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:1812.11251 · Physical Review A

Conditional states and entropy in qudit-qubit systems

M. Bilkis, N. Canosa, R. Rossignoli, N. Gigena

Published Jun 25, 2019• 1 citation
arXivHTMLPDF

Abstract

We examine, in correlated mixed states of qudit-qubit systems, the set of all conditional qubit states that can be reached after local measurements at the qudit based on rank-1 projectors. While for a similar measurement at the qubit the conditional postmeasurement qudit states lie on the surface of an ellipsoid, for a measurement at the qudit we show that the set of postmeasurement qubit states can form more complex solid regions. In particular, we show the emergence, for some classes of mixed states, of sets which are the convex hull of solid ellipsoids and which may lead to conelike and trianglelike shapes in limit cases. We also analyze the associated measurement-dependent conditional entropy, providing a full analytic determination of its minimum and of the minimizing local measurement at the qudit for the previous states. Separable rank-2 mixtures are also discussed.

DOI
10.1103/physreva.99.062119

Related papers

Generalized conditional entropy optimization for qudit-qubit states

N. Gigena, R. Rossignoli

We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the postmeasurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest nontrivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.

arXiv:1409.0807•Physical Review A•Oct 16, 2014•12 citations

Generalized conditional entropy in bipartite quantum systems

N. Gigena, R. Rossignoli

We analyze, for a general concave entropic form, the associated conditional entropy of a quantum system A + B , obtained as a result of a local measurement on one of the systems ( B ). This quantity is a measure of the average mixedness of A after such measurement, and its minimum over all local measurements is shown to be the associated entanglement of formation between A and a purifying third system C . In the case of the von Neumann entropy, this minimum determines also the quantum discord. For classically correlated states and mixtures of a pure state with the maximally mixed state, we show that the minimizing measurement can be determined analytically and is universal , i.e., the same for all concave forms. While these properties no longer hold for general states, we also show that in the special case of the linear entropy, an explicit expression for the associated conditional entropy can be obtained, whose minimum among projective measurements in a general qudit–qubit state can be determined analytically, in terms of the largest eigenvalue of a simple 3 × 3 correlation matrix. Such minimum determines the maximum conditional purity of A , and the associated minimizing measurement is shown to be also universal in the vicinity of maximal mixedness. Results for X states, including typical reduced states of spin pairs in XY chains at weak and strong transverse fields, are also provided and indicate that the measurements minimizing the von Neumann and linear conditional entropies are typically coincident in these states, being determined essentially by the main correlation. They can differ, however, substantially from that minimizing the geometric discord.

arXiv:1308.3000•Journal of Physics A Mathematical and Theoretical•Dec 11, 2013•11 citations

Brief theory of multiqubit measurement

Constantin V. Usenko

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.

arXiv:2401.13122•International Journal of Quantum Information•May 31, 2024•0 citations

Quantum correlations and least disturbing local measurements

R. Rossignoli, N. Canosa, L. Ciliberti

We examine the evaluation of the minimum information loss due to an unread local measurement in mixed states of bipartite systems, for a general entropic form. Such a quantity provides a measure of quantum correlations, reducing for pure states to the generalized entanglement entropy, while in the case of mixed states it vanishes just for classically correlated states with respect to the measured system, as the quantum discord. General stationary conditions are provided, together with their explicit form for general two-qubit states. Closed expressions for the minimum information loss as measured by quadratic and cubic entropies are also derived for general states of two-qubit systems. As an application, we analyze the case of states with maximally mixed marginals, where a general evaluation is provided, as well as $X$ states and the mixture of two aligned states.

arXiv:1112.1587•Physical Review A•Nov 28, 2011•19 citations

Conditionalq-entropies and quantum separability: a numerical exploration

Josep Amengual i Batle, Angel Ricardo Plastino, M. Casas +1 more

We revisit the relationship between quantum separability and the sign of the relative q -entropies of composite quantum systems. The q -entropies depend on the density matrix eigenvalues p i through the quantity ω q = ∑ i p q i . Rényi's and Tsallis' measures constitute particular instances of these entropies. We perform a systematic numerical survey of the space of mixed states of two-qubit systems in order to determine, as a function of the degree of mixture, and for different values of the entropic parameter q , the volume in state space occupied by those states characterized by positive values of the relative entropy. Similar calculations are performed for qubit–qutrit systems and for composite systems described by Hilbert spaces of larger dimensionality. We pay particular attention to the limit case q → ∞. Our numerical results indicate that, as the dimensionalities of both subsystems increase, typical mixed states of composite quantum systems tend, as far as their relative q -entropies are concerned, to behave in a classical way.

arXiv:quant-ph/0207129•Journal of Physics A Mathematical and General•Nov 19, 2002•51 citations

Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States

Marco Enríquez, Zbigniew Puchała, Karol Życzkowski

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.

arXiv:1506.04788•Entropy•Jul 20, 2015•16 citations
We examine, in correlated mixed states of qudit-qubit systems, the set of all conditional qubit states that can be reached after local measurements at the qudit based on rank-1 projectors. While for a similar measurement at the qubit the conditional postmeasurement qudit states lie on the surface of an ellipsoid, for a measurement at the qudit we show that the set of postmeasurement qubit states can form more complex solid regions. In particular, we show the emergence, for some classes of mixed states, of sets which are the convex hull of solid ellipsoids and which may lead to conelike and trianglelike shapes in limit cases. We also analyze the associated measurement-dependent conditional entropy, providing a full analytic determination of its minimum and of the minimizing local measurement at the qudit for the previous states. Separable rank-2 mixtures are also discussed.
We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the postmeasurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest nontrivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.
We analyze, for a general concave entropic form, the associated conditional entropy of a quantum system A + B , obtained as a result of a local measurement on one of the systems ( B ). This quantity is a measure of the average mixedness of A after such measurement, and its minimum over all local measurements is shown to be the associated entanglement of formation between A and a purifying third system C . In the case of the von Neumann entropy, this minimum determines also the quantum discord. For classically correlated states and mixtures of a pure state with the maximally mixed state, we show that the minimizing measurement can be determined analytically and is universal , i.e., the same for all concave forms. While these properties no longer hold for general states, we also show that in the special case of the linear entropy, an explicit expression for the associated conditional entropy can be obtained, whose minimum among projective measurements in a general qudit–qubit state can be determined analytically, in terms of the largest eigenvalue of a simple 3 × 3 correlation matrix. Such minimum determines the maximum conditional purity of A , and the associated minimizing measurement is shown to be also universal in the vicinity of maximal mixedness. Results for X states, including typical reduced states of spin pairs in XY chains at weak and strong transverse fields, are also provided and indicate that the measurements minimizing the von Neumann and linear conditional entropies are typically coincident in these states, being determined essentially by the main correlation. They can differ, however, substantially from that minimizing the geometric discord.
Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.
We examine the evaluation of the minimum information loss due to an unread local measurement in mixed states of bipartite systems, for a general entropic form. Such a quantity provides a measure of quantum correlations, reducing for pure states to the generalized entanglement entropy, while in the case of mixed states it vanishes just for classically correlated states with respect to the measured system, as the quantum discord. General stationary conditions are provided, together with their explicit form for general two-qubit states. Closed expressions for the minimum information loss as measured by quadratic and cubic entropies are also derived for general states of two-qubit systems. As an application, we analyze the case of states with maximally mixed marginals, where a general evaluation is provided, as well as XXX states and the mixture of two aligned states.
We revisit the relationship between quantum separability and the sign of the relative q -entropies of composite quantum systems. The q -entropies depend on the density matrix eigenvalues p i through the quantity ω q = ∑ i p q i . Rényi's and Tsallis' measures constitute particular instances of these entropies. We perform a systematic numerical survey of the space of mixed states of two-qubit systems in order to determine, as a function of the degree of mixture, and for different values of the entropic parameter q , the volume in state space occupied by those states characterized by positive values of the relative entropy. Similar calculations are performed for qubit–qutrit systems and for composite systems described by Hilbert spaces of larger dimensionality. We pay particular attention to the limit case q → ∞. Our numerical results indicate that, as the dimensionalities of both subsystems increase, typical mixed states of composite quantum systems tend, as far as their relative q -entropies are concerned, to behave in a classical way.
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.