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A non-expert introduction to the fields of quantum entanglement, quantum interference, and quantum computing. Topics include the concept of entangled quantum states, tensor product of Hilbert spaces and operators, quantum interference experiments, and the role of quantum mechanics in computing and communication. The document also discusses the historical context and significance of these phenomena.
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NATIONALLABORATORY
Report No..- ORNLITM-2000/
MANAGED BY UT-BAll-ELLE FOR THE DEPARTMENT OF ENERGY
ENTANGLEMENT AND QUANTUM COMPUTATION:
AN OVERVIEW
R. B. Perez
April 2000
Prepared by the Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 managed by UT-Battelle, LLC (^). for U.S. Department of Energy under contract DE-AC05000R
UT-BATTELLE ORNL-27* (4-00)
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ACKNOWLEDGMENT.. ............................................................................................................... (^)
Y ABSTRACT.. (^) ................................................................................. .............................................. (^) c ii
I INTRODUCTION ......................................................... .......................................................... (^) I
BRIEF REVIEW OF QUANTUM MECHANICS FUNDAMENTALS. ........... ... ... ....._..t...
HILBERT SPACE.. ............. .............................................................................................^7 OPERATORS ................................................................................................................. .; 2.2.1 Linear Operators.. ................................................................................................. 2.2.2 Inverse Operators .................................................................................................. (^3) 2.2.3 Adjoint Operators. ................................................................................................. q 2.2.4 Hermitian (Self Adjoint) Operators.. .................................................................... 2.2.5 Unitary Operators ..................................................................................................^7 EIGENVALUE EQUATIONS ........................................................................................ 4 GROUP THEORY (FOR POETS). ................................................................................. 4 TRANSITION OPERATORS ........................................................................................ (^5) GENERATING OPERATORS OF SU(N). ..................................................................... PURE AND MIXED STATES.. ...................................................................................... THE DENSITY OPERATOR ......................................................................................... THE LIOUVILLE EQUATION ......................................................................................
The authors are thankful to Dr. V. Protopopescu for many suggestions and criticisms as ~vell as for his careful reading of the manuscript. We are indebted to Sherry Abercrombie for her patience and diligence in typing the many equations and text of this report.
V
.’ (^).
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The quantum mechanical state of a system is said to be entangled when it arises from the superposition of states of identifiable correlated subsystems that can not be factorized. Entanglement is a pure quantum mechanical feature without a classical analogue.
In 1999 one of us, J. T. Mihalczo, suggested the use of nuclear reactions to produce entangled massive particles as an alternative to the already existing sources of entangled photons. Besides the High Flux Reactor, ORNL will be the site of the Spallation Source. both facilities providing the availability of the needed intense neutron sources able to induce a variety of nuclear reactions as a source of massive particles. (^) It then appeared of some interest to know more about the fields of entanglement, quantum logic. and their possible future applications. Motivated by these ideas, it was decided to compile information on those fields, mostly to educate ourselves, in an area of physics showing counterintuitive “spooky” behavior. This report, like Gallia, is divided into three parts.
(a) Compilation of the quantum mechanical formalism needed to understand entanglement and non-locality in quantitative terms (the casual reader may skip most of this part). Most of this analysis has been based on the SU(n) formalism as expounded in Reference
(b) Sections 8 up to 10 deal with the subject of quantum interference and entanglement of massive particles. It also includes discussions on the Einstein-Podolski-Rosen paradox. Bell’s inequalities, and the problem of measurement in quantum mechanics. A brief review of some interference measurements using massive particles is included as well.
(c) Due to the fact that the mathematical and physical background dealt with in parts (a) and (b) of the report is equally applicable to the fields of quantum logic and quantum information processing, the last part of this overview intends to provide a general description of the subjects of quantum cryptography, teleportation of quantum states. and quantum computers. The urgency for ORNL to participate in what is now a worldwide project on quantum logic is discussed in Section 1.5.
Clearly, this is not an in-depth review of the above-mentioned fields; it is rather an introduction for non-experts in quantum logic that, like ourselves, acquired some interested curiosity. As an aid, a relatively extensive list of references is provided.
i
This section contains a brief review of the quantum mechanical tools used in the following sections of this report. The fundamental ideas have been extracted from references 1. 2. and
2.1 HILBERT SPACE
A Hilbert space is a complete normed space H in which the norm is given by a scalar product. A generic element of H, gi, will be represented by the Dirac, ket. I gi >. and its complex conjugated by the bra notations < gi 1. then the scalar product is written as < gi j g, > = II gi l12.
A rule A that transforms a vector, gt E HI, to another vector, g2 E HZ,
2.2.1 Linear Operators
~(aIg,+azgZ)=al~g,+a2~g,
2.2.2 Inverse Operators
Inverse operators, A-‘, defined by
2.2.3. Adjoint Operators
5 2.2.4.^ Hermitian^ (Self Adjoint)^ Operators
(2.1)
( i = unit operator ) (2.3)
2.2.5 Unitary Operators
m 2.3 EIGENVALUE EQUATIONS
The eigenvalue equation for the operator A is
A / g, > = a, I g, > (^) (2.7)
where, a, , are the eigenvalues and jgi > the eigenvectors. Orthonormality is represented by the inner product
-Gig,>=&, (2.8)
A set M of elements is called a multiplicative group when:
(a) given a pair of elements, A, B E M, the product A B E M (AB f BA);
(b) (AB) C = A (BC);
a (2.9)
(c) There is a unit element, 1 E M, such that 1A = Al = A;
(d) Given A E M, there is an inverse element A“ E M such that AA-’ = A-IA = 1. (2.10)
In the case that condition (d) is not fulfilled, the membership is called a semigroup. The group of operators, U, (with elements Ui j) that transforms n-dimensional vectors,
v = (VI, V2’ ... v,,^ ) , according^ to
j=l
is called the unitary group, U(n), if
UU’ =U+U=l(i.e.^2 UijUij=C^ UI,,U,k=CYik)*^ (2.12) j=l (^) .I
Hence, /det (U) 1= 1; if det (U) = 1, the unitary group is called a special unitary group, SU(n).
In view of the condition, det U(n) = 1. and the hermiticity of A, there are s = nz - 1
independent generating operators. R, (j = 1, 2.. s) and the rank of the group. r, is equal to
(n - 1).
The generating operators, i, (j = 1, 2 ,.. .s) are Hermitian operators defined by the relations
[i,.i,l=2ik f,,k /ik k=l
The quantities fijk, are the so-called structure constants. The A - operators obey the trace relations:
T,{ i;}=o; Tr&ik:.=2’,k- (3 33)-.--
The SU(n) generators are expressed in terms of transition operators:
with
~,jk = fi,, + 6, ; Gjk = i ( i)jk - 6kj) (2.24)
a^2 we=- (^) i--!(!+l) (i?, + ... e,, -^1 Fg+,, !+I 1
(l<j<k<n; l<e<n-1).
A given Herrnitian operator, A, acting on an n-dimensional Hilbert space,,can be represented in terms of the SU(n) generators:
; Aj=T,CAi,). (2.26)
An example is worked out in Appendix A. . In particular, because the ihre o^ p^ erators together^ with^ the unit operator^ form a complete^ set of commuting operators, one can rewrite (2.26) in the form
“=i A, i+; 2 At $; A,,=T,{A}; AE=T,{Awp). (2.27) P+l
The eigenvalue equations associated with the iv, operators are
s (^) $1 v>=;i^ [(b’)/^ v >^ (l<v<n, lIP<n-1) (^) (2.28)
($+I) = e 2 we I !(!+I)^1
. 7 v=e+1 (2.30)
we(“)=O^ ;^ e+l^ <v^ In.^ (2.3 1)
When a quantum system can be described by a single state vector, /w>, given in terms of an orthonormal basis by
/Y>=C c, In> n
the corresponding state is called a “pure” state.
In the presence of fluctuations, due for instance to interaction with the environment, the system may find itself in various state 1~” > occurring with probability, pV, in this instance, the state of the system is called a mixed state.
For a pure state the density operator is defined as
(^3) It coincides with the projection operator on the state lY>.
i Its matrix^ elements,^ pi,^ define the density^ matrix.^ From (2.33) and (2.32)
p,jk=‘jIb/k>=
The real valued expectation values. A,. form the so-called coherence vector A, (A, ; j = l...s).
Upon operating on (2.33) with, i A $ , and after use is made of the Schrodinger equation:
d A i ft - / Y > = H / Y > , satisfied by the state vector 1 Y > and its complex conjugate one dt obtains the quantum mechanical version of the Liouville 0 equation
where A is the Hamiltonian operator.
3 For an isolated physical system, the Liouville equation is the mathematical representation of the quantum mechanical principle of the deterministic evolution of the system; i.e., if at t = c,,
. the density^ operator^ is p(tO) at a later time, one has p (t) given by
/G(t)=tJ(t - t,,)b(t,,>~-‘(t - to) (2.44)
where U(t - to) is a unitary operator. Note that the process described by Eq. (2.44) is reversible and as such it does not change the system’s entropy.
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