We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.
We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on n ⊗ n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on 3 ⊗ 3 with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on 3 ⊗ 3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator M 1 ⊗ M 2 → N 1 ⊗ N 2 , with M 1 M 2 = N 1 N 2 . This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022–5).
Needless to emphasize, the home of these quantum states and observables are the linear vector spaces or alternatively the Hilbert spaces. This chapter has crucial recipes relating to operations like inverse of matrices, tridiagonalization, QR decomposition, Gram-Schmidt orthogonalization, a wide spectrum of functions and powers of matrices, trace norm, Hilbert-Schmidt norm, singular value decomposition and so on. The recipies in this chapter can be not only used in quantum information theory but it can also be used as standalone recipies for any person working in linear algebra as well.
The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a [Formula: see text]-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis [Formula: see text] for the operator algebra [Formula: see text] of a Hilbert space [Formula: see text] of finite dimension [Formula: see text] or, after choosing an orthonormal basis for [Formula: see text], for the ⋆-algebra [Formula: see text] of complex matrices of order [Formula: see text]. Illustrations are given for the techniques. It is shown that the Schwinger basis [Formula: see text] of unitary operators can give for [Formula: see text], a product of primes [Formula: see text] and [Formula: see text], the ideal number [Formula: see text] of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all [Formula: see text], though with more overlaps or angles depending on the factorization of [Formula: see text] as a product of primes or their powers like [Formula: see text] with [Formula: see text], all primes, [Formula: see text] for [Formula: see text], or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like [Formula: see text] or [Formula: see text], where [Formula: see text] is the Galois field of size [Formula: see text] and [Formula: see text] is the ring of integers modulo [Formula: see text]. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.