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arXiv:0907.0533 · Physical Review A

Complete characterization of post-selected quantum statistics using weak measurement tomography

Holger Friedrich Hofmann

Published Jan 6, 2010• 59 citations
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Abstract

The reconstruction of quantum states from a sufficient set of experimental data can be achieved with arbitrarily weak measurement interactions. Since such weak measurements have negligible back action, the quantum state reconstruction is also valid for the postselected subensembles usually considered in weak measurement paradoxes. It is shown that postselection can then be identified with a statistical decomposition of the initial density matrix into transient density matrices conditioned by the anticipated measurement outcomes. This result indicates that it is possible to ascribe the properties determined by the final measurement outcome to each individual quantum system before the measurement has taken place. The ``collapse'' of the pure state wave function in a measurement can then be understood in terms of the classical ``collapse'' of a probability distribution as new information becomes available.

DOI
10.1103/physreva.81.012103

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The reconstruction of quantum states from a sufficient set of experimental data can be achieved with arbitrarily weak measurement interactions. Since such weak measurements have negligible back action, the quantum state reconstruction is also valid for the postselected subensembles usually considered in weak measurement paradoxes. It is shown that postselection can then be identified with a statistical decomposition of the initial density matrix into transient density matrices conditioned by the anticipated measurement outcomes. This result indicates that it is possible to ascribe the properties determined by the final measurement outcome to each individual quantum system before the measurement has taken place. The collapse'' of the pure state wave function in a measurement can then be understood in terms of the classical collapse'' of a probability distribution as new information becomes available.
The time-symmetric formalism endows the weak measurement and its outcome, the weak value, with many unique features. In particular, it allows a direct tomography of quantum states without resorting to complicated reconstruction algorithms and provides an operational meaning to wave functions and density matrices. Here, we propose and experimentally demonstrate the direct tomography of a measurement apparatus by taking the backward direction of weak measurement formalism. Our protocol works rigorously with the arbitrary measurement strength, which offers improved accuracy and precision. The precision can be further improved by taking into account the completeness condition of the measurement operators, which also ensures the feasibility of our protocol for the characterization of the arbitrary quantum measurement. Our work provides new insight on the symmetry between quantum states and measurements, as well as an efficient method to characterize a measurement apparatus.
We discuss the concept of connection states (or connection matrices) that describe posterior ensembles, post-selected according to the outcomes of a quantum measurement. Connection matrices allow one to obtain results of any weak and some non-weak pre- and post-selected measurements, in the same manner as density matrices allow one to predict the results of conventional quantum measurements. Connection matrices are direct extensions of the density matrices and are generally non-Hermitian, which we show to be a direct consequence of quantum complementarity. This implies that the ultimate reason for unusual weak values is quantum complementarity. We show that connection matrices can be determined experimentally. We also show that retrodictive states are a special case of connection states. We propose a new method of tomography of quantum detectors.
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from an appropriate weak measurement. The density matrix (with all of its elements) represents all that is directly accessible from a general measurement.
Weak measurements carried out between quantum state preparation and post-selection result in complex values for self-adjoint operators, corresponding to complex conditional probabilities for the projections on specific eigenstates. In this paper it is shown that the complex phases of these weak conditional probabilities describe the dynamic response of the system to unitary transformations. Quantum mechanics thus unifies the statistical overlap of different states with the dynamical structure of transformations between these states. Specifically, it is possible to identify the phase of weak conditional probabilities directly with the action of a unitary transform that maximizes the overlap of initial and final states. This action provides a quantitative measure of how much quantum correlations can diverge from the deterministic relations between physical properties expected from classical physics or hidden variable theories. In terms of quantum information, the phases of weak conditional probabilities thus represent the logical tension between sets of three quantum states that is at the heart of quantum paradoxes.
Quantum tomography is a cornerstone of quantum information science, enabling the reconstruction of states and channels from experimental data. Here we introduce a new paradigm, temporal state tomography (TST), for reconstructing quantum processes across multiple times. Our approach is based on temporal quasiprobability distributions (TQDs), which, in the informationally complete setting, provide a complete description of multi-time quantum processes and uniquely determine temporal states. We formulate TST as a unified framework for reconstructing both density operators and quantum channels within a single scheme. We show that any TQD can be obtained via classical post-processing of measurement outcomes generated by a fixed set of quantum instruments, thereby establishing a direct operational route to accessing TQDs experimentally. For informationally complete TQDs, the associated temporal state can be reconstructed via a temporal Bloch-type representation. Leveraging this correspondence, we derive the sample complexity of TST, thereby quantifying its statistical efficiency.
We present a general formalism for characterizing 2-time quantum states, describing pre- and postselected quantum systems. The most general 2-time state is characterized by a ``density vector'' that is independent of measurements performed between the preparation and postselection. We provide a method for performing tomography of an unknown 2-time density vector. This procedure, which cannot be implemented by weak or projective measurements, brings new insight to the fundamental role played by Kraus operators in quantum measurements. Finally, after showing that general states and measurements are isomorphic, we show that any measurement on a 2-time state can be mapped to a measurement on a preselected bipartite state.