arXiv:1304.5134 · Physica Scripta

Characterization of fiducial states in prime dimensions via mutually unbiased bases

Dardo Goyeneche, Roberto Salazar, A. Delgado

Published Mar 1, 2013• 0 citations

Abstract

In this work we present some new properties of fiducial states in prime dimensions. We parameterize fiducial operators on eigenvectors bases of displacement operators, which allows us to find a manifold $Ω$ of hermitian operators satisfying $\mathrm{Tr}(ρ)=\mathrm{Tr}(ρ^2)=1$ for any $ρ$ in $Ω$. This manifold contains the complete set of fiducial pure states in every prime dimension. Indeed, any quantum state $ρ\geq0$ belonging to $Ω$ is a fiducial pure state. Also, we present an upper bound for every probability associated to mutually unbiased decomposition of fiducial states. This bound allows us to prove that every fiducial state tends to be mutually unbiased to the maximal set of mutually unbiased bases in higher prime dimensions. Finally, we show that any $ρ$ in $Ω$ minimizes an entropic uncertainty principle related to the second order Rényi entropy.

DOI: 10.1088/0031-8949/2013/t153/014031

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arXiv:1304.5134 · Physica Scripta

Characterization of fiducial states in prime dimensions via mutually unbiased bases

Dardo Goyeneche, Roberto Salazar, A. Delgado

Published Mar 1, 2013• 0 citations

arXiv HTML PDF

Abstract

Ω

of hermitian operators satisfying

\mathrm{Tr}(ρ)=\mathrm{Tr}(ρ^2)=1

for any

ρ

Ω

. This manifold contains the complete set of fiducial pure states in every prime dimension. Indeed, any quantum state

ρ\geq0

belonging to

Ω

is a fiducial pure state. Also, we present an upper bound for every probability associated to mutually unbiased decomposition of fiducial states. This bound allows us to prove that every fiducial state tends to be mutually unbiased to the maximal set of mutually unbiased bases in higher prime dimensions. Finally, we show that any

ρ

Ω

minimizes an entropic uncertainty principle related to the second order Rényi entropy.

DOI: 10.1088/0031-8949/2013/t153/014031

Related papers

Mutually unbiased frames

Fabián Caro-Pérez, Victor Gonzalez Avella, Dardo Goyeneche

In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d -dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d . These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d -dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d . These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.

arXiv:2110.08293QuantumNov 3, 202214 citations

Unextendible mutually unbiased bases in prime-squared dimensions

Vishakh Hegde, Prabha Mandayam

A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ( $d=p^{2}$ ) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1$ MUBs that saturate both a Shannon ( $H_{1}$ ) and a collision ( $H_{2}$ ) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1$ MUBs that saturate both a Shannon ($H_{1}$) and a collision ($H_{2}$) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.

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Michel R. P. Planat, Haret Codratian Rosu

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Jun 16, 200533 citations

Quantum phase uncertainty in mutually unbiased measurements and Gauss sums

Michel R. P. Planat, Haret Codratian Rosu

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse $1/\sqrt{d}$ , with $d$ the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality $d+1$ have been derived for prime power dimensions $d=p^m$ using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse $1/\sqrt{d}$, with $d$ the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality $d+1$ have been derived for prime power dimensions $d=p^m$ using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.

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We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $logN$ because there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L\ensuremath{\ge}3$ measurements. In the case of a prime power dimension, $N={p}^{k}$ , and the number of measurements $L=N+1$ , the upper bound for the average entropy becomes minimal for a collection of mutually unbiased bases. An analogous approach is used to study entanglement with respect to $L$ different splittings of a composite system linked by bipartite quantum gates. We show that, for any two-qubit unitary gate $U\ensuremath{\in}\text{U}(4)$ there exist states being mutually separable or mutually entangled with respect to both splittings (related by $U$ ) of the composite system. The latter statement follows from the fact that the real projective space $\mathbb{R}{P}^{3}\ensuremath{\subset}\mathbb{C}{P}^{3}$ is nondisplaceable by a unitary transformation. For $L=3$ splittings the maximal sum of $L$ entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $logN$ because there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L\ensuremath{\ge}3$ measurements. In the case of a prime power dimension, $N={p}^{k}$, and the number of measurements $L=N+1$, the upper bound for the average entropy becomes minimal for a collection of mutually unbiased bases. An analogous approach is used to study entanglement with respect to $L$ different splittings of a composite system linked by bipartite quantum gates. We show that, for any two-qubit unitary gate $U\ensuremath{\in}\text{U}(4)$ there exist states being mutually separable or mutually entangled with respect to both splittings (related by $U$) of the composite system. The latter statement follows from the fact that the real projective space $\mathbb{R}{P}^{3}\ensuremath{\subset}\mathbb{C}{P}^{3}$ is nondisplaceable by a unitary transformation. For $L=3$ splittings the maximal sum of $L$ entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.

arXiv:1506.07709Physical Review ASep 9, 201522 citations

Mutually Unbiased Product Bases

Daniel McNulty

A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type. \nA pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type. \n

Apr 1, 20130 citations

Ω

of hermitian operators satisfying

\mathrm{Tr}(ρ)=\mathrm{Tr}(ρ^2)=1

for any

ρ

Ω

. This manifold contains the complete set of fiducial pure states in every prime dimension. Indeed, any quantum state

ρ\geq0

belonging to

Ω

ρ

Ω

minimizes an entropic uncertainty principle related to the second order Rényi entropy.

d=p^{2}

) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of

p+1

MUBs that saturate both a Shannon (

H_{1}

) and a collision (

H_{2}

) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse

1/\sqrt{d}

, with

d

the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality

d+1

have been derived for prime power dimensions

d=p^m

using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.

We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size

N

L

orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For

L=2

the maximal average entropy saturates at

logN

because there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L\ensuremath{\ge}3$ measurements. In the case of a prime power dimension,

N={p}^{k}

, and the number of measurements

L=N+1

, the upper bound for the average entropy becomes minimal for a collection of mutually unbiased bases. An analogous approach is used to study entanglement with respect to

L

different splittings of a composite system linked by bipartite quantum gates. We show that, for any two-qubit unitary gate $U\ensuremath{\in}\text{U}(4)$ there exist states being mutually separable or mutually entangled with respect to both splittings (related by

U

) of the composite system. The latter statement follows from the fact that the real projective space $\mathbb{R}{P}^{3}\ensuremath{\subset}\mathbb{C}{P}^{3}$ is nondisplaceable by a unitary transformation. For

L=3

splittings the maximal sum of

L

entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.