Structural approach to unambiguous discrimination of two mixed quantum states

Kleinmann, Matthias; Kampermann, Hermann; Bruß, Dagmar

doi:10.1063/1.3298683

Cited by 16 publications

(16 citation statements)

References 36 publications

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“…By (i),M opt i can be expressed as (25). However, since C 1 = C 2 impliesτ opt 0 = 0,M opt i becomes (26). Therefore P I (C 1 ) can be written as [0, ρ 11 +ρ 22 δ C1,C2 ].…”

Section: Proof Of Lemma Ii2 Assume That When {Qmentioning

confidence: 99%

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

Kwon

2017

Quantum Inf Process

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In this paper we consider the optimal discrimination of two mixed qubit states for a measurement that allows a fixed rate of inconclusive results. Our strategy is to transform the problem of two qubit states into a minimum error discrimination for three qubit states by adding a specific quantum state ρ0 and a prior probability q0, which behaves as an inconclusive degree. First, we introduce the beginning and the end of practical interval of inconclusive result, q (0) 0 and q (1) 0 , which are key ingredients in investigating our problem. Then we obtain the analytic form of them. Next, we show that our problem can be classified into two cases q0 = q 0 . In fact, by maximum confidences of two qubit states and non-diagonal element of ρ0, the our problem is completely understood. We provide an analytic solution of our problem when q0 = q (1) 0 , we rather supply the numerical method to find the solution, because of the complex relation between inconclusive degree and corresponding failure probability. Finally we confirm our results using previously known examples.

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“…By (i),M opt i can be expressed as (25). However, since C 1 = C 2 impliesτ opt 0 = 0,M opt i becomes (26). Therefore P I (C 1 ) can be written as [0, ρ 11 +ρ 22 δ C1,C2 ].…”

Section: Proof Of Lemma Ii2 Assume That When {Qmentioning

confidence: 99%

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

Kwon

2017

Quantum Inf Process

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“…• If η 1 ∈ [ 1 1+λ 2 , 1] the optimal measurement is a projective measurement, which either unambiguously identify state |ψ 1 � or produce an inconclusive result 15) and gives the probability of discrimination:…”

Section: Unambiguous Discrimination Of Two Known Pure Statesmentioning

confidence: 99%

“…This is because any USD measurement {E k } can be turned into proper USD measurement with the same operators E � k and the same probability of success. Kleinmann [15] shows that a POVM {E k } is a proper USD measurement if and only if E 0 acts as identity on S ⊥ , E 0 ≥ 0, I − E 0 ≥ 0, and γ 1 (I − E 0 )γ 2 = 0. Hence, the inconclusive element E 0 uniquely determines the proper USD measurement.…”

Section: Unambiguous Discrimination Of Two Known Pure Statesmentioning

confidence: 99%

“…For proper USD measurements the first two Raynal's reduction theorems are show [15] to commute and to be idempotent. Hence, only one application of each of them is needed.…”

Section: Unambiguous Discrimination Of Two Known Pure Statesmentioning

confidence: 99%

“…In particular, in [14] these authors found commutators, which reveal two dimensional block diagonal structure in the reduced states. Moreover, they succeeded [15,16] to rewrite the necessary and sufficient optimality conditions by Eldar, Stojnic, and Hassibi [17] into an operational form, which enabled them to prove uniqueness of the optimal measurement within the set of proper USD measurements. Finally, they completely classify and derive the solutions in four dimensional Hilbert space.…”

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confidence: 99%

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Quantum theory of unambiguous measurements

Sedlák¹

2009

Acta Physica Slovaca. Reviews and Tutorials

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In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.

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A minimal set of measurements for qudit-state tomography based on unambiguous discrimination

Kwon

2018

Quantum Inf Process

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Structural approach to unambiguous discrimination of two mixed quantum states

Cited by 16 publications

References 36 publications

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

Quantum theory of unambiguous measurements

A minimal set of measurements for qudit-state tomography based on unambiguous discrimination

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