Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2111.07726

Complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities

Donghoon Ha, Younghun Kwon

Published Nov 15, 2021• 0 citations
arXivHTMLPDF

Abstract

In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condition, we completely analyze minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities.

DOI
10.48550/arxiv.2111.07726

Related papers

Algebraic approach to minimum-error discrimination of qubit states

Xin Lü, Shi-Hai Dong

Abstract We introduce a general algebraic approach for qubit state discrimination. The primary challenge in such problems lies in determining the nonzero elements of an optimal positive operator-valued measure (POVM). Our work establishes an explicit algebraic criterion to verify whether an optimal POVM for discriminating arbitrary qubit states contains zero elements. Based on this condition, we propose an exhaustive algorithm to obtain analytical solutions for discriminating any number of mixed qubit states with arbitrary prior probabilities.

Journal of Physics A Mathematical and Theoretical•Mar 3, 2026•0 citations

Geometric bloch vector solution to minimum-error discriminations of mixed qubit states

Mahdi Rouhbakhsh N., Seyed Arash Ghoreishi

Abstract We investigate minimum-error (ME) discrimination for mixed qubit states using a geometric approach. By analyzing positive operator-valued measure (POVM) solutions and introducing Lagrange operator $$\Gamma $$ Γ , we develop a four-step structured instruction to find $$\Gamma $$ Γ for N mixed qubit states. Our method covers optimal solutions for two, three, and four mixed qubit states, including a novel result for four qubit states. We introduce geometric-based POVM classes and non-decomposable subsets for constructing optimal solutions, enabling us to find all possible answers for the general problem of minimum-error discrimination for N mixed qubit states with arbitrary a priori probabilities.

arXiv:2108.12299•Quantum Information Processing•Aug 24, 2023•5 citations

Optimal discrimination of single-qubit mixed states

Graeme Weir, Stephen Mark Barnett, Sarah Croke

We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities. It has long been known that the minimum-error probability is given by the trace of the Lagrange operator $\mathrm{\ensuremath{\Gamma}}$. The remarkable feature of our approach is the central role played not by $\mathrm{\ensuremath{\Gamma}}$, but by its inverse.

arXiv:1704.01035•Physical Review A•Aug 14, 2017•20 citations

Quantum Contextual Advantage Depending on Nonzero Prior Probabilities in State Discrimination of Mixed Qubit States

Jaehee Shin, Donghoon Ha, Younghun Kwon

Recently, Schmid and Spekkens studied the quantum contextuality in terms of state discrimination. By dealing with the minimum error discrimination of two quantum states with identical prior probabilities, they reported that quantum contextual advantage exists. Meanwhile, if one notes a striking observation that the selection of prior probability can affect the quantum properties of the system, it is necessary to verify whether the quantum contextual advantage depends on the prior probabilities of the given states. In this paper, we consider the minimum error discrimination of two states with arbitrary prior probabilities, in which both states are pure or mixed. We show that the quantum contextual advantage in state discrimination may depend on the prior probabilities of the given states. In particular, even though the quantum contextual advantage always exists in the state discrimination of two nonorthogonal pure states with nonzero prior probabilities, the quantum contextual advantage depends on prior probabilities in the state discrimination of two mixed states.

Entropy•Nov 26, 2021•16 citations

Optimal discrimination of quantum states

Graeme Weir

Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.

•Jan 1, 2018•4 citations

Analysis of optimal unambiguous discrimination of three pure quantum states

Donghoon Ha, Younghun Kwon

We consider unambiguous discrimination of three pure quantum states. Necessary and sufficient conditions to decide which states should be detected for optimal measurement of unambiguous discrimination are provided in terms of inner products and the geometric phase $\mathrm{\ensuremath{\Phi}}$. We get the optimal measurement and the optimal failure probability when the optimal unambiguous discrimination does not require the detection of every given quantum state. When at least two quantum states are orthogonal to each other, we supply the optimal measurement and optimal failure probability in analytic form. When all three quantum states are not orthogonal to each other and $\mathrm{\ensuremath{\Phi}}\ensuremath{\ne}0$, we find an analytic condition to determine the zero and nonzero elements for an optimal positive operator valued measure. We explain how to determine the solution in a geometric manner. Using the known solution of a case where the mutual inner products are real, we check the necessary and sufficient conditions when $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$ and analyze the property of the singular point when $\mathrm{\ensuremath{\Phi}}=0$ and the relation between the optimal points.

Physical Review A•Jun 9, 2015•25 citations
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condition, we completely analyze minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities.
Abstract We introduce a general algebraic approach for qubit state discrimination. The primary challenge in such problems lies in determining the nonzero elements of an optimal positive operator-valued measure (POVM). Our work establishes an explicit algebraic criterion to verify whether an optimal POVM for discriminating arbitrary qubit states contains zero elements. Based on this condition, we propose an exhaustive algorithm to obtain analytical solutions for discriminating any number of mixed qubit states with arbitrary prior probabilities.
Abstract We investigate minimum-error (ME) discrimination for mixed qubit states using a geometric approach. By analyzing positive operator-valued measure (POVM) solutions and introducing Lagrange operator Γ\Gamma Γ Γ , we develop a four-step structured instruction to find Γ\Gamma Γ Γ for N mixed qubit states. Our method covers optimal solutions for two, three, and four mixed qubit states, including a novel result for four qubit states. We introduce geometric-based POVM classes and non-decomposable subsets for constructing optimal solutions, enabling us to find all possible answers for the general problem of minimum-error discrimination for N mixed qubit states with arbitrary a priori probabilities.
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities. It has long been known that the minimum-error probability is given by the trace of the Lagrange operator $\mathrm{\ensuremath{\Gamma}}$. The remarkable feature of our approach is the central role played not by $\mathrm{\ensuremath{\Gamma}}$, but by its inverse.
Recently, Schmid and Spekkens studied the quantum contextuality in terms of state discrimination. By dealing with the minimum error discrimination of two quantum states with identical prior probabilities, they reported that quantum contextual advantage exists. Meanwhile, if one notes a striking observation that the selection of prior probability can affect the quantum properties of the system, it is necessary to verify whether the quantum contextual advantage depends on the prior probabilities of the given states. In this paper, we consider the minimum error discrimination of two states with arbitrary prior probabilities, in which both states are pure or mixed. We show that the quantum contextual advantage in state discrimination may depend on the prior probabilities of the given states. In particular, even though the quantum contextual advantage always exists in the state discrimination of two nonorthogonal pure states with nonzero prior probabilities, the quantum contextual advantage depends on prior probabilities in the state discrimination of two mixed states.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We consider unambiguous discrimination of three pure quantum states. Necessary and sufficient conditions to decide which states should be detected for optimal measurement of unambiguous discrimination are provided in terms of inner products and the geometric phase $\mathrm{\ensuremath{\Phi}}$. We get the optimal measurement and the optimal failure probability when the optimal unambiguous discrimination does not require the detection of every given quantum state. When at least two quantum states are orthogonal to each other, we supply the optimal measurement and optimal failure probability in analytic form. When all three quantum states are not orthogonal to each other and $\mathrm{\ensuremath{\Phi}}\ensuremath{\ne}0$, we find an analytic condition to determine the zero and nonzero elements for an optimal positive operator valued measure. We explain how to determine the solution in a geometric manner. Using the known solution of a case where the mutual inner products are real, we check the necessary and sufficient conditions when $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$ and analyze the property of the singular point when $\mathrm{\ensuremath{\Phi}}=0$ and the relation between the optimal points.