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arXiv:1211.4292

Weak measurements with completely mixed probe states

Shuhei Tamate, Toshihiro Nakanishi, Masao Kitano

Published Nov 19, 2012• 0 citations
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Abstract

Weak measurements with mixed probe states were investigated in the presence of noise in the probe system. We show that the completely mixed state can be used as a probe state for weak measurements of imaginary weak values. The completely mixed probe state has some advantages in terms of noise tolerance. We also experimentally demonstrated weak measurements of polarization rotation via unpolarized light, which corresponds to weak measurements with completely mixed probe states in a two-state system.

DOI
10.48550/arxiv.1211.4292

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Weak measurements with mixed probe states were investigated in the presence of noise in the probe system. We show that the completely mixed state can be used as a probe state for weak measurements of imaginary weak values. The completely mixed probe state has some advantages in terms of noise tolerance. We also experimentally demonstrated weak measurements of polarization rotation via unpolarized light, which corresponds to weak measurements with completely mixed probe states in a two-state system.
Weak measurements allow a probe system to encode the transitionary weak value of a desired observable on a signal system that is both pre- and post-selected. Such interactions alter the state and measurement statistics of the probe system. We establish that by encoding the imaginary part of a weak value onto an initially entangled probe, we can modify its entanglement content, provided the probe observable can distinguish between system states of different von Neumann entropies. Apart from fundamental interest, this result illustrates the utility of the imaginary weak value as a calculational tool in certain entanglement concentration protocols.
The weak value, the average result of a weak measurement, has proven useful for probing quantum and classical systems. Examples include amplifying small signals, investigating quantum paradoxes, and elucidating fundamental quantum phenomena such as geometric phase. A key characteristic of the weak value is that it can be complex, in contrast to a standard expectation value. However, typically only either the real or imaginary component of the weak value is determined in a given experimental setup. Weak measurements can be used to, in a sense, simultaneously measure noncommuting observables. This principle was used in the direct measurement of the quantum wave function. However, the wave function's real and imaginary components, given by a weak value, are determined in different setups or on separate ensembles of systems, putting the procedure's directness in question. To address these issues, we introduce and experimentally demonstrate a general method to simultaneously read out both components of the weak value in a single experimental apparatus. In particular, we directly measure the polarization state of an ensemble of photons using weak measurement. With our method, each photon contributes to both the real and imaginary parts of the weak-value average. On a fundamental level, this suggests that the full complex weak value is a characteristic of each photon measured.
Weak measurement has proven useful for metrology, for probing quantum paradoxes, and for the direct measurement of quantum states. Weak measurements of polarization qubits with temporal pointers provide a direct route to weak-value characterization, including in regimes where the pointer is only partially coherent. We derive a closed-form expression for the post-selected temporal shift that interpolates between the incoherent and coherent (ideal) weak-measurement limits through a single parameter ννν, the pointer visibility. The expression is experimentally tested for a range of polarization states spanning three trajectories corresponding to rotations about the principal axes of the Poincaré sphere. Two of the trajectories provide parameter-free benchmarks, with the real part of the weak value following cos⁡2θ\cos^2θcos2θ and 1/21/21/2. In contrast, the weak value along the third trajectory is sensitive to incoherence and is well described by the single-parameter partial-coherence model, from which a reduced pointer visibility is obtained. Time shifts are extracted from the smoothed derivative of the pulse leading edge with internal calibration, reaching few-picosecond precision on 10 ns optical pulses using commercial components. The same apparatus also gives qualitative access to the imaginary quadrature through the frequency centroid, showing that temporal weak measurements remain accessible and quantitative in partially coherent pulsed systems.
The weak values and weak measurement formalism were initially limited to pure states, which were later extended to mixed states, leading to intriguing applications in quantum information processing tasks. Weak values are considered to be abstract properties of systems describing a complete picture between successive measurements in the two-state vector formalism (TSVF). The remarkable achievements of the weak value formalism in experimental quantum mechanics have persuaded most quantum physicists that it is impeccable. However, we explore a scenario where the formalism of weak values for mixed states is employed in a quantum communication protocol, but discover that it generates inaccurate outcomes. This reinforces our previous conclusion that the weak values may not be elements of the reality of weak measurements, contrary to what the proponents of weak values proposed.
A state of light which is a superposition of the vacuum and the one-photon number state is the simplest state containing phase information. Recently we have shown how a field in such a state might be generated and here we explore its usefulness as a probe for measuring unknown states of light. We find that this probe can be used reasonably simply both to determine completely some pure states of light and to measure the diagonal and nearest off-diagonal elements of the density matrix in the number state basis and hence to obtain the mean sine and cosine of the phase of an unknown mixed state. We suggest further how a field in a superposition of the vacuum and the two-photon number state might be generated and how this can be used as a probe, both to measure the off-diagonal matrix elements second nearest to the diagonal of a mixed state density matrix and to measure the variance of the cosine and the sine of the phase. We also examine the experimentally more likely case where the probe fields are in mixed states and show how the same information about the unknown state can still be retrieved.