A concept of the generalized quantum measurement is introduced as the transformation, which establishes a correspondence between the initial states of the object system and final states of the object--measuring device (meter) system with the help of a classical informational index, unambiguously linked to the classically compatible set of states of the object--meter system. It is shown that the generalized measurement covers all the key known quantum measurement concepts--standard projective, entangling, fuzzy and the generalized measurement with the partial or complete destruction of the initial information contained in the object. A special class of partially-destructive measurements that map the continual set of the states in finite-dimensional quantum systems to that one of the infinite-dimensional quantum systems is considered. Their informational essence and some information characteristics are discussed in detail.
We provide explicit quantum circuits for the non-destructive deterministic discrimination of Bell states in the Hilbert space $C^{d^{n}}$, where $d$ is qudit dimension. We discuss a method for generalizing this to non-destructive measurements on any set of orthogonal states distributed among $n$ parties. From the practical viewpoint, we show that such non-destructive measurements can help lower quantum communication complexity under certain conditions.
It has become common practice to model large spin ensembles as an effective pseudospin with total angular momentum J = N x j, where j is the spin per particle. Such approaches (at least implicitly) restrict the quantum state of the ensemble to the so-called symmetric Hilbert space. Here, we argue that symmetric states are not generally well-preserved under the type of decoherence typical of experiments involving large clouds of atoms or ions. In particular, symmetric states are rapidly degraded under models of decoherence that act identically but locally on the different members of the ensembl…
In this contribution we group the operator basis for d^2 dimensional Hilbert space in a way that enables us to relate bases of entangled states with single particle mutually unbiased state bases (MUB), each in dimensionality d. We utilize these sets of operators to show that an arbitrary density matrix for this d^2 dimensional Hilbert space system is analyzed by via d^2+d+1 measurements, d^2-d of which involve those entangled states that we associate with MUB of the d-dimensional single particle constituents. The number $d^2+d+1$ lies in the middle of the number of measurements needed for bipa…
We show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of a "momentum-like" variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an "…
We show how to reconstruct a process theory of local systems starting from a global theory of reversible processes on a single global system, by using the purification principle. In such a process theory, local systems are not given, but rather `emerge' as the global system is decomposed into subsystems. Local systems thus have specific identities and their composition is naturally limited by structural constraints, a behaviour which we formalise by defining symmetric partially-monoidal categories. We reconstruct quantum theory from the global theories of unitary groups acting on projective Hi…
A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial $\mathit{IN}$ attributes and $\mathit{OUT}$ class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from $\mathit{IN}$ Hilbert space into $\mathit{OUT}$ Hilbert space is then built from an optimization problem of transferring maximal possible probability from $\mathit{IN}$ to $\mathit{OUT}$, this leads to the formulation of a new algebraic problem. Constructed Knowle…
This paper addresses the problem of how much information we can extract without disturbing a statistical experiment, which is a family of partially known normal states on a von Neumann algebra. We define the classical part of a statistical experiment as the restriction of the equivalent minimal sufficient statistical experiment to the center of the outcome space, which, in the case of density operators on a Hilbert space, corresponds to the classical probability distributions appearing in the maximal decomposition by Koashi and Imoto [Phys.~Rev.~A $\textbf{66}$, 022318 (2002)]. We show that we…
We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a re…
We provide exact analytical solutions for a two dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis-Riesenfeld method of invariants. We conclude that the latter…
Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of states. We present a principle that gives a definite distinction between the operations that preserve the states of the system and those that disturb the states. The principle is derived by alternately applying a fundamental property of classical signals and a fundamental property of quantum ones. The principle can be cast into a simple form by using a decomposition of the relevant Hilbert space, which is uniquely determined by the set of possible states. The decomposition implies the cl…
Motivated by previous works on a Floquet version of the PXP model [Mukherjee {\it et al.} Phys. Rev. B 102, 075123 (2020), Mukherjee {\it et al.} Phys. Rev. B 101, 245107 (2020)], we study a one-dimensional spin-$1/2$ lattice model with three-spin interactions in the same constrained Hilbert space (where all configurations with two adjacent $S^z=\uparrow$ spins are excluded). We show that this model possesses an extensive fragmentation of the Hilbert space which leads to a breakdown of thermalization upon unitary evolution starting from a large class of simple initial states. Despite the non-i…
We formulate a conclusive teleportation protocol for a system in d-dimensional Hilbert space utilizing the positive operator valued measurement at the sending station. The conclusive teleportation protocol ensures some perfect teleportation events when the channel is only partially entangled, at the expense of lowering the overall average fidelity. We find the change of the fidelity as optimizing the conclusive teleportation events and discuss how much information remains in the inconclusive parts of the teleportation.
The axion particle, a consequence of an elegant hypothesis that resolves the strong-CP problem of quantum chromodynamics, is a plausible origin for cosmological dark matter. In searches for axionic dark matter that detect the conversion of axions to microwave photons, the quantum noise associated with microwave vacuum fluctuations will soon limit the rate at which parameter space is searched. Here we show that this noise can be partially overcome either by squeezing the quantum vacuum using recently developed Josephson parametric devices, or by using superconducting qubits to count microwave p…
The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a general quantum channel, in each timestep followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unit…
We significantly advance the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 =(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinant…
It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a para…
We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P(alpha) is put forth, yielding for generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and colleagues by applying Zeilberger's algorithm ("creative telescoping") t…
Fate of emergent Hilbert space fragmentation and information insulation upon softening of hard dual constraints at PXP-chain junctions
Department of Physics, School of Mathematical Sciences Ramakrishna Mission Vivekananda Educational and Research Institute（拉马克里希纳使命维韦卡南达教育与研究机构数学科学学院物理系）
We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy. That is: 1. A finitary probability space Δ^n (=all probability measures on {1,...,n}) can be fully and faithfully represented by the pair consisting of the abstraction D^n (=the object up to isomorphism) of a partially ordered set (Δ^n,\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via a systematic purel…
