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Open Systems & Information Dynamics

Measurement of Purity, the Simplest Nonlinear Functional of the Density Matrix

Tohru Tanaka, Hiromichi Nakazato

Published Mar 12, 2014• 2 citations
Publisher

Abstract

After reviewing the earlier ideas to measure the purity of a quantum system, a new scheme for the measurement of purity is presented, which aims at providing a simple, direct and efficient procedure without resorting to quantum state tomography. The scheme is also illustrated in the Bloch-vector representation for a two-level quantum system (qubit) and for a general D-level quantum system. As a variant, a simple example implemented in a one-dimensional scattering process to measure the purity of the target qubit is also given.

DOI
10.1142/s1230161214400095

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After reviewing the earlier ideas to measure the purity of a quantum system, a new scheme for the measurement of purity is presented, which aims at providing a simple, direct and efficient procedure without resorting to quantum state tomography. The scheme is also illustrated in the Bloch-vector representation for a two-level quantum system (qubit) and for a general D-level quantum system. As a variant, a simple example implemented in a one-dimensional scattering process to measure the purity of the target qubit is also given.
Abstract The purity and fidelity of quantum operations are of great importance in characterizing the quality of quantum operations. The currently available definitions of the purity and fidelity of quantum operations are based on the average over all possible input pure quantum states, i.e. they are state-dependent (SD). In this paper, without resorting to quantum states, we define the state-independent (SI) purity and fidelity of a general quantum operation (evolution) in virtue of a new density matrix formalism for quantum operations, which is extended from the quantum state level to quantum operation level. The SI purity and fidelity gain more intrinsic physical properties of quantum operations than state-dependent ones, such as the purity of a one-qubit amplitude damping channel (with damping rate 1) is 1/2, which is in line with the fact that the channel is still a nonunitary operation described by two Kraus operators rather than a unitary one. But the state-dependent Haar average purity is 1 in this case. So the SI purity and fidelity proposed here can help the experimentalists to exactly quantify the implementation quality of an operation. As a byproduct, a new measure of the operator entanglement is proposed for a quantum evolution (unitary or nonunitary) in terms of the linear entropy of its density matrix on the orthonormal operator bases (OOBs) in Hilbert–Schmidt space.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
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.
Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as ${\ensuremath{\sum}}_{\mathbit{a},\mathbit{b}}{|\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathbit{a}\mathbit{b}}|}^{2}\ensuremath{\sim}{5}^{N}/{N}_{\mathrm{tot}}$ for full tomography, and ${\mathrm{\ensuremath{\Delta}}\ensuremath{\mu}}^{2}\ensuremath{\sim}{7}^{N}/{N}_{\mathrm{tot}}^{2}$ for purity estimation, for NNN qubits and Ntot{N}_{\mathrm{tot}}Ntot​ measurements performed.
There is a well-known mathematical similarity between two-dimensional classical polarization optics and two-level quantum systems, where the Poincar\'e and Bloch spheres are identical mathematical structures. This analogy implies that the classical degree of polarization and quantum purity are in fact the same quantity. We make extensive use of this analogy to analyze various measures of polarization for higher dimensions proposed in the literature, in particular the N=3N=3N=3 case, illustrating interesting relationships that emerge as well as the advantages of each measure. We also propose a different class of measures of entanglement based on the purity of subsystems.
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.