Quantum Mixed State Research Articles

An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

We show that Ω(rd/ϵ) copies of an unknown rank-r, dimension-d quantum mixed state are necessary in order to learn a classical description with 1–ϵ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).

Quantum nonlocal test of “X” state based on geometric interpretation of CHSH inequality

Quantum nonlocal correlation is one of the important features that distinguish the quantum theory from classical theory. As a typical quantum mixed state, the study of quantum nonlocal correlation based on the “X” state is of great importance for the verification of the correctness of quantum theory and the application of quantum information theory. In this work, with the traditional Clauser-Horne-Shimony-Holt (CHSH) inequality testing for quantum nonlocal correlations, we propose a strategy for testing the quantum nonlocal correlations based on the geometric interpretation of the “X” state. By using the geometric interpretation of the “X” state, which is described by the transform of Bloch sphere, it is possible to investigate the optimal selection of measurement settings. The maximum value of CHSH inequalities can also obtained from the physical images. Finally, the range of parameters for a successful quantum nonlocal correlation testing based on the CHSH inequality for the “X” state is studied. The results show that when <inline-formula><tex-math id="M1">\begin{document}$f = 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M1.png"/></alternatives></inline-formula>, the “X” state will be reduced to a normal pure entangled state, and the quantum nonlocal correlation testing results are in full agreement with the traditional ones. This result proves the correctness of the geometric interpretation strategy proposed in this work. When<inline-formula><tex-math id="M2">\begin{document}$f \lt 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M2.png"/></alternatives></inline-formula>, only some of the “X” states can be used for e successfully testing the quantum nonlocal correlations. It is also found that the range of fidelity values that can successfully test the quantum nonlocal correlations will be further increased by increasing the values of <i>r</i>. In particular, when <i>r</i> = 1, the range of fidelity value will reach a largest one (e.g. <inline-formula><tex-math id="M3">\begin{document}$f \gt 0.781$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M3.png"/></alternatives></inline-formula>). The results in this work can provide the reference for experimentally testing the quantum nonlocal correlation by using the “X” state.

Necessary conditions on effective quantum entanglement catalysts

Quantum catalytic transformations play important roles in the transformation of quantum entangled states under local operations and classical communications (LOCC). The key problems in catalytic transformations are the existence and the bounds on the catalytic states. We present the necessary conditions of catalytic states based on a set of points given by the Schmidt coefficients of the entangled source and target states. The lower bounds on the dimensions of the catalytic states are also investigated. Moreover, we give a detailed protocol of quantum mixed state transformation under entanglement-assisted LOCC.

Separability discrimination and decomposition of m -partite quantum mixed states

The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then, employing the $E$-truncated $K$-moment method, we obtain an optimization model for discriminating separability. Moreover, applying semidefinite relaxation method, we get a hierarchy of semidefinite relaxation optimization models and propose an $E$-truncated $K$-moment and semidefinite relaxations (ETKM-SDR) algorithm for detecting the separability of mixed states. The algorithm can also be used for symmetric and non-symmetric decomposition of separable mixed states. By numerical examples, we find that not all symmetric separable states have symmetric decompositions. The algorithm can be used for studying properties of mixed states in the future.

Shadow Tomography of Quantum States

We introduce the problem of shadow tomography: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability ...

Special Section on the Fiftieth Annual ACM Symposium on Theory of Computing (STOC 2018)

This issue of SICOMP contains 10 specially selected papers from the Fiftieth Annual ACM Symposium on Theory of Computing, otherwise known as STOC 2018, held June 25 to 29 in Los Angeles, California...

Testing quantum nonlocality with high probability using quantum mixed state based on hardy-type paradox

Quantum nonlocality is an important phenomenon predicted by quantum mechanics. It is also one of the most important characteristics that quantum theory is different from classical theory. Therefore, it is of great significance to test the quantum nonlocality with higher successful probability. In this paper, a testing logic based on Hardy-type paradox is proposed and its applicability is proved. Such a logic can be used to test the quantum nonlocality for both the quantum mixed state and the quantum pure state with a high successful probability. It is found that, for quantum pure states, the probability of successfully testing the quantum nonlocality first increases and then decreases with the increase of entanglement degree of quantum states. The maximum successful probability of the testing the quantum pure state is over 39%. Furthermore, taking the Werner-like state, a quantum mixed state for example, the high successful probability of testing the quantum nonlocality is investigated by using the proposed logic. It is found that with the increase of the purity of the quantum mixed state, the successful probability of testing the quantum nonlocal correlation will increase. Finally, the conditions and the range of testing quantum nonlocality with high successful probability for Werner states are given. It is found that for r = 0.599997, the Werner-like quantum mixed state has a maximum range (i.e. \begin{document}${\rm Tr}({{\rho}^2}) \geqslant 0.874696$\end{document} ) of successfully testing the quantum nonlocality.

Non-Markovianity of geometrical qudit decoherence

In the following paper, we generalize the geometrical framework of qubit decoherence to higher dimensions. The quantum mixed state is represented by the probability distribution, which is the K\"ahler function on the projective Hilbert space. The Markovian master equation for density operators turns out to be equivalent to the Fokker-Planck equation for quantum probability distributions. Several examples are analyzed, featuring different generalizations of the Pauli channel.

Upper bounds for the Poincaré recurrence time in quantum mixed states

In this paper we use geometric techniques to provide upper bounds for the Poincaré recurrence time of a quantum mixed state with a discrete spectrum of energies. We obtain two types of upper bounds. One of them depends on the uncertainty in the energy or on the average of the gap of energies and extends previous results obtained for pure states. The other upper bound depends only on the number of relevant states. The first upper bound tends to zero at the classical limit, while the other bound is related with the number of relevant states and survives at the classical limit.

Selective Probing of Non-Native Cardiolipin-Bound Conformations of Ferricytochrome C via Ferrocyanide-Mediated Photoreduction

Upon binding to cardiolipin (CL), ferricytochrome c replaces its native Met80 ligand and ultimately gains peroxidase activity, a vital step towards its dissociation from the inner mitochondrial membrane and its subsequent initiation of apoptosis. Multiple binding studies involving ferricytochrome c and CL-containing liposomes have resulted in a variety of partially conflicting binding models, but little quantitative insight into the co-existence of native and partially unfolded conformations of ferricytochrome c has been obtained thus far. While multiple groups have put forth binding models to explain this phenomenon, further validation of these models and characterization of the involved conformational states is still outstanding. Resonance Raman (RR) spectroscopy is perfect for this task due to the high sensitivity of the heme vibrational modes to ligation, oxidation, and spin state changes of the heme iron. By utilizing ferrocyanide-mediated photoreduction, a normally undesired phenomenon that is inhibited for ferricytochrome c conformations lacking Met80 ligation, we selectively probed the non-native states of ferricytochrome c bound to CL-containing liposomes both in the presence and absence of NaCl. The thus elucidated concentrations of these non-native states at different lipid concentrations and ionic strengths were consistent with a previously reported model, where different conformations of the protein were calculated based on a global thermodynamic fit to a proposed [CL] dependent equilibrium between a native-like (prone to photoreduction) and a partially unfolded bound state for which photoreduction is inhibited. RR spectral evidence suggests a dominance of hexacoordinated low spin states even at high lipid content. Only trace evidence for the presence of a high spin state could be inferred from Raman spectra, but the presence of a charge transfer band at 625 nm in the optical spectrum might point to the presence of a quantum-mixed state.

Self-consistent Bogoliubov–de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields

A self-consistent Bogoliubov deGennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc2, characterized by a well-defined Landau--Bloch band structure in the quasi-particle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3, accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.

Probe the Heme Iron Ligand and Conformational Change of Misfolded States of Cytochrome C through EPR Spectroscopy

Cytochrome c is a heme-protein existing in the membrane of mitochrondria which plays a pivotal role in apoptosis. Besides its important biological functions, it is a model system for protein folding and unfolding studies. Recently, Soffer et al. [Soffer et al. Biochemistry 2013, 52, 1397−1408] claimed that a meta-stable misfolded state of horse heart ferricytochreome c could be obtained after exposing the protein to the ferricyanide solution at very alkaline pH for an extended period of time, and it did not undergo any remarkable change in its secondary and tertiary structures upon adjusting the solution to folding promoting conditions at neutral pH. Yet the axial coordination of the heme iron as well as the electronic structures for the misfolded protein remained undetermined. For this purpose, in this work, electron paramagnetic resonance (EPR) spectra of the misfolded protein in the solution at different pH values were recorded, and DFT calculation was conducted to predict the EPR g-tensor at different axial ligand configurations. Our EPR data indicate that the misfolded protein at very alkaline condition exhibits a hexacoordinated low spin state, while at pH 5 the protein adapt the configuration similar to the native-like low-spin ferric heme complex which possesses a histidine/methionine coordination environment. For pH value between 6 and 7, the EPR spectra appear to be the superposition of the two species. With the aid of DFT calculation, also checked by the EPR and other spectroscopic data, the possible hexacoordinated low spin state with a hydroxyl ion as the proximal ligand and the pentacoordinated quantum mixed state of the heme iron for misfolded protein at different folding conditions were discussed. This work was partially supported by Hundred Talents Program of Chinese Academy of Sciences.

Hamiltonian Formulation of Statistical Ensembles and Mixed States of Quantum and Hybrid Systems

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.

ESR and magnetic studies of octahedral [Fe(III)(Cl)(pcd)(H2O)(DMSO)] (pcd = pyridine-2,6-dicarboxylato) compound showing Fe(III) species with different spin states in solution

By direct synthesis has been obtained the octahedral complex [Fe(III)(Cl)(pdc)(H2O)(DMSO)] 1 (pdc=pyridyn-2,6-dicarboxylato), which shows a supramolecular structure and a clear and large molecular distortion. UV–Vis spectrum of 1 shows at 741nm, 836nm (ε=100M−1cm−1) and 970nm (ε=1600M−1cm−1) typical iron d–d transitions. FIR spectrum of 1 shows vibrations υFeO, υFeN and υFeCl. NMR 1H spectroscopy is typical of paramagnetic compounds. The susceptibility data fit to a Hamiltonian, Hex=−2JS1·S2, using two models, give close exchange interaction constant values of: J=−1.803cm−1, g=1.810 and J=−1.102cm−1, g=1.832. X-band ESR spectra of powder sample of 1 at 300K/77K show broad signals at H≈56.70mT/64.53mT, characteristic of ferromagnetic absorptions. X-band spectra of 1 at 77K in methanol solution at different concentrations show typical signals of Fe(III) species with high spin, S=5/2, quantum mixed spin state (qms), 5/2, 3/2, which are present in ESR spectra of peroxidase enzymes, and a low spin S=1/2. 1 has been used as catalyst in the guaiacol test, characteristic of peroxidase enzymes, giving positive by visual observation and by UV–Vis spectroscopy. By ERS spectroscopy has been observed the characteristic π-cationic-radical ferryl, enzyme covalently modified, when the peroxidase enzyme is active. The presence of Fe(III) species with qms observed by ESR spectra of 1, and its catalytic activity, both characteristic of peroxidase enzymes, are a significant conclusion of the work.

The (Not Completely Irreversible) Population of a Misfolded State of Cytochrome c under Folding Conditions

This paper reports the discovery of a (meta)stable partially unfolded state of horse heart ferricytochrome c that was obtained after exposing the protein to a solution with an alkaline pH of 11.5 for 1 week. Thereafter, the protein did not undergo any detectable change in its secondary and tertiary structure upon adjusting the solution to folding promoting conditions at neutral pH. Spectroscopic data suggest that the misfolded protein exhibits a hexacoordinated low-spin state with a hydroxyl ion as the likely ligand. Below pH 6, a new ligation state emerges with the spectroscopic characteristics of a pentacoordinated quantum mixed state of the heme iron. Gel electrophoresis revealed substantial formation of soluble dimers and trimers at submillimolar concentrations, whereas monomers were dominant at lower, micromolar concentrations. Ultraviolet circular dichroism spectra indicate that oxidized monomers are pre-molten globule to globule-like with a substantial fraction of secondary (helical) structure reminiscent of alkaline state V. The oligomers contain even more helical structure, which suggests domain swapping as the underlying mechanism of their formation. A substantial fraction of the submillimolar mixture of monomers and oligomers underwent a reduction of the heme iron. Its dependence on pH suggests the coupling to a proton transfer process. Altogether, our data indicate a partially unfolded ferricytochrome c conformation with spectroscopic characteristics reminiscent of the recently discovered alkaline isomer V(b), which is stabilized under folding conditions by exposing the protein to a very alkaline pH for an extended period of time.

Spin dynamics on photoexcited state of functionality π-radical via quantum-mixed state: Theoretical study of the spin polarized state generation using the mechanism via quantum-mixed state

A theoretical study of the generation of the dynamic spin polarization (DSP) in the phtotoexcited states utilizing the spin dynamics of the functionality π-radical is reported. The functionality π-radical can be regarded as a model compound of “trapped trion” in the molecular level. The spin dynamics from the quantum-mixed (QM) state to the strongly exchange-coupled (SC) pure quartet high-spin state of a functionality π-radical (A–D–R) was investigated theoretically by the stochastic Liouville equations (SLE). A possibility of the dynamic spin polarization utilizing the unique spin dynamics of the triad system is proposed based on the knowledge obtained by the numerical calculation of the spin dynamics and of the magnetic field dependence of the population transfer from the QM states to the SC states.

Misfolded States of Cytochrome C at Native Conditions

Cytochrome c is a highly flexible heme protein known to partially unfold at a pH above 8, generally reversibly. This study reveals the unexpected conformational change that cytochrome c undergoes upon exposure to potassium ferricyanide (∼0.2 μM) at pH 11.5 for an extended period of time. A partially unfolded state was formed which is in equilibrium with an aggregated, soluble state at pH 6 to 8. UV-CD data indicate that the secondary structure remains mostly intact. The heme is hexacoordinated low spin (pH > 6), likely with two axial histidines (His18, and His33). Below pH 6, a new ligation state emerges, for which spectroscopic evidence suggests a pentacoordinated quantum mixed state of the heme iron, previously found only in ferricytochrome c' and in class 3 peroxidases. Our data indicates the population of a frustrated misfolded state, occupied as a result of an alternate folding pathway, along which the glass transition precedes folding. Gel electrophoresis revealed that the protein is predominantly monomeric at low concentration (0.05 mM), while significant amounts of soluble dimers and trimers are formed at higher concentration, where we also observed a substantial fraction of reduced cytochrome c, which we assigned to the monomeric species based on results from size exclusion chromatography. We speculate that cytochrome c might adopt a similar state on the surface of liposomes and on the inner membrane of mitochondria have been found to acquire peroxidase activity, for which this state may be a prerequisite.

Entanglement of a Quantum Field with a Dispersive Medium

In this Letter we study the entanglement of a quantum radiation field interacting with a dielectric medium. In particular, we describe the quantum mixed state of a field interacting with a dielectric through plasma and Drude models and show that these generate very different entanglement behavior, as manifested in the entanglement entropy of the field. We also present a formula for a "Casimir" entanglement entropy, i.e., the distance dependence of the field entropy. Finally, we study a toy model of the interaction between two plates. In this model, the field entanglement entropy is divergent; however, as in the Casimir effect, its distance-dependent part is finite, and the field matter entanglement is reduced when the objects are far.

Theoretical study of dynamic electron-spin-polarization via the doublet-quartet quantum-mixed state (II). Population transfer and magnetic field dependence of the spin polarization

The population transfer to the spin-sublevels of the unique quartet (S = 3/2) high-spin state of the strongly exchange-coupled (SC) radical-triplet pair (for example, an Acceptor-Donor-Radical triad (A-D-R)) via a doublet-quartet quantum-mixed (QM) state is theoretically investigated by a stochastic Liouville equation. In this work, we have treated the loss of the quantum coherence (de-coherence) due to the de-phasing during the population transfer and neglected the effect of other de-coherence mechanisms. The dependences on the magnitude of the exchange coupling or the fine-structure parameter of the QM state are investigated. The dependence on the velocity of the population transfer (by the electron transfer or the energy-transfer) from the QM state to the SC quartet state is also clarified. It is revealed that the de-coherence during the population transfer mainly originates from the fine-structure term of the QM state in the doublet-triplet exchange coupled systems. This de-coherence leads to the unique dynamic electron polarization (DEP) on the high-field spin sublevels of the SC state, which is similar to the unique DEP pattern of the photo-excited triplet states of the reaction centers of photosystems I and II. The magnetic field dependence of the population transfer leading to the populations of the spin-sublevels of the SC states is also calculated. The possibility of the control of energy transport, spin transport and information technology by using the QM state is discussed based on these results. The knowledge obtained in this work is useful in the spin dynamics of any doublet-triplet exchange coupled systems.

Theoretical study of dynamic electron-spin-polarization via the doublet-quartet quantum-mixed state and time-resolved ESR spectra of the quartet high-spin state

The mechanism of the unique dynamic electron polarization of the quartet (S = 3/2) high-spin state via a doublet-quartet quantum-mixed state and detail theoretical calculations of the population transfer are reported. By the photo-induced electron transfer, the quantum-mixed charge-separate state is generated in acceptor-donor-radical triad (A-D-R). This mechanism explains well the unique dynamic electron polarization of the quartet state of A-D-R. The generation of the selectively populated quantum-mixed state and its transfer to the strongly coupled pure quartet and doublet states have been treated both by a perturbation approach and by exact numerical calculations. The analytical solutions show that generation of the quantum-mixed states with the selective populations after de-coherence and/or accompanying the (complete) dephasing during the charge-recombination are essential for the unique dynamic electron polarization. Thus, the elimination of the quantum coherence (loss of the quantum information) is the key process for the population transfer from the quantum-mixed state to the quartet state. The generation of high-field polarization on the strongly coupled quartet state by the charge-recombination process can be explained by a polarization transfer from the quantum-mixed charge-separate state. Typical time-resolved ESR patterns of the quantum-mixed state and of the strongly coupled quartet state are simulated based on the generation mechanism of the dynamic electron polarization. The dependence of the spectral pattern of the quartet high-spin state has been clarified for the fine-structure tensor and the exchange interaction of the quantum-mixed state. The spectral pattern of the quartet state is not sensitive towards the fine-structure tensor of the quantum-mixed state, because this tensor contributes only as a perturbation in the population transfer to the spin-sublevels of the quartet state. Based on the stochastic Liouville equation, it is also discussed why the selective population in the quantum-mixed state is generated for the "finite field" spin-sublevels. The numerical calculations of the elimination of the quantum coherence (de-coherence and/or dephasing) are demonstrated. A new possibility of the enhanced intersystem crossing pathway in solution is also proposed.