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arXiv:1911.06609 · Journal of Physics Communications

Direct measurement methods of density matrix of an entangled quantum state

Yusuf Turek

Published Jun 3, 2020• 4 citations
arXivHTMLPDF

Abstract

Abstract In general, the state of a quantum system represented by the density operator and its determination is a fundamental problem in quantum theory. In this study, two theoretical methods such as using postselected measurement characterized by modular value and sequential measurements of triple products of complementary observables to direct measurement of matrix elements of density operator of a two photon entangled quantum state are introduced. The similarity and possible feasibilities of those two methods are discussed by considering the previous theoretical and experimental works.

DOI
10.1088/2399-6528/ab9938

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Abstract In general, the state of a quantum system represented by the density operator and its determination is a fundamental problem in quantum theory. In this study, two theoretical methods such as using postselected measurement characterized by modular value and sequential measurements of triple products of complementary observables to direct measurement of matrix elements of density operator of a two photon entangled quantum state are introduced. The similarity and possible feasibilities of those two methods are discussed by considering the previous theoretical and experimental works.
One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.
One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction.Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states.The scheme relies on measuring a sequence of three observables, each complementary to the last.The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong.We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix.The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons.This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.
The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to revealing quantum properties relies on the reconstruction of entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of the matrix components of the measurement operators, which makes it possible to simplify the characterization process. We propose a general framework to directly obtain individual matrix elements of the measurement operators by sequentially measuring two noncompatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the required information. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix elements. The investigation of the measurement precision indicates the good feasibility of our protocol for arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix components of the measurement operators.
Recent work by Lundeen et al. [Nature (London) 474, 188 (2011)] directly measured the wave function by weakly measuring a variable followed by a normal (i.e., "strong") measurement of the complementary variable. We generalize this method to mixed states by considering the weak measurement of various products of these observables, thereby providing the density matrix an operational definition in terms of a procedure for its direct measurement. The method only requires measurements in two bases and can be performed in situ, determining the quantum state without destroying it.
New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only O(d) different operations, compared to d^{2} linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.