Brendon Lloyd Higgins

Entanglement-free Heisenberg-limited phase estimation

Measurement underpins all quantitative science. A key example is the measurement of optical phase, used in length metrology and many other applications. Advances in precision measurement have consistently led to important scientific discoveries. At the fundamental level, measurement precision is limited by the number N of quantum resources (such as photons) that are used. Standard measurement schemes, using each resource independently, lead to a phase uncertainty that scales as 1/—known as the standard quantum limit. However, it has long been conjectured that it should be possible to achieve a precision limited only by the Heisenberg uncertainty principle, dramatically improving the scaling to 1/N (ref. 3). It is commonly thought that achieving this improvement requires the use of exotic quantum entangled states, such as the NOON state. These states are extremely difficult to generate. Measurement schemes with counted photons or ions have been performed with Nu2009≤u20096 (refs 6–15), but few have surpassed the standard quantum limit and none have shown Heisenberg-limited scaling. Here we demonstrate experimentally a Heisenberg-limited phase estimation procedure. We replace entangled input states with multiple applications of the phase shift on unentangled single-photon states. We generalize Kitaev’s phase estimation algorithm using adaptive measurement theory to achieve a standard deviation scaling at the Heisenberg limit. For the largest number of resources used (N = 378), we estimate an unknown phase with a variance more than 10u2009dB below the standard quantum limit; achieving this variance would require more than 4,000 resources using standard interferometry. Our results represent a drastic reduction in the complexity of achieving quantum-enhanced measurement precision.

Entanglement-enhanced measurement of a completely unknown optical phase

We demonstrate a method for achieving phase measurements with accuracy beyond the standard quantum limit using entangled states. A sophisticated feedback scheme means that no initial estimate of the phase is required.

Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements

We derive, and experimentally demonstrate, an interferometric scheme for unambiguous phase estimation with precision scaling at the Heisenberg limit that does not require adaptive measurements. That is, with no prior knowledge of the phase, we can obtain an estimate of the phase with a standard deviation that is only a small constant factor larger than the minimum physically allowed value. Our scheme resolves the phase ambiguity that exists when multiple passes through a phase shift, or NOON states, are used to obtain improved phase resolution. Like a recently introduced adaptive technique (Higgins et al 2007 Nature 450 393), our experiment uses multiple applications of the phase shift on single photons. By not requiring adaptive measurements, but rather using a predetermined measurement sequence, the present scheme is both conceptually simpler and significantly easier to implement. Additionally, we demonstrate a simplified adaptive scheme that also surpasses the standard quantum limit for single passes.

How to perform the most accurate possible phase measurements

We present the theory of how to achieve phase measurements with the minimum possible variance in ways that are readily implementable with current experimental techniques. Measurements whose statistics have high-frequency fringes, such as those obtained from NOON states, have commensurately high information yield (as quantified by the Fisher information). However this information is also highly ambiguous because it does not distinguish between phases at the same point on different fringes. We provide schemes to eliminate this phase ambiguity in a highly efficient way, providing phase estimates with uncertainty that is within a small constant factor of the Heisenberg limit, the minimum allowed by the laws of quantum mechanics. These techniques apply to NOON state and multi-pass interferometry, as well as phase measurements in quantum computing. We have reported the experimental implementation of some of these schemes with multi-pass interferometry elsewhere. Here we present the theoretical foundation, and also present some new experimental results. There are three key innovations to the theory in this paper. First, we examine the intrinsic phase properties of the sequence of states (in multiple time modes) via the equivalent two-mode state. Second, we identify the key feature of the equivalent state that enables the optimal scaling of the intrinsic phase uncertainty to be obtained. This enables us to identify appropriate combinations of states to use. The remaining difficulty is that the ideal phase measurements to achieve this intrinic phase uncertainty are often not physically realizable. The third innovation is to solve this problem by using realizable measurements that closely approximate the optimal measurements, enabling the optimal scaling to be preserved. We consider both adaptive and nonadaptive measurement schemes.

Experimental quantum key distribution with simulated ground-to-satellite photon losses and processing limitations

Quantum key distribution (QKD) has the potential to improve communications security by offering cryptographic keys whose security relies on the fundamental properties of quantum physics. The use of a trusted quantum receiver on an orbiting satellite is the most practical near-term solution to the challenge of achieving long-distance (global-scale) QKD, currently limited to a few hundred kilometers on the ground. This scenario presents unique challenges, such as high photon losses and restricted classical data transmission and processing power due to the limitations of a typical satellite platform. Here we demonstrate the feasibility of such a system by implementing a QKD protocol, with optical transmission and full post-processing, in the high-loss regime using minimized computing hardware at the receiver. Employing weak coherent pulses with decoy states, we demonstrate the production of secure key bits at up to 56.5 dB of photon loss. We further illustrate the feasibility of a satellite uplink by generating secure key while experimentally emulating the varying channel losses predicted for realistic low-Earth-orbit satellite passes at 600 km altitude. With a 76 MHz source and including finite-size analysis, we extract 3374 bits of secure key from the best pass. We also illustrate the potential benefit of combining multiple passes together: while one suboptimal upper-quartile pass produces no finite-sized key with our source, the combination of three such passes allows us to extract 165 bits of secure key. Alternatively, we find that by increasing the signal rate to 300 MHz it would be possible to extract 21570 bits of secure finite-sized key in just a single upper-quartile pass.

Adaptive Measurements in the Optical Quantum Information Laboratory

Adaptive techniques make practical many quantum measurements that would otherwise be beyond current laboratory capabilities. For example, they allow discrimination of nonorthogonal states with a probability of error equal to the Helstrom bound, measurement of the phase of a quantum oscillator with accuracy approaching (or in some cases attaining) the Heisenberg limit (HL), and estimation of phase in interferometry with a variance scaling at the HL, using only single qubit measurement and control. Each of these examples has close links with quantum information, in particular, experimental optical quantum information: the first is a basic quantum communication protocol; the second has potential application in linear optical quantum computing; the third uses an adaptive protocol inspired by the quantum phase estimation algorithm. We discuss each of these examples and their implementation in the laboratory, but concentrate upon the last, which was published most recently [Higgins et al. , Nature, vol. 450, p. 393, 2007].

Mixed state discrimination using optimal control

We present theory and experiment for the task of discriminating two nonorthogonal states, given multiple copies. We implement several local measurement schemes, on both pure states and states mixed by depolarizing noise. We find that schemes which are optimal (or have optimal scaling) without noise perform worse with noise than simply repeating the optimal single-copy measurement. Applying optimal control theory, we derive the globally optimal local measurement strategy, which outperforms all other local schemes, and experimentally implement it for various levels of noise.

Multiple-copy state discrimination: Thinking globally, acting locally

We theoretically investigate schemes to discriminate between two nonorthogonal quantum states given multiple copies. We consider a number of state discrimination schemes as applied to nonorthogonal, mixed states of a qubit. In particular, we examine the difference that local and global optimization of local measurements makes to the probability of obtaining an erroneous result, in the regime of finite numbers of copies

Entanglement-free, Heisenberg-limited phase measurement

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Entanglement-enhanced measurement of a completely unknown phase

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