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Reversible Weak Quantum Measurements

Reversible Weak Quantum Measurements. 09 December 2008 Presented by: Adam Reiser Anuj Das Neil Kumar George O’Quinn. Agenda. Strong and Weak Measurements Hardy’s Paradox Reversibility of Weak Measurements Experiments and Possible Applications. Strong and Weak Measurements.

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Reversible Weak Quantum Measurements

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  1. Reversible Weak Quantum Measurements 09 December 2008 Presented by: Adam Reiser Anuj Das Neil Kumar George O’Quinn

  2. Agenda • Strong and Weak Measurements • Hardy’s Paradox • Reversibility of Weak Measurements • Experiments and Possible Applications

  3. Strong and Weak Measurements

  4. Strong Measurement • First formulated 1943 by von Neumann • “Measurement” in quantum mechanics by default refers to strong measurement • Associated with a physical observable corresponding to a linear Hermitian operator • An irreversible projection of a superposition onto one of its eigenstates • Example: the Stern-Gerlach apparatus • Intrinsic angular momentum assumes an eigenstate upon interaction with a magnet • This interaction is a strong measurement

  5. Stern-Gerlach Device • Particles forced to assume an eigenstate (wave collapses) • Direction of beam indicates the eigenstate of spin *(images available under GNU Free Documentation License)

  6. Weak Measurement • Measurement is continuous • Partial wavefunction collapse can be undone • Underlying eigenstates are hidden • Demonstrated for both phase and charge qubits

  7. Weak measurement of a charge qubit • a quantum point contact is a narrow channel (approx. 10-6 m) between two conducting regions that serves as an extremely sensitive charge detector

  8. Charge Qubits • Strong measurement characterized by average currents corresponding to qubit's eigenstates • Weak measurement yields average current... • Note: This is an average, not an eigenstate!

  9. Undoing a weak measurement Given a weak measurement r0 • The goal is to measure a new state ru(t) = -r0 • This fails if r(t) crosses origin • Probability of success calculable • Note that probability of reversal scales relative to certainty of original measurement r0 • | r0 | >> 1: high confidence of measured state

  10. Phase Qubits • Similar to flux qubits – Josephson junction + SQUID • Controlled by an external flux øe

  11. Detection of Quantum States Lower barrier øe to initiate measurement • Upper energy state tunnels with rate Γ • Lower energy state remains

  12. Φ℮ qubit SQUID Weak Measurement (Phase) • For Γ•t >> 1, collapse to one state occurs • But if the barrier is raised after a finite time t ~ Γ-1 • If tunneling occurs, qubit is destroyed • If not – phase accumulates, qubit remains

  13. Hardy’s Paradox

  14. B A Photon detectors Inner arm Outer arm C E Inner arm PD2 F Quartz plates D 2 1 Outer arm Quartz D1 PD1 Hardy’s Paradox • Double Interferometer Experiment • Shared inner arm permits destructive interference • Two emitters, two detectors, lots of splitters

  15. The Paradox If we have a coincidence detection at only one output, there are three possibilities: • Both particles traveled the outer path: • But then we should have two output detections • One traveled the outer path, one the inner path: • Again, we should have two detections • Both particles traveled the inner path: • But then they should have annihilated through interference • P[O1|C1C2] = P[O2|C1C2] = 1 • P[O1O2|C1C2] = 0

  16. C E N/2 B B E F C N/2 F N/2 Possible Outcomes

  17. What Does This Mean? Photon detectors Inner arm Outer arm Inner arm PD2 Quartz plates 2 Outer arm 1 PD1 Weak Measurement We can’t tell the state of a qubit undergoing unitary evolution Weak measurement allows us to take measurements within the photon paths

  18. Reversibility of Quantum Measurements

  19. Quantum vs. Classical Measurement • In quantum physics, we seem to have a significant difference from classical mechanics to contend with because of measurements having only certain probabilistic outcomes. • Information about the current state can be garnered from past measurements of identically configured quantum states. • However, information from future measurements may tell a fundamentally different story. • This makes quantum state description time-asymmetric.

  20. Quantum States Defined by Future and Past • Until now, we have essentially been using the familiar ket notation in order to describe quantum states: |Ψ›= U (t1,t) |a› • Due to time-asymmetry, it appears we need another state as well: ‹Φ| =‹b| U† (t,t2) • We can generalize this as follows: • Note that ‹Φ||Ψ › is not an inner product!

  21. Measuring a Weak Value • We use the following definition to find the weak value of a measurement C: • Note: a weak measurement of a purely pre-selected system is its expectation value. • Note: Cw may actually be well outside the acceptable range of eigenvalues associated with C.

  22. Pre-Selection and Post-Selection

  23. Aharonov, Vaidman Experiment

  24. Reversing a Measurement • Involves using a device that will only take a full measurement (collapse state to |1>) with probability p. • If the device does not react, the state partially collapses to |0>. • A pi-pulse, second measurement, then another pi-pulse can be used to restore original state (w/ prob. 1-p).

  25. Formal Quantum State Reconstruction • A more complicated method of quantum state reconstruction involves the creation of a superoperator based off of a Gaussian distribution of weakly measured states.

  26. Experiments and Possible Applications Using Weak Measurements

  27. Realization of a Measurement of a "Weak Value”Ritchie (1991) • Performed a weak measurement using a Gaussian beam of light polarized at 45 degrees, and sent through a polarizing beam splitter • The two beams were placed close enough together to cause overlapping Gaussians • Post-selected using a polarization filter close to 45 degrees

  28. Quantum State Reconstruction via Continuous MeasurementSilberfarb (2005) • Reconstructed a quantum state using weak continuous measurement of an ensemble average • All members of the ensemble are evolved identically in such a way as to map new information onto the measured quantity • This process provides enough information to estimate the density matrix with minimal disturbance to the system

  29. Measurement of Quantum Weak Values of Photon PolarizationPryde (2005) • Used a nondeterministic entangling circuit to allow a photon to make a weak measurement of the polarization of another photon • Used pre- and post- selection to eliminate extraneous results • Non-classical interference between two photons is required to allow the weak measurement to take place

  30. Coherent State Evolution in a Superconducting Qubit from Partial-Collapse MeasurementKatz (2006) • Weak measurement with tomography • Uses quantum-state tomography to investigate state evolution • A Josephson junction is used to produce the superconducting phase qubit • Measurement is carried out by lowering the energy barrier to cause the probability of tunneling by the |1> to increase

  31. Reversal of the Weak Measurement of a Quantum State in a Superconducting Phase Qubit Katz (2008) • The system is prepared as before, however two steps are added • A p-pulse is applied to swap the states • Another weak measurement is performed

  32. Photon detectors Outer arm Inner arm Inner arm PD2 Quartz plates 2 1 Outer arm Quartz PD1 Direct Observation of Hardy’s Paradox by Joint Weak Measurement with an Entangled Photon Pair Yokota (2008) • This experiment showed that Hardy’s paradox exists, but does not solve the paradox • Used photons replicate thought experiment • Two interferometers were used, with inner arms overlapped at the middle half beam splitter • If the photons meet at the beam splitter they interfere with each other

  33. Quantum Error Correction • Traditional methods required extra qubits so that strong measurements could be used without destroying the original state • Use of continuous feedback through weak measurements could possibly be used to detect defects

  34. Possible Implications for Quantum Cryptography • The main aspect of security in quantum cryptography is the complete collapse of the quantum state if it is intercepted • Weak measurements could possibly be used to gain information about a system without collapsing the quantum state • Caveat: weak measurement and reversal would introduce a slight delay that could be detected

  35. References • Aharonov, Vaidman. The Two-State Vector Formalism: an Updated Review • Aharonov, Albert, Vaidman. How The Result of a Component of the Spin of a Spin-1/2 Particle Can Turn Out to be 100. • Silberfarb, Jessen, Deutsch. Quantum State Reconstruction via Continuous Measurement. • Korotkov, Jordan. Undoing a weak quantum measurement of a solid-state qubit. • Ahnhert. Weak Measurement in Quantum Mechanics. Electronic Structure Discussion Group. • Katz, et al. Reversal of the Weak Measurement of a Quantum State in a Superconducting Phase Qubit. Phys. Rev. Lett. 101, 200401 (2008) – Published November 10, 2008. • D. L. Christoph Bruder, Physics 1, 34 (2008). • A. N. Korotkov and A. N. Jordan, Phys. Rev. Lett. 97, 166805 (2006). • images provided under GFDL from commons.wikimedia.org

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