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

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

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)

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

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

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!

Undoing a weak measurement Given a weak measurement r 0 The goal is to measure a new state r u (t) = -r 0 This fails if r(t) crosses origin Probability of success calculable Note that probability of reversal scales relative to certainty of original measurement r 0 | r 0 | >> 1: high confidence of measured state

Phase Qubits Similar to flux qubits – Josephson junction + SQUID Controlled by an external flux ø e

Detection of Quantum States Lower barrier ø e to initiate measurement Upper energy state tunnels with rate Γ Lower energy state remains

Weak Measurement (Phase) For Γt >> 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 qubit SQUID Φ℮

Hardy’s Paradox

Quartz plates Quartz 1 2 Photon detectors PD 2 PD 1 Outer arm Inner arm D1D1 D E F B C A Hardy’s Paradox Double Interferometer Experiment –Shared inner arm permits destructive interference –Two emitters, two detectors, lots of splitters

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[O 1 |C 1 C 2 ] = P[O 2 |C 1 C 2 ] = 1 –P[O 1 O 2 |C 1 C 2 ] = 0

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

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

Reversibility of Quantum Measurements

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.

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 (t 1,t) |a› Due to time-asymmetry, it appears we need another state as well: ‹ Φ | =‹b| U † (t,t 2 ) We can generalize this as follows: Note that ‹Φ||Ψ › is not an inner product!

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: C w may actually be well outside the acceptable range of eigenvalues associated with C.

Pre-Selection and Post-Selection

Aharonov, Vaidman Experiment

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).

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.

Experiments and Possible Applications Using Weak Measurements

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

Quantum State Reconstruction via Continuous Measurement Silberfarb (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

Measurement of Quantum Weak Values of Photon Polarization Pryde (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

Coherent State Evolution in a Superconducting Qubit from Partial-Collapse Measurement Katz (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

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  -pulse is applied to swap the states Another weak measurement is performed

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 Quartz plates Quartz 1 2 Photon detectors PD 2 PD 1 Outer arm Inner arm

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

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

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, (2008) – Published November 10, D. L. Christoph Bruder, Physics 1, 34 (2008). A. N. Korotkov and A. N. Jordan, Phys. Rev. Lett. 97, (2006). images provided under GFDL from commons.wikimedia.org