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Physical Review A

Determining an unknown state of a high-dimensional quantum system with a single, factorized observable

Ya‐Fei Yu, Hua Wen, Hua Li, Xinhua Peng

Published Mar 25, 2011• 5 citations
Publisher

Abstract

One can uniquely identify an unknown state of a quantum system $S$ by measuring a ``quorum'' consisting of a complete set of noncommuting observables. It is also possible to determine the quantum state by repeated measurements of a single and factorized observable, when the system $S$ is coupled to an assistant system $A$ whose initial state is known. This is because that the redistribution of the information about the unknown quantum state into the composite system $A+S$ results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. Here we focus on quantum state tomography of high-dimensional quantum systems (e.g., a spin greater than 1/2) via a single observable. We determine the condition for the best determination and the upper bound to achieve the most robust measurements. From the experimental view we require a suitable interaction Hamiltonian to maximize the measure efficiency. For this we numerically investigate a three-level system. Moreover, the error analysis for the different-dimensional quantum states shows that the present measurement method is still very effective in determining an unknown state of a high-dimensional quantum system.

DOI
10.1103/physreva.83.032318

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One can uniquely identify an unknown state of a quantum system SSS by measuring a ``quorum'' consisting of a complete set of noncommuting observables. It is also possible to determine the quantum state by repeated measurements of a single and factorized observable, when the system SSS is coupled to an assistant system AAA whose initial state is known. This is because that the redistribution of the information about the unknown quantum state into the composite system A+SA+SA+S results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. Here we focus on quantum state tomography of high-dimensional quantum systems (e.g., a spin greater than 1/2) via a single observable. We determine the condition for the best determination and the upper bound to achieve the most robust measurements. From the experimental view we require a suitable interaction Hamiltonian to maximize the measure efficiency. For this we numerically investigate a three-level system. Moreover, the error analysis for the different-dimensional quantum states shows that the present measurement method is still very effective in determining an unknown state of a high-dimensional quantum system.
This paper shows how one can reconstruct an unknown nnn-qubit state by only a single apparatus. Its core is a redistribution of the information within an extended Hilbert space by coupling the unknown system with an assistant system through only a pairwise interaction, which results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. In such an interaction configuration, which is more feasible in experiments, the quality of the measurement only depends on that of the subsystem consisting a of one-qubit system coupled to a one-qubit assistant. We analyze in detail the performance of this procedure and experimentally implement it for reconstructing an unknown two-qubit state using a four-nuclear-spin system.
The unknown state rho of a quantum system S is determined by letting it interact with an auxiliary system A, the initial state of which is known. A one-to-one mapping can thus be realized between the density matrix rho and the probabilities of the occurrence of the eigenvalues of a single and factorized observable of S+A, so that rho can be determined by repeated measurements using a single apparatus. If S and A are spins, it suffices to measure simultaneously their z components after a controlled interaction. The most robust setups are determined in this case for an initially pure or a completely disordered state of A. They involve an Ising or anisotropic Heisenberg coupling and an external field.
Experimental determination of an unknown quantum state usually requires several incompatible measurements. However, it is also possible to determine the full quantum state from a single, repeated measurement. For this purpose, the quantum system whose state is to be determined is first coupled to a second quantum system (the ``assistant'') in such a way that part of the information in the quantum state is transferred to the assistant. The actual measurement is then performed on the enlarged system including the original system and the assistant. We discuss in detail the requirements of this procedure and experimentally implement it on a simple quantum system consisting of nuclear spins.
The density matrix of a two-level system (spin, atom) is usually determined by measuring the three noncommuting components of the Pauli vector. This density matrix can also be obtained via the measurement data of two commuting variables, using a single apparatus. This is done by coupling the two-level system to a mode of radiation field, where the atom-field interaction is described with the Jaynes-Cummings model. The mode starts its evolution from a known coherent state. The unknown initial state of the atom is found by measuring two commuting observables: the population difference of the atom and the photon number of the field. We discuss the advantages of this setup and its possible applications.
Given the state of a quantum system, one can calculate the expectation value of any observable of the system. However, the inverse problem of determining the state by performing different measurements is not a trivial task. In various experimental setups it is reasonably straightforward to reconstruct the state of a quantum system employing linear tomographic technique. In this way the elements of the density matrix can be linearly related to a set of measured quantities. But since different observables of a quantum system are not commuting with each other, one often has to perform series of successive measurements of observables which cannot be done simultaneously. Simultaneous measurement of observables cost less time and energy and is more beneficial. In this paper we review the strategy of quantum state tomography with simultaneous measurement of commuting observables. This can be done by introducing an assistant system of which the state is known. We show that the interaction between the assistant and the system of interest within different frame works allows the reconstruction of the state of the system. Specifically, we consider a two-level system and reconstruct its initial state by introducing an assistant which can be either another two-level system or a single cavity mode of the electromagnetic field.
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointer) regardless of the size of the system. The system is coupled to a "pointer" of single qubit, and the wavefunction of the system is "reaped" onto the pointer upon the measurement of the system. Subsequently, standard two-state tomography on the pointer and classical post-processing are used to reconstruct the quantum state of the system. We also developed an efficient and scalable iterative maximum likelihood algorithm to estimate states from statistically incomplete data.