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

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 post-selected sub-ensembles usually considered in weak measurement paradoxes. It is shown that post-selection 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 wavefunction in a measurement can then be understood in terms of the classical "collapse" of a probability distribution as new information becomes available.

On the role of complex phases in the quantum statistics of weak measurements

“…As discussed in [19], weak measurements can be efficiently represented by measurement operators for the actual outcomes µ of the weak measurement,Ê…”

“…It is therefore possible to treat the post-selection of a final result | f as a condition that is completely independent of the measurement dynamics (for details see [19]). According to Bayesian statistics, the conditional probability for the weak measurement outcome µ is then…”

“…However, there has recently been renewed interest in weak measurements, not only motivated by new experimental possibilities [4][5][6], but also because weak measurements may clarify fundamental issues in quantum mechanics by indicating possible experimental resolutions of quantum paradoxes [5,[7][8][9][10][11][12][13][14][15][16]. Such resolutions of quantum paradoxes are usually based on the interpretation of weak measurement statistics in terms of negative probabilities whose averages can reproduce the experimentally observed violation of inequalities [17][18][19]. A more detailed analysis of the strange features of weak measurements may therefore lead to a better understanding of the essential differences between quantum and classical statistics.…”

Conditional average in a quantum system with postselection

“…whereρ iM f /Tr[ρ iM f ] is the transient density operator (the connection-state density operator) [23,24,36]. Furthermore, the joint quasi-probability P(a, f |i) can also be expressed as…”

Quantum State Estimation Using Weak Measurements

“…Other examples of quantum state tomography with weak measurements can be found in refs [21][22][23]. An approach to perform quantum state tomography using weak measurement POVMs was introduced by Hofmann 24 . In this paper, we present some of our results on state estimation by 'weak' measurements and a more detailed account for the qubit case is available in our recent paper 25 .…”