functions: Nogues et al, Phys. Rev. A 62, 054101 (2000) (cavity QED) Leibfried et al, PRL 77, 4281 (1996) (trapped ion) Single-photon process tomography: White et al., PRA 65, 012301 (2002) James et al., PRA 65, 052312 (2001) Ancilla-assisted photon-polarisation tomography: Altepeter et al., PRL 90, 193601 (2003) Phase-space tomography on single-photon fields: Lvovsky et al., PRL 87, 050402 (2001) Two-photon process tomography: Mitchell et al., PRL. 91, 120402 (2003) Applications of process tomography: Weinstein et al., PRL 86, 1889 (2001) (in NMR experiment) Boulant et al., quant-ph/0211046 (interpreting superoperators) White et al., quant-ph/0308115 (for 2-photon gates)
