scholarly journals Entanglement cost of generalised measurements

2003 ◽  
Vol 3 (5) ◽  
pp. 405-422
Author(s):  
R. Jozsa ◽  
M. Koashi ◽  
N. Linden ◽  
S. Popescu ◽  
S. Presnell ◽  
...  
Keyword(s):  
Quantum Measurements ◽  
Great Circle ◽  
Bloch Sphere ◽  
Numerical Techniques ◽  
Bipartite Entanglement ◽  
Von Neumann ◽  
Tensor Product Space ◽  
Large N ◽  
Dimension 2 ◽  
The Cost

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_{\min}(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any $d$-dimensional POVM by E_{\min} (M)/\log_2 d, we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.

scholarly journals Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements

2014 ◽  
Vol 21 (04) ◽  
pp. 1450010
Author(s):  
Toru Fuda
Keyword(s):  
Quantum Measurements ◽  
Von Neumann Entropy ◽  
Projection Operators ◽  
Mixed States ◽  
Convergence Conditions ◽  
Von Neumann ◽  
Zeno Effect ◽  
The Difference ◽  
Arbitrary Precision ◽  
Continuous Quantum Measurements

By carrying out appropriate continuous quantum measurements with a family of projection operators, a unitary channel can be approximated in an arbitrary precision in the trace norm sense. In particular, the quantum Zeno effect is described as an application. In the case of an infinite dimension, although the von Neumann entropy is not necessarily continuous, the difference of the entropies between the states, as mentioned above, can be made arbitrarily small under some conditions.

SPREADING OF N DIFFUSING SPECIES WITH DEATH AND BIRTH FEATURES

Fractals ◽  
1996 ◽  
Vol 04 (02) ◽  
pp. 161-168 ◽  
Author(s):  
S. HAVLIN ◽  
A. BUNDE ◽  
H. LARRALDE ◽  
Y. LEREAH ◽  
M. MEYER ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Smooth Surface ◽  
Center Of Mass ◽  
Direct Measure ◽  
Dimensional Lattice ◽  
Random Walker ◽  
Random Walkers ◽  
Large N ◽  
Dimension 2 ◽  
The Mean

The number of distinct sites visited by a random walker after t steps is of great interest, as it provides a direct measure of the territory covered by a diffusing particle. We review the analytical solution to the problem of calculating SN(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattice, for d=1, 2, 3 in the limit of large N. There are three distinct time regimes for SN(t). A remarkable transition, for dimension ≥2, in the geometry of the set of visited sites is found. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough. We also review the results for a model for migration and spreading of populations and diseases. The model is based on N diffusing species, where each species has a probability α- of dying (or recovery from a disease) and a probability α+ to give birth (or to infect another species). It is found analytically that when α+ ≈ α- ≠ 0, after a crossover time t× ~ N/2α-, the territory covered by the population is localized around its center of mass while the center of mass diffuses regularly. When α+ > α-, the localization breaks down after a second crossover time and the species diffuse and spread around their center of mass. These results may explain the phenomena of migration and spreading of diseases and population appearing in nature.

scholarly journals ENTANGLEMENT PROPERTIES OF QUANTUM MANY-BODY WAVE FUNCTIONS

2009 ◽  
Vol 23 (20n21) ◽  
pp. 4041-4057
Author(s):  
J. W. CLARK ◽  
A. MANDILARA ◽  
M. L. RISTIG ◽  
K. E. KÜRTEN
Keyword(s):  
Quantum Phase Transitions ◽  
Body Wave ◽  
Wave Functions ◽  
Von Neumann Entropy ◽  
Many Body ◽  
Lattice Systems ◽  
Bipartite Entanglement ◽  
Von Neumann ◽  
Many Body Systems ◽  
The One

The entanglement properties of correlated wave functions commonly employed in theories of strongly correlated many-body systems are studied. The variational treatment of the transverse Ising model within correlated-basis theory is reviewed, and existing calculations of the one- and two-body reduced density matrices are used to evaluate or estimate established measures of bipartite entanglement, including the Von Neumann entropy, the concurrence, and localizable entanglement, for square, cubic, and hypercubic lattice systems. The results discussed in relation to the findings of previous studies that explore the relationship of entanglement behaviors to quantum critical phenomena and quantum phase transitions. It is emphasized that Jastrow-correlated wave functions and their extensions contain multipartite entanglement to all orders.

scholarly journals Measurement-induced randomness and state-merging

2018 ◽  
Vol 16 (02) ◽  
pp. 1850018
Author(s):  
Indranil Chakrabarty ◽  
Abhishek Deshpande ◽  
Sourav Chatterjee
Keyword(s):  
Quantum Information Processing ◽  
Classical System ◽  
Conditional Entropy ◽  
Von Neumann Entropy ◽  
Quantum Mechanical ◽  
Total System ◽  
Joint Entropy ◽  
Von Neumann ◽  
Classical Counterpart ◽  
The Cost

In this work we introduce the randomness which is truly quantum mechanical in nature arising as an act of measurement. For a composite classical system, we have the joint entropy to quantify the randomness present in the total system and that happens to be equal to the sum of the entropy of one subsystem and the conditional entropy of the other subsystem, given we know the first system. The same analogy carries over to the quantum setting by replacing the Shannon entropy by the von Neumann entropy. However, if we replace the conditional von Neumann entropy by the average conditional entropy due to measurement, we find that it is different from the joint entropy of the system. We call this difference Measurement Induced Randomness (MIR) and argue that this is unique of quantum mechanical systems and there is no classical counterpart to this. In other words, the joint von Neumann entropy gives only the total randomness that arises because of the heterogeneity of the mixture and we show that it is not the total randomness that can be generated in the composite system. We generalize this quantity for N-qubit systems and show that it reduces to quantum discord for two-qubit systems. Further, we show that it is exactly equal to the change in the cost quantum state merging that arises because of the measurement. We argue that for quantum information processing tasks like state merging, the change in the cost as a result of discarding prior information can also be viewed as a rise of randomness due to measurement.

scholarly journals Quantum random state generation with predefined entanglement constraint

2014 ◽  
Vol 12 (05) ◽  
pp. 1450030 ◽  
Author(s):  
Anmer Daskin ◽  
Ananth Grama ◽  
Sabre Kais
Keyword(s):  
Error Correction ◽  
Weighted Graph ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Bipartite Entanglement ◽  
Von Neumann ◽  
Communication Algorithms ◽  
Equivalent Quantum ◽  
State Generation ◽  
Random Quantum States

Entanglement plays an important role in quantum communication, algorithms, and error correction. Schmidt coefficients are correlated to the eigenvalues of the reduced density matrix. These eigenvalues are used in von Neumann entropy to quantify the amount of the bipartite entanglement. In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states. We also show that it is possible to adjust the entanglement between subsystems by changing the quantum gates corresponding to the Schmidt coefficients. In this manner, random quantum states with predefined bipartite entanglement amounts can be generated using random Schmidt basis. This provides a technique for generating equivalent quantum states for given weighted graph states, which are very useful in the study of entanglement, quantum computing, and quantum error correction.

On the general quantum theory of measurement

1969 ◽  
Vol 65 (1) ◽  
pp. 111-122 ◽  
Author(s):  
Arthur I. Fine
Keyword(s):  
Quantum Theory ◽  
Theory Of Measurement ◽  
Quantum Measurements ◽  
Initial State ◽  
Von Neumann ◽  
General Quantum ◽  
Von Neumann Measurements ◽  
Quantum Theory Of Measurement

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

scholarly journals Limits to Perception by Quantum Monitoring with Finite Efficiency

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1527
Author(s):  
Luis Pedro García-Pintos ◽  
Adolfo del Campo
Keyword(s):  
Quantum Measurements ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Actual State ◽  
Ising Chain ◽  
Trace Distance ◽  
Gaussian States ◽  
Von Neumann ◽  
Additional Information ◽  
Continuous Quantum Measurements

We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and the actual state of the system. These bounds are expressed solely in terms of the purity and von Neumann entropy of the state assigned by the agent, and are shown to characterize how an agent’s perception of the system is altered by access to additional information. We apply our results to Gaussian states and to the dynamics of a system embedded in an environment illustrated on a quantum Ising chain.

What are Quantum Measurements?

2020 ◽  
pp. 63-73
Author(s):  
Gershon Kurizki ◽  
Goren Gordon
Keyword(s):  
Quantum State ◽  
Physical Reality ◽  
Human Mind ◽  
Quantum Measurements ◽  
Projection Operators ◽  
Von Neumann ◽  
Single Location ◽  
Quantum Darwinism ◽  
Wavefunction Collapse

Chapter 4 introduces a great QM mystery: the notion of quantum measurements. Henry is in a superposition of versions localized in several places, but when Eve measures Henry’s position she (as a classical observer) either sees Henry or she does not. Physical reality is made of such measurements. Eve’s measurement projects or collapses Henry’s superposition state to a single location. The meaning of quantum-state or wavefunction “collapse” and the role of the observer have been at the heart of the historical debate concerning the interpretation of QM. Whereas Von Neumann and Wigner stressed the inseparability of the observed (measured) world from the human mind, alternative “observer-free” views were suggested, such as Everett’s many-world interpretation or Zurek’s quantum Darwinism that replaces the observer by the environment. In the appendix to this chapter the notion of probability amplitudes is elucidated, new notations for operators are introduced and projection operators are presented.

scholarly journals A Cache System Design for CMPs with Built-In Coherence Verification

VLSI Design ◽  
2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Mamata Dalui ◽  
Biplab K. Sikdar
Keyword(s):  
Cellular Automata ◽  
Chip Multiprocessors ◽  
Cache Coherence ◽  
Attractor State ◽  
Von Neumann ◽  
Memory Block ◽  
Cache Line ◽  
The Cost ◽  
Effective Design ◽  
Cache System

This work reports an effective design of cache system for Chip Multiprocessors (CMPs). It introduces built-in logic for verification of cache coherence in CMPs realizing directory based protocol. It is developed around the cellular automata (CA) machine, invented by John von Neumann in the 1950s. A special class of CA referred to as single length cycle 2-attractor cellular automata (TACA) has been planted to detect the inconsistencies in cache line states of processors’ private caches. The TACA module captures coherence status of the CMPs’ cache system and memorizes any inconsistent recording of the cache line states during the processors’ reference to a memory block. Theory has been developed to empower a TACA to analyse the cache state updates and then to settle to an attractor state indicating quick decision on a faulty recording of cache line status. The introduction of segmentation of the CMPs’ processor pool ensures a better efficiency, in determining the inconsistencies, by reducing the number of computation steps in the verification logic. The hardware requirement for the verification logic points to the fact that the overhead of proposed coherence verification module is much lesser than that of the conventional verification units and is insignificant with respect to the cost involved in CMPs’ cache system.

scholarly journals Geometry of color perception. Part 2: perceived colors from real quantum states and Hering’s rebit

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
M. Berthier
Keyword(s):  
State Space ◽  
Color Perception ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Full Detail ◽  
Von Neumann ◽  
Real Dimension ◽  
Quantum Description ◽  
Color Opponency ◽  
Dimension 2

Abstract Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space $\mathcal{P}$ P of perceived colors. We show that $\mathcal{P}$ P is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein’s hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering’s disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

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