Detection theory and quantum mechanics

Helstrom, Carl W.

doi:10.1016/s0019-9958(67)90302-6

Cited by 219 publications

(185 citation statements)

References 9 publications

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“…This limiting case corresponds to the standard two-outcome discrimination game [1][2][3], so the minimized cost will be equal to the usual HB. (ii) When k → 0, there is no penalty for non-guesses, so it is always better to decline (p d = 1) to produce min(C) → 0.…”

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confidence: 99%

Violating the modified Helstrom bound with nonprojective measurements

2015

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We consider the discrimination of two pure quantum states with three allowed outcomes: a correct guess, an incorrect guess, and a non-guess. To find an optimum measurement procedure, we define a tunable cost that penalizes the incorrect guess and non-guess outcomes. Minimizing this cost over all projective measurements produces a rigorous cost bound that includes the usual Helstrom discrimination bound as a special case. We then show that nonprojective measurements can outperform this modified Helstrom bound for certain choices of cost function. The Ivanovic-Dieks-Peres unambiguous state discrimination protocol is recovered as a special case of this improvement. Notably, while the cost advantage of the latter protocol is destroyed with the introduction of any amount of experimental noise, other choices of cost function have optima for which nonprojective measurements robustly show an appreciable, and thus experimentally measurable, cost advantage. Such an experiment would be an unambiguous demonstration of a benefit from nonprojective measurements.A fundamental consequence of quantum mechanics is the inability to perfectly distinguish between two nonorthogonal quantum states. Any attempt to guess which state is which after making a measurement will have an unavoidable probability of error that is bounded from below, as shown originally by Helstrom [1, 2] and in related work by Holevo [3]. This lower bound, known as the Helstrom bound (HB), grows with the overlap of the two states being discriminated.The HB can be circumvented, however, if a third option is added to the guessing game. Ivanovic, Dieks, and Peres showed that if one can also decline to guess after a measurement, then it is possible to reduce the probability of error to zero while still retaining a significant chance of guessing correctly [4][5][6]. Intriguingly, to maximize the correct guess probability in such "Unambiguous State Discrimination" (USD), it is not sufficient to use standard projective measurements; instead, one must use generalized (nonprojective) measurements [7,8].This advantage of nonprojective measurements in state discrimination is so surprising that it has become a featured example in modern quantum information textbooks (e.g., [9,10]), and has led to considerable research, both in theory [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and experiment [28][29][30][31][32][33][34][35][36][37] (reviewed, e.g., in [38,39]). Most of this work has focused on the extreme cases of zero declining (as with the HB) or zero error (as with USD), with fewer papers considering intermediate cases that minimize the declining probability given a fixed nonzero error rate [23][24][25][26][27]. Moreover, to our knowledge all but one [27] of these few works have neglected the effect that experimental imperfections will have upon the accessible minima. We are thus not aware of any paper that discusses a rigorous bound suitable to experimentally demonstrate that nonprojective measurements have a definitive advantage over projective...

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confidence: 99%

Violating the modified Helstrom bound with nonprojective measurements

2015

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“…However, it is one of the innermost consequences of the laws of quantum mechanics that nonorthogonal states cannot be discriminated with certainty. Optimal detection strategies were first investigated by Helstrom [20,21] and Holevo [22] and a lot of attention has since been devoted to the development of optimal and near-optimal receivers for binary coherent states [23][24][25][26][27][28][29][30][31][32] and for the discrimination of larger signal alphabets [33][34][35][36][37][38][39][40][41]. An overview over different receiver schemes is provided in Appendix B.…”

Section: Binary Coherent-state Cloningmentioning

confidence: 99%

“…An intermediate regime where both erroneous and inconclusive results are allowed has also been considered [29] and the minimal probability of error for a fixed probability of inconclusive results has been derived for pure [51] and mixed states [52]. In the following, we always assume Comparison of the error probability of the homodyne detector (red, dotted), the Kennedy receiver (green, dashed) [24], the optimized displacement receiver (purple, dash-dotted) [29], and the Helstrom bound [20,21].…”

Section: Appendix B: Binary Coherent State Receiversmentioning

confidence: 99%

Optimally cloned binary coherent states

et al. 2017

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Binary coherent state alphabets can be represented in a two-dimensional Hilbert space. We capitalize this formal connection between the otherwise distinct domains of qubits and continuous variable states to map binary phase-shift keyed coherent states onto the Bloch sphere and to derive their quantum-optimal clones. We analyze the Wigner function and the cumulants of the clones, and we conclude that optimal cloning of binary coherent states requires a nonlinearity above second order. We propose several practical and near-optimal cloning schemes and compare their cloning fidelity to the optimal cloner.

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“…where {P (b), P (e)} are the prior probabilities for Bob's and Eve's bit values, and P (b | e) is the conditional probability for Bob's bit value to be b given that Eve's is e. Alice's bits are equally likely to be 0 or 1, and Eve's conditional error probabilities satisfy [10] …”

Section: +45mentioning

confidence: 99%

Attacking quantum key distribution with single-photon two-qubit quantum logic

Shapiro

Wong

2006

Phys. Rev. A

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The Fuchs-Peres-Brandt (FPB) probe realizes the most powerful individual attack on BennettBrassard 1984 quantum key distribution (BB84 QKD) by means of a single controlled-NOT (CNOT) gate. This paper describes a complete physical simulation of the FPB-probe attack on polarizationbased BB84 QKD using a deterministic CNOT constructed from single-photon two-qubit quantum logic. Adding polarization-preserving quantum nondemolition measurements of photon number to this configuration converts the physical simulation into a true deterministic realization of the FPB attack.

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Detection theory and quantum mechanics

Cited by 219 publications

References 9 publications

Violating the modified Helstrom bound with nonprojective measurements

Violating the modified Helstrom bound with nonprojective measurements

Optimally cloned binary coherent states

Attacking quantum key distribution with single-photon two-qubit quantum logic

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