Some applications of uncertainty relations in quantum information

Majumdar, A. S.; Pramanik, Tanumoy

doi:10.1142/s0219749916400220

Cited by 9 publications

(5 citation statements)

References 69 publications

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“…Even the seminal error-disturbance relation by Heisenberg, while causing problems in terms of rigorous interpretation, has been examined in various different ways [24][25][26][27].Studies devoted to uncertainty relations are often motivated by a broad network of potential applications. In terms of quantum information, for instance, the uncertainty relations have found themselves [28] as important ingredients in security proofs of quantum key distribution [29,30]. In experimental studies within the field of quantum optics, they have been used in identification of quantum correlations such as entanglement [31][32][33] and Einstein-Podolsky-Rosen-steering [34][35][36][37][38].…”

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confidence: 99%

Uncertainty relations for characteristic functions

Rudnicki¹,

Tasca²,

Walborn³

2016

Phys. Rev. A

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We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is saturated in two extremal cases for wavefunctions described by periodic Dirac combs. We further discuss a broad spectrum of applications of the ChUR, in particular, we constrain quantum optical measurements involving general detection apertures and provide the uncertainty relation that is relevant for Loop Quantum Cosmology. A method to measure the characteristic function directly using an auxiliary qubit is also briefly discussed

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confidence: 99%

Uncertainty relations for characteristic functions

Rudnicki¹,

Tasca²,

Walborn³

2016

Phys. Rev. A

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“…If players A and B play the game using the shared state ρ AB then the maximum probability P max of winning the game overall strategy is given by [43,44]…”

Section: B Revisiting the Nonlocality Of Two-qubit Entangled States D...mentioning

confidence: 99%

Detection of the genuine non-locality of any three-qubit state

Garg¹,

Adhikari²

2023

Preprint

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It is known that the violation of Svetlichny inequality by any three-qubit state described by the density operator ρABC witness the genuine non-locality of ρABC. But it is not an easy task as the problem of showing the genuine non-locality of any three-qubit state reduces to the problem of a complicated optimization problem. Thus, the detection of genuine non-locality of any three-qubit state may be considered as a challenging task. Therefore, we have taken the different approach and derive the lower and upper bound of the expectation value of the Svetlichny operator with respect to any three-qubit state to study this problem. The expression of the obtained bounds depend on whether the reduced two-qubit entangled state detected by the CHSH witness operator or not. It may be expressed in terms of the following quantities such as (i) the eigenvalues of the product of the given three-qubit state and the composite system of single qubit maximally mixed state and reduced two-qubit state and (ii) the non-locality of reduced two-qubit state. We then achieve the inequality whose violation may detect the genuine non-locality of any three-qubit state. Few examples are cited to support our obtained results. Lastly, we discuss its possible implementation in the laboratory.

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“…In this subsection, we will define the strength of the non-locality of two-qubit entangled state r AB ent in terms of the maximum probability of winning of a game played between two distant players which are sharing an entangled state r AB ent . Let us consider an XOR game played between two distant players Alice (A) and Bob (B) [34,38]. In this game, the winner is decided by the XOR of the answers Å = + ( ) a b a b mod2 , where a, b ä {0, 1} and it denotes the answers given by the players A and B, when the referee asks them randomly selected questions (s, t) ä S × T, where S and T denote finite non-empty sets.…”

Section: A Definition Of the Strength Of The Non-locality Of Two-qubi...mentioning

confidence: 99%

Strength of the non-locality of two-qubit entangled states and its applications

Garg

Adhikari

2023

Phys. Scr.

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Non-locality is a feature of quantum mechanics that cannot be explained by local realistic theory. It can be detected by the violation of Bell's inequality. In this work, we have considered the evaluation of Bell's inequality with the help of the XOR game. In the XOR game, a two-qubit entangled state is shared between the two distant players. It may generate a non-local correlation between the players which contributes to the maximum probability of winning of the game. We have aimed to determine the strength of the non-locality through XOR game. Thus, we have defined a quantity $S_{NL}$ called the strength of non-locality, purely on the basis of the maximum probability of winning of the XOR game. We have also derived the relation between the introduced quantity $S_{NL}$ and the quantity $M$ introduced in \cite{horo3}, to study the non-locality of a two-qubit entangled state problem in depth. The quantity $M$ may be defined as the sum of the two largest eigenvalues of the correlation matrix of the given entangled state and it determines whether the given entangled state under probe is non-local. Further, we have explored the non-locality of any two-qubit entangled state, whose non-locality cannot be detected by the CHSH inequality.  Interestingly, we have found that the newly defined quantity $S_{NL}$ fails to detect non-locality for the entangled state, when the witness operator corresponding to $CHSH$ operator cannot detect the entangled state. To overcome this problem, we have modified the definition of the strength of non-locality and have shown that the modified definition may detect the non-locality of such entangled states, which were earlier undetected by $S_{NL}$. Furthermore, we have provided two applications of the strength $S_{NL}$ of the non-locality such as (i) establishment of a link between the two-qubit non-locality determined by $S_{NL}$ and the three-qubit non-locality determined by the Svetlichny operator and (ii) determination of the upper bound of the power of the controller in terms of $S_{NL}$ in controlled quantum teleportation.

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Some applications of uncertainty relations in quantum information

Cited by 9 publications

References 69 publications

Uncertainty relations for characteristic functions

Uncertainty relations for characteristic functions

Detection of the genuine non-locality of any three-qubit state

Strength of the non-locality of two-qubit entangled states and its applications

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