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arXiv:quant-ph/0012124 · Physical Review Letters

Simultaneous Minimum-Uncertainty Measurement of Discrete-Valued Complementary Observables

Alexei S. Trifonov, Gunnar G. Bjork, Jonas Söderholm

Published May 14, 2001• 27 citations
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Abstract

We have made the first experimental demonstration of the simultaneous minimum uncertainty product between two complementary observables for a two-state system (a qubit). A partially entangled two-photon state, where each of the photons carries (partial) information of the initial state, was used to perform such a measurement.
DOI
10.1103/physrevlett.86.4423

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Different quantum information schemes, such as eavesdropping in quantum cryptography, dictate the necessity of extracting information about pair conjugate observables from a single copy of a quantum system. Mathematically, quantum measurements are usually described by an uncertainty relation. The difference between the simultaneous measurement uncertainty relation form those known from the textbooks on quantum mechanics is the additional uncertainty associated with the measurement procedure itself in contrast to the state preparation uncertainties described by the Schroedinger-Robertson type uncertainty relations. We present here an overview of our approach based on the state estimation theory and maximum likelihood strategy. We make a theoretical analysis and an experimental verification of minimum-uncertainty product of the two-states quantum system simultaneous measurement based on partially entangled photon pairs.
In quantum mechanics, measurements disturb the state of the system being measured. This disturbance is largest for complementary properties (e.g. position and momentum) and hence limits the precision with which such properties can be determined simultaneously. Often, this fact is conflated with Heisenberg's uncertainty principle, which refers to an uncertainty relation between complementary properties that is intrinsic to quantum states. In this thesis, the distinction between these two fundamental characteristics of quantum mechanics is made clear. At the intersection of the two are "joint measurements", which circumvent measurement disturbance to simultaneously determine complementary properties. They have applications in quantum metrology and enable a direct measurement of quantum states. The focus of this thesis is on the latter. The thesis is structured in the following way. The first chapter serves as an introduction to joint measurements. It surveys the seminal works in the field, doing so in a chronological manner to provide some historical context. The remainder of the thesis discusses two strategies to experimentally achieve joint measurements. The first strategy is to sequentially measure the complementary properties, making these measurements weak so that they do not disrupt each other. The second strategy is to first clone the system being measured, and then measure each complementary property on a separate clone. Both strategies are experimentally demonstrated on polarized photons, but can be readily extended to other systems.
Two observables A and B of an n-level system (i.e., a quantum system with n-dimensional state space) are called complementary, if knowledge of the measured value of A implies maximal uncertainty of the measured value of B, and vice versa. Such observables exist for all n, but no classification (up to equivalence) of all possible pairs of complementary observables is known except for n\ensuremath{\le}4. Complementary observables are conjectured to satisfy an ``entropic'' uncertainty relation of the strongest possible form. This relation has been verified for n\ensuremath{\le}4 by explicit calculations. A recent attempt of substantiating the widespread interpretation of uncertainty relations in terms of mutual disturbances between measurements is criticized.
The state of a quantum-mechanical system after a simultaneous measurement of conjugate variables such as coordinate and momentum is derived. In the case of the minimum-uncertainty, or ideal, simultaneous measurement, the state of the system after the measurement is a coherent state corresponding to that representing a minimum-uncertainty wave packet. An operator formalism to describe simultaneous measurements of noncommuting observables is introduced which parallels that familiar for a single measurement, and an expression for the joint probability distribution predicting the results of an ideal simultaneous measurement is derived and applied to the example of a harmonic oscillator. It is found that the minimum uncertainty in a simultaneous measurement of noncommuting variables has two causes: (1) the unavoidable perturbations introduced by the measuring process, and (2) the unavoidable lack of precision in the state of the system itself. For position and momentum measurements, these two independent uncertainties contribute to give a net minimum uncertainty of $\ensuremath{\Delta}{q}^{\ensuremath{'}}\ensuremath{\Delta}{p}^{\ensuremath{'}}=\ensuremath{\hbar}$. It is also shown that the formalism of simultaneous measurement leads to a well-defined quantum-mechanical phase space.
The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for N(N ≥ 2) observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicate the existence of stringent lower bounds.
The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Minimum uncertainty states are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all minimum uncertainty states of any given set of observables and determine the minimum entanglement required. It turns out all minimum uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a precise connection between minimum uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.