Maximum Likelihood Estimation for a Group of Physical Transformations

Chiribella, Giulio; Mauro, Giacomo; Volta, A.; Perinotti, Paolo; Sacchi, Massimiliano F.

doi:10.1142/s0219749906002018

Cited by 21 publications

(39 citation statements)

References 28 publications

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“…Distinguishability can be quantified using maximum likelihood or fidelity measures [22][23][24][25][26]. Because we also want the reference frame states to become ideal (perfectly distinguishable) in the classical limit, we want this distinguishability to scale with D s R (see [23,[25][26][27] regarding asymptotic measures). A useful choice of reference frame states for a group G on D s R dimensions are the maximum likelihood states [23], denoted |g or |s R ; g (the latter following the notation |s R ; ψ(g) ), as these states are optimal for a range of operational tasks involving reference frames.…”

Section: Quantum Reference Frame Statesmentioning

confidence: 99%

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Changing quantum reference frames

2014

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We consider the process of changing reference frames in the case where the reference frames are quantum systems. We find that, as part of this process, decoherence is necessarily induced on any quantum system described relative to these frames. We explore this process with examples involving reference frames for phase and orientation. Quantifying the effect of changing quantum reference frames provides a theoretical description for this process in quantum experiments, and serves as a first step in developing a relativity principle for theories in which all objects including reference frames are necessarily quantum.

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Section: Quantum Reference Frame Statesmentioning

confidence: 99%

“…It is straightforward to generalise the machinery to cases where dim M (q) = dim N (q) ; see Refs. [15,22,23,[28][29][30]. For more details regarding properties of these states see Ref.…”

Section: Quantum Reference Frame Statesmentioning

confidence: 99%

Changing quantum reference frames

2014

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Section: Figmentioning

confidence: 99%

“…For n ≥ 3, the only cases where a solution is known are those of pure states with a high degree of symmetry. This includes the symmetric states [16], generated by the action of a unitary operation U satisfying U n = 1 1, and, more generally, states that are generated by a group of unitaries [17,18].Interestingly, the change point problem does not fall into any of the above categories. In spite of this, we show that the problem can be completely solved in the asymptotic regime: in the limit of long sequences, the maximum probability of success takes the elegant form…”

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confidence: 99%

Quantum Change Point

Sentís¹,

Bagán²,

Calsamiglia³

et al. 2016

Phys. Rev. Lett.

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Sudden changes are ubiquitous in nature. Identifying them is crucial for a number of applications in biology, medicine, and social sciences. Here we take the problem of detecting sudden changes to the quantum domain. We consider a source that emits quantum particles in a default state, until a point where a mutation occurs that causes the source to switch to another state. The problem is then to find out where the change occurred. We determine the maximum probability of correctly identifying the change point, allowing for collective measurements on the whole sequence of particles emitted by the source. Then, we devise online strategies where the particles are measured individually and an answer is provided as soon as a new particle is received. We show that these online strategies substantially underperform the optimal quantum measurement, indicating that quantum sudden changes, although happening locally, are better detected globally.The detection of sudden changes in a sequence of random variables is a pivotal topic in statistics, known as the change point problem [1][2][3]. The problem has widespread applications, including the study of stock market variations [4], protein folding [5], and landscape changes [6]. In general, identifying change points plays a crucial role in all problems involving the analysis of samples collected over time [2,7] because such analysis requires the stability of the system parameters [8]. If changes are correctly detected, the sample can be conveniently divided in subsamples, which can then be analyzed by the standard statistical techniques. The detection of change points can also be viewed as a border problem [9], namely a problem where one wants to draw a separation between two (or more) different configurations -a task that plays a central role in machine learning [10].The simplest example of a change point problem is that of a coin with variable bias. Imagine that a game of Heads or Tails is played with a fair coin, but after a few rounds one player suspects that the other has replaced the coin with a biased one. After inspection of the coin, the suspicion is confirmed: the coin has now a bias. Can we identify when the coin was changed based only on the the sequence of outcomes? This classical problem has a natural extension to the quantum realm, illustrated in Figure 1: A source is promised to prepare quantum particles in some default state. At some point, however, the source undergoes a mutation and starts to produce copies of a different state. Given the sequence of particles emitted by the source, the problem is to find out when the change took place. In the basic version of the problem, the initial and final states are known, as in the classical example of the coin. No prior information is given about the location of the change: a priori, every point of the sequence is equally likely to be the change point. For simplicity, we assume the quantum states to be pure. FIG. 1:The quantum change point problem. A quantum source emits particles in a default state |0 , until the p...

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“…This is important because in practice the usefulness of unambiguous discrimination can be undermined by the fact that the inconclusive outcome occurs too frequently. To this purpose, we introduce the notion of generalized t-designs, which includes as special cases the unitary t-designs of [43][44][45][46] and all the examples where the unknown gates form a group [47][48][49][50]. Relative to gate identification, generalized t-designs have three important features:…”

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confidence: 99%

Identification of a reversible quantum gate: assessing the resources

2013

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We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies, we consider unambiguous gate discrimination-where errors are not tolerated but inconclusive outcomes are allowed-and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce the notion of generalized t-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized t-design we give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies and those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous 4 Identifying an unknown unitary evolution, available as a black box, is a fundamental problem in quantum theory [1][2][3][4][5][6][7][8][9], with a wide range of applications in quantum information and computation. In quantum computation, the problem is known as oracle identification [10][11][12][13][14] and is the core of paradigmatic quantum algorithms such as Grover's [15] and . In quantum information processing, the identification of an unknown unitary gate plays a key role in the stabilizer formalism of quantum error correction [17,18] and in its generalization to unitary error bases [19][20][21][22][23], in the security analysis of quantum cryptographic protocols that encode secret data into unitary gates [24][25][26][27][28][29], New Journal of Physics 15 (2013) 103019 (http://www.njp.org/)

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Maximum Likelihood Estimation for a Group of Physical Transformations

Cited by 21 publications

References 28 publications

Changing quantum reference frames

Changing quantum reference frames

Quantum Change Point

Identification of a reversible quantum gate: assessing the resources

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