Maximizing the Hilbert space for a finite number of distinguishable quantum states
Andrew D. Greentree, S. G. Schirmer, F. Green, Lloyd C. L. Hollenberg, A. R. Hamilton, R. G. Clark
备注：
4 pages, 3 figures, submitted for publication
发表信息：
Phys. Rev. Lett. 92, 097901 (2004), also appeared in Virtual Journal of Quantum Information, March 04 issue, and Virtual Journal of Nanoscale Science and Technology, March 15, 04, issue.
We consider a quantum system with a finite number of distinguishable quantum states, which may be partitioned freely by a number of quantum particles, assumed to be maximally entangled. We show that if we partition the system into a number of qudits, then the Hilbert space dimension is maximized when each quantum particle is allowed to represent a qudit of order $e$. We demonstrate that the dimensionality of an entangled system, constrained by the total number quantum states, partitioned so as to maximize the number of qutrits will always exceed the dimensionality of other qudit partitioning. We then show that if we relax the requirement of partitioning the system into qudits, but instead let the particles exist in any given state, that the Hilbert space dimension is greatly increased.
The entanglement between two bosons or fermions can be accessed if there exists an auxiliary degree of freedom which can be used to label and effectively distinguish the two particles. For some types of entanglement between two indistinguishable particles one can observe duality, i.e., if the entanglement is present in the Hilbert space H and an auxiliary Hilbert space H' is used to label the particles, then if we used H as a label the entanglement would be present in H'. For distinguishable particles this effect does not occur because of superselection rules which prevent superpositions of di…
Bennett et al. \cite{BDF+99} identified a set of orthogonal {\em product} states in the $3\otimes 3$ Hilbert space such that reliably distinguishing those states requires non-local quantum operations. While more examples have been found for this counter-intuitive ``nonlocality without entanglement'' phenomenon, a complete and computationally verifiable characterization for all such sets of states remains unknown. In this Letter, we give such a characterization for the $3\otimes 3$ space.
We consider a fixed quantum measurement performed over $n$ identical copies of quantum states. Using a rigorous notion of distinguishability We consider a fixed quantum measurement performed over $n$ identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon's 12th theorem, we show that in the case of a single qubit the number of distinguishable states is $W(α_1,α_2,n)=|α_1-α_2|\sqrt{\frac{2n}{πe}}$, where $(α_1,α_2)$ is the angle interval from which the states are chosen. In the general case of an $N$-dimensional Hilbert space and an area $Ω$ of the dom…
Two-qubit states occupy a large and relatively unexplored Hilbert space. Such states can be succinctly characterized by their degree of entanglement and purity. In this letter we investigate entangled mixed states and present a class of states that have the maximum amount of entanglement for a given linear entropy.
According to classical physics particles are basic building blocks of the world. Classical particles are distinguishable objects, individuated by physical characteristics. By contrast, in quantum mechanics the standard view is that particles of the same kind ("identical particles") are in all circumstances indistinguishable from each other. This indistinguishability doctrine is motivated by the (anti)symmetrization postulates together with the assumption (``factorism'') that each single particle is represented in exactly one factor space in the tensor product Hilbert space of a many-particles …
We develop a framework to systematically investigate the influence of many-particle interference on the dynamics of generic $-$ possibly interacting $-$ bosonic systems. We consider mixtures of bosons which belong to several distinguishable species, allowing us to tune the level of many-particle interference, and identify the corresponding signatures in the time-dependent expectation values of observables. Interference contributions to these expectation values can be classified based on the number of interfering particles. Interactions are shown to generate a series of additional, higher-order…
Quantum correlations between two free spinless dissipative distinguishable particles (interacting with a thermal bath) are studied analytically using the quantum master equation and tools of quantum information. Bath-induced coherence and correlations in an infinite-dimensional Hilbert space are shown. We show that for temperature T > 0 the time-evolution of the reduced density matrix cannot be written as the direct product of two independent particles. We have found a time-scale that characterizes the time when the bath-induced coherence is maximum before being wiped out by dissipation (purit…
The study of quantum systems evolving from initial states to distinguishable, orthogonal final states is important for information processing applications such as quantum computing and quantum metrology. However, for most unitary evolutions and initial states the system does not evolve to an orthogonal quantum state. Here, we ask what proportion of quantum states evolves to nearly orthogonal systems as a function of the dimensionality of the Hilbert space of the system, and numerically study the evolution of quantum states in low-dimensional Hilbert spaces. We find that, as well as the speed o…
Two qubit quantum computations are viewed as two player, strictly competitive games and a game-theoretic measure of optimality of these computations is developed. To this end, the geometry of Hilbert space of quantum computations is used to establish the equivalence of game-theoretic solution concepts of Nash equilibrium and mini-max outcomes in games of this type, and quantum mechanisms are designed for realizing these mini-max outcomes.
A model of quantum noisy channel with input encoding by a classical random vector is described. An equation of optimality is derived to determine a complete set of wave functions describing quantum decodings based on quasi-measurements maximizing the classical amount of transmitted information. A solution of this equation is found for the Gaussian multimode case with input Gaussian distribution. It is described by the overcomplete family of coherent vectors describing an optimal quasimeasurement of the canonical annihilation amplitudes in the output Hilbert space. It is found that the optimal …
Particle distinguishability is a significant challenge for quantum technologies, in particular photonics where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantisation, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum information inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes clear that the HOM experiment is an instance of a (mix…
We consider the question of perfect local distinguishability of mutually orthogonal bipartite quantum states, with the property that every state can be specified by a unitary operator acting on the local Hilbert space of Bob. We show that if the states can be exactly discriminated by one-way LOCC where Alice goes first, then the unitary operators can also be perfectly distinguished by an orthogonal measurement on Bob's Hilbert space. We give examples of sets of N<=d maximally entangled states in $d \otimes d $ for d=4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly fo…
A method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles. The invariants are organized into infinite families, in particular, the generalization to higher dimensional single particle Hilbert spaces is straightforward. Many well-known invariants and their generalizations are also included.
We discuss quantum correlations in systems of indistinguishable particles in relation to entanglement in composite quantum systems consisting of well separated subsystems. Our studies are motivated by recent experiments and theoretical investigations on quantum dots and neutral atoms in microtraps as tools for quantum information processing. We present analogies between distinguishable particles, bosons and fermions in low-dimensional Hilbert spaces. We introduce the notion of Slater rank for pure states of pairs of fermions and bosons in analogy to the Schmidt rank for pairs of distinguishabl…
Supmech, the universal mechanics developed in the previous two papers, accommodates both quantum and classical mechanics as subdisciplines (a brief outline is included for completeness); this feature facilitates, in a supmech based treatment of quantum measurements, an unambiguous treatment of the apparatus as a quantum system approximated well by a classical one. Taking explicitly into consideration the fact that observations on the apparatus are made when it has `settled down after the measurement interaction' and are restricted to macroscopically distinguishable pointer readings, the unwant…
A fundamental resource in any communication and computation task is the amount of information that can be transmitted and processed. Information encoded in a classical system is limited by the dimension d_c of the system, i.e., the number of distinguishable states. A system with d_c=2^n classical states can carry n bits of classical information. Information encoded in a quantum system is limited by the dimension d_q of the Hilbert space of the system, i.e., the number of perfectly distinguishable quantum states. A system with d_q=2^n perfectly distinguishable quantum states can carry n qubits …
The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterised by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other N-m qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3 qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller t…
A subspace of a multipartite Hilbert space is completely entangled if it contains no product states. Such subspaces can be large with a known maximum size, S, approaching the full dimension of the system, D. We show that almost all subspaces with dimension less than or equal to S are completely entangled, and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n does not exceed D-S. This condition holds for almost all sets of states of all multipartite systems, and reveals something surprising. The criterion is identical for se…
Is the most random pattern random? Maximizing localization in a two-dimensional lattice with engineered disorder
Albert-Ludwigs-Universität Freiburg（弗莱堡大学）；National Institute of Information and Communications Technology（信息通信研究机构）；Tokyo University of Science（东京科学大学）
Recently, there is a growing interest in study quantum mechanics from the information geometry perspective, where a quantum state is depicted with a point in the projective Hilbert space. By taking quantum Fisher information (QFI) as the metric of projective Hilbert spaces, estimating a small parameter shift is equivalent to distinguishing neighboring quantum states along a given curve. Henceforth, information geometry plays a significant role in the single parameter estimation. However, the absence of high dimensional measures limits its applications in studying the multi-parameter systems. I…
Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered $n$-tuple of group elements is to be determined by querying elements of the tuple. For any finite group $G$, we give an algorithm to find the product of two elements of $G$ with a single quantum query with probability $2/|G|$. This generalizes Deutsch's Algorithm from $Z_…
We propose an extended quantum theory, in which the number K of parameters necessary to characterize a quantum state behaves as fourth power of the number N of distinguishable states. As the simplex of classical N-point probability distributions can be embedded inside a higher dimensional convex body of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in N dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensional…
For several types of correlations: mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions and non-Gaussian correlations in fermionic systems we estimate the fraction of non-correlated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of non-correlated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence a state randomly chosen from the set of den…
Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d-1)$-photon Fock state. These states can be generated from a classical…
No Reference-Free Generalization in Quantum Machine Learning
Institute for Convergence Research and Education in Advanced Technology（融合研究与先进教育技术研究所）；Yonsei University（延世大学）；Department of Quantum Information（量子信息系）
Quantum mechanics in configuration space in context
School of Physics and Astronomy, University of Leeds（利兹大学物理与天文学院）；Umm Al-Qura University（乌姆古拉大学）；School of Philosophy, Religion and History of Science, University of Leeds（利兹大学哲学、宗教与科学史学院）
Quantum computing experiments are moving into a new realm of increasing size and complexity, with the short-term goal of demonstrating an advantage over classical computers. Boson sampling is a promising platform for such a goal, however, the number of involved single photons was up to five so far, limiting these small-scale implementations to a proof-of-principle stage. Here, we develop solid-state sources of highly efficient, pure and indistinguishable single photons, and 3D integration of ultra-low-loss optical circuits. We perform an experiment with 20 single photons fed into a 60-mode int…
Entanglement plays a central role in numerous fields of quantum science. However, as one departs from the typical "Alice versus Bob" setting into the world of indistinguishable fermions, it is not immediately clear how the concept of entanglement is defined among these identical particles. Our endeavor to recover the notion of subsystems, or mathematically speaking, the tensor product structure of the Hilbert space, lead to two natural pictures of defining fermionic entanglement: the particle picture and the mode picture. In the particle picture, entanglement characterizes the deviation of a f…
We ask to what extent an isolated quantum system can eventually "contract" to be contained within a given Hilbert subspace. We do this by starting with an initial random state, considering the probability that all the particles will be measured in a fixed subspace, and maximizing this probability over all time. This is relevant, for example, in a cosmological context, which may have access to indefinite timescales. We find that when the subspace is much smaller than the entire space, this maximal probability goes to $1/2$ for real initial wave functions, and to $π^2/16$ when the initial wave f…
