Purity-bounded uncertainty relations in multidimensional space -- generalized purity
Mikalai Karelin
备注：
12 pages, 2 figures. draft version, to appear in J. Phys. A Partially based on a poster "Multidimensional uncertainty relations for states with given generalized purity" presented on X Intl. Conf. on Quantum Optics'2004 (Minsk, Belarus, May 30 -- June 3, 2004) More actual report is to be presented on ICSSUR-2005, Besançon, France and on EQEC'05, Munich. V. 5: amended article after referees' remarks
发表信息：
J. Phys. A: Math. Gen., vol. 38, pp. 6393-6403 (2005)
Uncertainty relations for mixed quantum states (precisely, purity-bounded position-momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in general multi-dimensional case. An expression for family of mixed states at the lower bound of uncertainty relation is obtained. It is shown, that in case of entropy-bounded uncertainty relations, lower-bound state is thermal, and a transition from one-dimensional problem to multi-dimensional one is trivial. Results of numerical calculation of the relation lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.
Bounded uncertainty relations provide the minimum value of the uncertainty assuming some additional information on the state. We derive analytically an uncertainty relation bounded by a pair of constraints, those of purity and Gaussianity. In a limiting case this uncertainty relation reproduces the purity-bounded derived by V I Man'ko and V V Dodonov and the Gaussianity-bounded one [Phys. Rev. A 86, 030102R (2012)].
We experimentally verify uncertainty relations for mixed states in the tomographic representation by measuring the radiation field tomograms, i.e. homodyne distributions. Thermal states of single-mode radiation field are discussed in details as paradigm of mixed quantum state. By considering the connection between generalised uncertainty relations and optical tomograms is seen that the purity of the states can be retrieved by statistical analysis of the homodyne data. The purity parameter assumes a relevant role in quantum information where the effective fidelities of protocols depend critical…
The position-momentum uncertainty relations containing the dependence of their quantum bounds on state purity parameter $μ$ are discussed in context of possibilities to influence on the potential barrier transparency by means of decoherence processes. The behavior of barrier transparency $D$ is shown to satisfy the condition $μ^{-1}\ln D=const$. The particular case of thermal state with temperature $T$ where the purity parameter is a function of temperature is considered. For large temperature the condition for the barrier transparency is shown to be $T\ln D=const$.
Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. They provide optimal lower bounds and imply the multidimensional variance-based uncertainty principle. The article concludes with an open conjecture.
Uncertainty equality for SU(N) observables enabling the experimentally friendly detection of k-inseparability via purity measurements
Università degli Studi di Salerno（萨莱诺大学）；Institute of Nanotechnology of the National Research Council of Italy, CNR-NANOTEC（意大利国家研究委员会纳米技术研究所）；Università del Salento（萨尔伦托大学）；INFN, Sezione di Napoli, Gruppo Collegato di Salerno（意大利国家核物理研究所那不勒斯分部萨莱罗协作组）
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two $n$-uples of canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots p_n)$ in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured varia…
The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, $l_1$ norm of coherence and coherence of formation. In stead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all optical setup, we prepare a family of quantum states whose purity can be fully…
In this work we determine and discuss the entropic uncertainty measures of Shannon type for all the discrete stationary states of the multidimensional harmonic systems directly in terms of the states' hyperquantum numbers, the dimensionality and the oscillator strength. We have found that these entropies have a monotonically increasing behavior when both the dimensionality and the population of the states are increasing
The products of weak values of quantum observables are shown to be of value in deriving quantum uncertainty and complementarity relations, for both weak and strong measurement statistics. First, a 'product representation formula' allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. We show this formula also leads to strong uncertainty relations for unitary operators, and underlies an interpretation of weak values as optimal (complex) estimates of quantum observables. Furthermore, we show that two incompatible obse…
The $D$-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large $D$. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. …
A First-Principles Thermodynamic Uncertainty Relation for Shortcuts to Adiabaticity
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；CONICET–Universidad de Buenos Aires（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（光子科学研究所，巴塞罗那科学技术学院）
How much of the uncertainty in predicting measurement outcomes for non-commuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two non-commuting observables into a classical component, and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits "purity-based" lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a non-trivial inter…
The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydroge…
Tighter entropic uncertainty relations in the presence of quantum memories for complete sets of mutually unbiased bases
Changsha University of Science and Technology（长沙理工大学）；Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Sicence and Technology（长沙理工大学数学建模与工程分析湖南省重点实验室）；Capital Normal University（首都师范大学）；Jining University（济宁学院）
We show how preparation uncertainty relations that are formulated as sums of variances may be tightened by using the quantum Fisher information to quantify quantum fluctuations. We apply this to derive stronger angular momentum uncertainty relations, which in the case of spin-$1/2$ turn into equalities involving the purity. Using an analogy between pure-state decompositions in the Bloch sphere and the moment of inertia of rigid bodies, we identify optimal decompositions that achieve the convex- and concave-roof decomposition of the variance. Finally, we illustrate how these results may be used…
Uncertainty-disturbance relations and applications
University of Science and Technology of China（中国科学技术大学）；Hefei National Research Center for Physical Sciences at the Microscale（合肥微尺度物质科学国家研究中心）；School of Physical Sciences, University of Science and Technology of China（中国科学技术大学物理学院）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit（马普-南锡-新国大-南科大国际联合研究实验室）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）；Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学广东省量子科学与工程重点实验室）；Hefei National Laboratory（合肥国家实验室）
Gaussian distribution of a quantum state with continuous spectrum is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\hat x$ and $\hat p$, quantum entropic uncertainty relation can take a suggestive form, where the standard deviations $σ_x$ and $σ_p$ are featured explicitly. From the construction, it follows in a transparent manner that: (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modifed form, $σ_xσ_p\geq\hbar e^{\cal N}/2$; (ii) the additional factor ${\cal N}$ qua…
The spreading properties of the stationary states of the quantum multidimensional harmonic oscillator are analytically discussed by means of the main dispersion measures (radial expectation values) and the fundamental entropy-like quantities (Fisher information, Shannon and Rényi entropies, disequilibrium) of its quantum probability distribution together with their associated uncertainty relations. They are explicitly given, at times in a closed compact form, by means of the potential parameters (oscillator strength, dimensionality, $D$) and the hyperquantum numbers $(n_r,μ_1,μ_2,\ldots,μ_{D-1…
The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values $\langle r^α\rangle$ and $\left\langle p^α\right\rangle$, respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary $D$-dimensional hydrogenic states in ter…
The uncertainty relation is a fundamental limit in quantum mechanics and is of great importance to quantum information processing as it relates to quantum precision measurement. Due to interactions with the surrounding environment, a quantum system will unavoidably suffer from decoherence. Here, we investigate the dynamic behaviors of the entropic uncertainty relation of an atom-cavity interacting system under a bosonic reservoir during the crossover between Markovian and non-Markovian regimes. Specifically, we explore the dynamic behavior of the entropic uncertainty relation for a pair of inc…
Recently, Xu et al. [Phys. Rev. A 86, 012113(2012)] explored the behavior of the entropic uncertainty relation under the influence of local unital and nonunital noisy channels for a class of Bell-diagonal states. We here reform their results and investigate the entropic uncertainty relation under the influence of unital and nonunital noisy channels with memory. Different types of noisy channels with memory, such as amplitude damping channel(nonunitary), phase-damping and depolarizing channels(unitary) have been taken into account. Some analytical or numerical results are presented. The effect …
Entropic uncertainty is a well-known concept to formulate uncertainty relations for continuous variable quantum systems with finitely many degrees of freedom. Typically, the bounds of such relations scale with the number of oscillator modes, preventing a straight-forward generalization to quantum field theories. In this work, we overcome this difficulty by introducing the notion of a functional relative entropy and show that it has a meaningful field theory limit. We present the first entropic uncertainty relation for a scalar quantum field theory and exemplify its behavior by considering few …
We investigate the connection between quasi-classical (pointer) states and generalized coherent states (GCSs) within an algebraic approach to Markovian quantum systems (including bosons, spins, and fermions). We establish conditions for the GCS set to become most robust by relating the rate of purity loss to an invariant measure of uncertainty derived from quantum Fisher information. We find that, for damped bosonic modes, the stability of canonical coherent states is confirmed in a variety of scenarios, while for systems described by (compact) Lie algebras stringent symmetry constraints must …
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system, but also the entropic uncertainty measures of Rényi type which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in $1/D$ in similar systems with a non-standard dimensiona…
Uncertainty relations are old, yet potentially rewarding to explore. By introducing a quantity called the uncertainty matrix, we provide a link between purity and observable incompatibility, and derive several stronger uncertainty relations in both forward and reverse directions for arbitrary quantum states, i.e., mixed as well as pure, and arbitrary incompatible quantum observables, none of which suffer from the problem of triviality. Besides the tightness, the interpretations of terms in these uncertainty relations may be of independent inter- est. We provide the possible generalization of s…
For configurational space of arbitrary dimension a strict form of the uncertainty principle has been obtained, which takes into account the dependence of inequality limit on the effective number of pure states present in given statistical mixture. It is shown that in a state with minimal uncertainty the density operators eigenfunctions coincide with the stationary wavefunctions of a multidimensional harmonic oscillator. The mixed state obtained has a permutational symmetry which is typical for a system of identical bosons.
The Rényi entropies of Coulomb systems $R_{p}[ρ], 0 < p < \infty$ are logarithms of power functionals of the electron density $ρ(\vec{r})$ which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, its analytical determination is a hard issue not yet solved except for the first lowest-lying energetic states of some specific systems. This is so even for the $D$-dimensional hydrogenic system, which is the main prototype of the multidimensional Coulomb many-body systems. Recently, the Rényi entropies of this system have been found in the two ex…
Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [Phys. Rev. Lett. 107, 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides a new insight into the structure of entangled quantum states. In case of two qubits the app…
