scholarly journals Classification of all alternatives to the Born rule in terms of informational properties

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 15 ◽  
Author(s):  
Thomas D. Galley ◽  
Lluis Masanes
Keyword(s):  
Quantum Theory ◽  
Two Dimensional ◽  
Born Rule ◽  
Mixed States ◽  
Structure And Dynamics ◽  
Finite Dimensional ◽  
Information Causality ◽  
Pure States ◽  
Probabilistic Theories

The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.

scholarly journals Any modification of the Born rule leads to a violation of the purification and local tomography principles

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 104 ◽  
Author(s):  
Thomas D. Galley ◽  
Lluis Masanes
Keyword(s):  
Quantum Theory ◽  
Born Rule ◽  
Mixed States ◽  
Many Worlds ◽  
Local Tomography ◽  
Local Measurements ◽  
State Tomography ◽  
Quantum Phenomena ◽  
The Many

Using the existing classification of all alternatives to the measurement postulates of quantum theory we study the properties of bi-partite systems in these alternative theories. We prove that in all these theories the purification principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This allows us to derive the measurement postulates of quantum theory from the structure of pure states and reversible dynamics, and the requirement that the purification principle holds. The violation of the purification principle implies that there is some irreducible classicality in these theories, which appears like an important clue for the problem of deriving the Born rule within the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the no-signalling principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.

Superselection Sectors, Pure States, Bose and Fermi Distributions by Completely Positive Quantum-Motions

2005 ◽  
Vol 12 (01) ◽  
pp. 23-35 ◽  
Author(s):  
Klaus Dietz
Keyword(s):  
Hilbert Spaces ◽  
Mixed States ◽  
Completely Positive ◽  
Absolute Value ◽  
Zero Modes ◽  
Finite Dimensional ◽  
Constants Of Motion ◽  
Pure States ◽  
Damped Oscillator

The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decoherence are constructed, the importance of zero-modes in the absolute value [Formula: see text] of V for the generation of pure states from arbitrary mixed states is illustrated. The universal rôle of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.

scholarly journals LOCAL UNITARY EQUIVALENT CLASSES OF SYMMETRIC N-QUBIT MIXED STATES

2013 ◽  
Vol 11 (08) ◽  
pp. 1350072 ◽  
Author(s):  
SAKINEH ASHOURISHEIKHI ◽  
SWARNAMALA SIRSI
Keyword(s):  
Mixed State ◽  
Classification Scheme ◽  
Ghz State ◽  
Mixed States ◽  
Separable States ◽  
State Classification ◽  
Pure States ◽  
Two Parameters ◽  
Special Case

Majorana representation (MR) of symmetric N-qubit pure states has been used successfully in entanglement classification. Generalization of this has been a long standing open problem due to the difficulties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric N-qubit mixed states based on the geometrical multiaxial representation (MAR) of the density matrix. In addition to the two parameters defined for the entanglement classification of the symmetric pure states based on MR, namely, diversity degree and degeneracy configuration, we show that another parameter called rank needs to be introduced for symmetric mixed state classification. Our scheme of classification is more general as it can be applied to both pure and mixed states. To bring out the similarities/differences between the MR and MAR, N-qubit GHZ state is taken up for a detailed study. We conclude that pure state classification based on MR is not a special case of our classification scheme based on MAR. We also give a recipe to identify the most general symmetric N-qubit pure separable states. The power of our method is demonstrated using several well-known examples of symmetric two-qubit pure and mixed states as well as three-qubit pure states. Classification of uniaxial, biaxial and triaxial symmetric two-qubit mixed states which can be produced in the laboratory is studied in detail.

scholarly journals A Gramian approach to entanglement in bipartite finite dimensional systems: the case of pure states

2020 ◽  
Vol 20 (13&14) ◽  
pp. 1081-1108
Author(s):  
Roman Gielerak ◽  
Marek Sawerwain
Keyword(s):  
Quantitative Measure ◽  
Point Of View ◽  
Gram Matrix ◽  
Matrix Structure ◽  
Matrix Approach ◽  
Mixed States ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Pure States ◽  
Reduced Density

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.

scholarly journals Reconstructing quantum theory from diagrammatic postulates

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 445
Author(s):  
John H. Selby ◽  
Carlo Maria Scandolo ◽  
Bob Coecke
Keyword(s):  
Quantum Theory ◽  
Information Gain ◽  
Quantum Systems ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Process Theories ◽  
Classical Quantum ◽  
The Right ◽  
Theoretic Description ◽  
Probabilistic Theories

A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

scholarly journals Some Remarks on the Entanglement Number

Quanta ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 22-36
Author(s):  
George Androulakis ◽  
Ryan McGaha
Keyword(s):  
Point Of View ◽  
Entanglement Measure ◽  
Mixed States ◽  
Interesting Point ◽  
Finite Dimensional ◽  
Entanglement Measures ◽  
Pure States ◽  
Tree Representations ◽  
Natural Way ◽  
Finite Dimensional Hilbert Space

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

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