The book is based on lecture courses on the theory of matrices and its applications that the author has given several times in the course of the last seventeen. ideas, results, and techniques in linear algebra and mainly matrix theory. PDF · Matrix Polynomials and Canonical Forms. Fuzhen Zhang. Pages matrices to be equal zero (the element on the intersection of the row i and column j is the only possible exclusion). Answer: for equality one need ajl = 0 for l = j;.
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Jan 6, PDF | 50+ minutes read | Set theory and Algebraic Structures, Vector Spaces, Matrix Theory, Determinants, Linear Systems and Factorization. Jul 18, to be the standard multiplication of matrices (which is associative). Then The subject of “matrix theory” involves the use of matrices and. There are several popular matrix applications for quantum theory. The book is is an essential tool in matrix theory as well as in operator theory. A typical.
The first thing in mind is then to calculate the Witten index again for the system of N D0-branes. Here we briefly review this procedure, leaving details to the original literature . We will concentrate on the bosonic variables, since introduction of fermionic variables is straightforward.
Thus equations of motion as well as constraints derived from the new action are the same as before. However, one cannot ignore this total derivative term at the quantum mechanical level. In particular, ref. We shall discuss this in the next section. If the similarity transformation operator U behaves in a reasonable way, the new theory obtained is identical to the original one. However, if the similarity transformation is singular, the new theory can be really a different theory.
To see that our similarity transformation is sometimes singular, we consider two cases separately. The exponential in U is cubic in X. A simpler example of this type is a single particle with a single coordinate.
Apparently this new wave function is no longer normalizable.
These examples are quite similar to the case of a charged particle on a circle when a constant gauge field is turned on. In this case the new wave function is not periodic any more, if the original one is. Similarly, the above demonstrations show that the boundary conditions for wave functions are changed under the action of U.
The 2nd Case: Another situation in which the physics is changed by a similarity transformation is when there are further constraints which reduce the physical Hilbert space to a smaller space on which the unitary operator is no longer well defined. In the next section we will show that the noncommutativity of D-brane worldvolume due to constant B field background can be understood in this way.
Before we examine the D-brane case, let us consider a toy model as a warm-up. Obviously the constraint kills some degree of freedoms of X i and the similarity transformation is no longer well defined in the con- strained matrix model.
In the next section, we will see that the constraints effecting matrix model compactification is quite similar in nature to the simple constraint we considered here. It is therefore important to first perform the similarity transformation and then impose the compactification constraints.
This simple example illustrates the same key reason why the similarity transformation 2. It should be clear from this example that this consideration can be applied to orbifolds as well as compactifications. The two cases differ only in 5 the Yang-Mills coupling, whose correct value can be obtained by treating the operation tr properly in each case.
Thus, we shall not distinguish explicitly between the two. Before going over to D0-branes, the U operator 2. If we take the fermion part of constraints 2.
Everything we said so far is classical. In fact, in order for U to be a unitary operator, it is necessary to use the Weyl ordering.
However, Ib can be nonvanishing only in the large N limit and is proportional to the conserved membrane charge. So long as we consider states with the same membrane charge, these terms have no observable effects and hence we will drop them from now on and use 3.
But they cannot always be omitted if we consider membrane processes with charge transfer. The main reason for this separation 3. The directions transverse to the brane will be denoted by X a , X b. These are exactly the commutation relations obtained by quantizing open strings on a D-brane. Here we want to emphasize again that nowhere we have resorted to string theory. It is satisfying to see that the star product in noncommutative gauge theory has a simple origin from matrix model.
Here a puzzle arises.
The reason why this naive 0 procedure is not correct is as follows. The similarity transformation is ill-defined after the constraints are imposed. This is just what happens in the warm-up example in Sec. Instead, if we perform the similarity transformation first and then impose the constraints for compactification as we were doing here in this section , we get the non- commutativity on the D-brane.
In the above we have given discussions in the Hamiltonian formulation. If we want to recover the field A0 , for consistency, it 4 This notation is very natural in noncommutative geometry for the following reason. Different level splittings differentiated by colours and different times differentiated by symbol type are studied.
Indeed, if the coupled qubit has no internal dynamics, the average state, in the p large dimension limit, is triply degenerate.
If the qubit has an internal Hamiltonian, the degeneracy is lifted: a systematic splitting of the eigenvalues and hence a deviation from Werner states happens on a time scale set by the mean level spacing of the environment. We also studied the concurrence of the eigenvector corresponding to the non- degenerate eigenvalue.
It remains very close to one, independently of the concurrences of the other eigenvectors, which fluctuate without a clear pattern.
We find the rather surprising result, that this property of the eigenfunction of the largest eigenvalue persists even in cases where the previous test shows, that we no longer deal with Werner states. Whether this property is characteristic of the average density matrix or whether it is common to many models will have to be analysed in future work. A random matrix theory of decoherence 15 5. Moreover, with quantitative entanglement witnesses , one can bound some of these non-linear quantities using linear observables.
Yet the very concept of a relevant near environment, important for many of the new insights obtained form an RMT treatment, suggests to consider small environments as well. This leads us to discuss the appropriateness of either approach in typical applications. One must recall, that the determination of any of these quantities is not achievable as a single measurement.
Two notes are important at this point: First, the problem discussed here appears for any type of averaging, not only for averaging over time evolutions. Second, for the GUE the ensemble a state average is implicit in the ensemble average, while this is not the case for a GOE , and thus has to be performed separately. This can indeed be the case e. The differentiation between both approaches is also relevant within quantum information tasks.
For the teleportation of an unknown state, however, it is important to know the success rate at each attempt, and thus the quality of the process should result from an average quality for individual density matrices corresponding to each try. At this point, to avoid confusion, we should recall that there is the option of obtaining a mixed density matrix without any entanglement by allowing probabilistic variations in the unitary time evolution of the central system itself in the absence of any significant coupling to the environment.
Clearly we then have an ensemble of pure states, whose average naturally will not be pure. Yet errors due to variations of the unitary A random matrix theory of decoherence 16 evolution are usually considered as a loss of fidelity. The purity of the mixed density matrix would then measure an average quantity closely related to fidelity. This explains our emphasis throughout this paper on the fact, that we need a fixed evolution for the central system.
It also shows, that the separation of external and internal perturbations for the study of the stability of quantum information tasks is somewhat artificial in a practical sense, though it is very useful for theoretical studies. The upshot of this discussion resides in the very fact, that for many quantum information tasks, we need repeated experiments. The decision, which average should be taken, is a subtle one. This emphasizes the importance of the result indicating, that the averages coincide in many cases.
Conclusions We have presented a random matrix theory of decoherence and entanglement. The need for such a theory derives from two facts.
On the one hand random matrix theory is known to provide a good generic description of properties of systems displaying, what is often known as quantum chaos or wave chaos. Indeed this statement is almost tautological as many authors nowadays omit any relation to classical chaos and use the relation to statistical properties of RMT as the definition of the latter. Fidelity decay is the other determining factor for the stability of quantum processes, in particular for those relating to quantum information.
RMT has proven very successful in this field even in describing experiments [14, 15, 16], and comparison with numerics for dynamical spin chains have also given encouraging results [32, 22, 33].
On the other hand the universal regime of exponential decay of coherence usually considered and derived in many ways including RMT [7, 8] is basically founded in the typical situation of very long Heisenberg times in the environment. Yet we argued, that situations with a near environment with fairly low level density and thus short Heisenberg times occur, and will become standard as quantum information systems with ever better isolation from the general environment are developed.
It is under these circumstances, that RMT can provide the generic model to which the behaviour of specific systems should be compared. After describing the RMT model family, proposed to a large extent in earlier work, we proceed to analyse a point, which has set RMT models apart from other models of decoherence. The ensemble of evolution operators creates for the central system an ensemble of density matrices rather than a single density matrix.
Consequently, properties such as entropy, purity and concurrence have so far been calculated as averages over that ensemble. However, we can also compute the average density matrix first, putting us on equal footing with other more conventional models.
The various properties mentioned above are then determined from that single density matrix. This has also the great advantage to produce lower order quantities that have a fair chance to be calculated exactly using super-symmetric techniques.
While we A random matrix theory of decoherence 17 have not yet achieved this goal, we have calculated the average density matrix in linear response approximation. This as well as numerics allowed us to compare purity, von Neumann entropy as well as concurrence of the average density matrix to the average of these quantities over the ensemble of density matrices.
The central finding is, that for large environments at constant Heisenberg time or mean level distance the difference between the two approaches converges to zero as the inverse of the dimension of the relevant Hilbert space of the environment.
This indicates, that for decoherence we can often use the average density matrix, and thus the RMT models are really on the same footing with usual descriptions. Yet we have to note, that, when describing the entanglement between two smaller systems, deviations are important and the question which average the behaviour of a single system should be compared to depends on the particular experimental situation. Consistently, if we specialize to a two-qubit central system, we find that average concurrence and the concurrence of the average density matrix will also approach the same limit for large environments.
We found the more surprising fact, that at least for small decoherence the eigenfunction of the dominant eigenvalue of the average density matrix remains to very good approximation a Bell state, if the initial state was a Bell state. We have thus shown that purity and other quantities measuring entanglement yield the same result in the large environment limit, whether calculated from the average density matrix or as an average over the ensemble of density matrices.
This was done analytically for purity in the linear response regime section 2 and numerically, for purity, von Neumann entropy and concurrence beyond the linear response regime section 4.
Taking into consideration the convexity of this quantities, this might imply that in the large dimension limit of the environment the measure for the density matrices becomes similar to a Dirac delta in the sense that all or a large class of convex functions could be calculated directly from the average density matrix.
However other results  suggest that typical states are far from the average expected state. This apparent contradiction as well as the distribution of the density matrices as such shall be studied in a later paper. This work can also be readily extended considering more realistic RMT ensembles that RMT allows more realistic models then the classical ensembles.
In particular the two-body random ensembles may play an important role particularly in their recent formulation for distinguishable spins . Acknowledgments We are grateful for discussions with J.
Paz, F. Leyvraz, T. Guhr, W. Schleich, T. Prosen, M. Znidaric, R. Blatt, H. Eisert and D. Appendix A.
The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields. He received his Ph. In addition to research papers, he is the author of the book Linear Algebra: Skip to main content Skip to table of contents. Advertisement Hide. Matrix Theory Basic Results and Techniques. Front Matter Pages i-xvii. Elementary Linear Algebra Review.
Pages Partitioned Matrices, Rank, and Eigenvalues.