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1 REPRESENTATION OF QUANTUM STATES AS POINTS IN A PROBABILITY SIMPLEX ASSOCIATED TO A SIC-POVM

José Ignacio Rosado

Published Aug 16, 2016• 7 citations
OpenAlex

Abstract

The quantum state of a $d$-dimensional system can be represented by the $d^2$ probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of $\R^{d^2-1}$ in a simplex, we will call this set of vectors $\mathcal{Q}$. Other way of represent a $d$-dimensional system is by the corresponding Bloch vector also in $\R^{d^2-1}$, we will call this set of vectors $\mathcal{B}$. In this paper it is proved that with the adequate scaling $\mathcal{B}=\mathcal{Q}$. Also we indicate some features of the shape of $\mathcal{Q}$.

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SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

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The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1

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Necessary condition for existence of conditional SIC-POVM (Mathematics for Uncertainty and Fuzziness)

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This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in $\mathbb{C}^{n}$ whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such $n^{2}$ vectors in $\mathbb{C}^{n}$ for any $n$ .But the existence is only proved when $n\leq 15$ and $n=19$ , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in $\mathbb{C}^{n}$ .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 $\rho\in M_{n}(\mathbb{C})$ is called a density matrix (or state) if $\rho\geq 0$ and $\prime b(\rho)=1.$For any density matrix $\rho$ , we can define a state $\hat{\rho}$ by $\hat{\rho}(X)=h(\rho X)$ .Conversely, any state is written by the above form.Therefore, there exists a one-t $(\succ$ one correspondence between density matrices and states.Definition 2.2 A set of positive operators $\{P_{i}\}_{i=1}^{k}\subset M_{n}(\mathbb{C})$ is called a positive operator valued measure (POVM) if $P_{i}\geq 0(1\leq i\leq k)$ and $\sum_{i=1}^{k}P=I.$If $P_{1}$ is a projection, then $\{P_{i}\}_{i=1}^{k}$ is called a projection valued measure $(PVM)$ .

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Geometry of Quantum States from Symmetric Informationally Complete Probabilities

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It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.

•Aug 6, 2013•3 citations
The quantum state of a ddd-dimensional system can be represented by the d2d^2d2 probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of Rd2−1\R^{d^2-1}Rd2−1 in a simplex, we will call this set of vectors Q\mathcal{Q}Q. Other way of represent a ddd-dimensional system is by the corresponding Bloch vector also in Rd2−1\R^{d^2-1}Rd2−1, we will call this set of vectors B\mathcal{B}B. In this paper it is proved that with the adequate scaling B=Q\mathcal{B}=\mathcal{Q}B=Q. Also we indicate some features of the shape of Q\mathcal{Q}Q.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.
The geometry of the Quantum State Space, described by Bloch vectors, is a very intricate one. A deeper understanding of this geometry could lead to the solution of some difficult problems in Quantum Foundations and Quantum Information such as the existence of SIC-POVMs and the cardinality of the maximal set of MUBs. In this paper we show that the geometry of quantum states can be described by the probability distributions that quantum states induce over the outcomes of symmetric POVMs not necessarily of arbitrary rank or informationally complete. We also describe the properties of these symmetric POVMs.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in Cn\mathbb{C}^{n}Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such n2n^{2}n2 vectors in Cn\mathbb{C}^{n}Cn for any nnn .But the existence is only proved when n≤15n\leq 15n≤15 and n=19n=19n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in Cn\mathbb{C}^{n}Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 ρ∈Mn(C)\rho\in M_{n}(\mathbb{C})ρ∈Mn​(C) is called a density matrix (or state) if ρ≥0\rho\geq 0ρ≥0 and ′b(ρ)=1.\prime b(\rho)=1.′b(ρ)=1.For any density matrix ρ\rhoρ , we can define a state ρ^\hat{\rho}ρ^​ by ρ^(X)=h(ρX)\hat{\rho}(X)=h(\rho X)ρ^​(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t (≻(\succ(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators {Pi}i=1k⊂Mn(C)\{P_{i}\}_{i=1}^{k}\subset M_{n}(\mathbb{C}){Pi​}i=1k​⊂Mn​(C) is called a positive operator valued measure (POVM) if Pi≥0(1≤i≤k)P_{i}\geq 0(1\leq i\leq k)Pi​≥0(1≤i≤k) and ∑i=1kP=I.\sum_{i=1}^{k}P=I.∑i=1k​P=I.If P1P_{1}P1​ is a projection, then {Pi}i=1k\{P_{i}\}_{i=1}^{k}{Pi​}i=1k​ is called a projection valued measure (PVM)(PVM)(PVM) .
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.