No-go theorems for quantum computing give a mathematical proof that quantum dynamic should be linear as well unitary.In this paper we analyze in a detailed way the role of linearity and unitarity for no-cloning theorem; also we introduce a nogo theorem for the square root of Not gate √Not which can never would work in the complete complex Hilbert space.
No-go theorems for quantum computing give a mathematical proof that quantum dynamic should be linear as well unitary.In this paper we analyze in a detailed way the role of linearity and unitarity for no-cloning theorem; also we introduce a nogo theorem for the square root of Not gate √Not which can never would work in the complete complex Hilbert space.
Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C2: the unitary and the anti-unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a xed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the required transformations 0 => 1 and 1 => 0 (i.e., logical quantum Not). The anti-unitary Not is a quantum realization of a gate which acts as a classical Not on any pair of mutually orthogonal vectors, each of which is a potential realization of the classical bits (i.e., universal quantum Not). Although the latter is not completely positive, one can give an approximated unitary realization of the gate by appending an ancilla. Finally, we consider the unitary and the anti-unitary operator realizations of two important genuine quantum gates that transform elements of the computational
basis of C2 into its superpositions: the square root of the identity and the square root of the Not.
The no-cloning theorem, the no-deleting theorem, and other no-go theorems play a crucial role in quantum mechanics and quantum information. In this paper, we show that it is impossible to conjugate or reflect an unknown quantum state or conjugate or reflect the bit-flip (the phase-flip) of an unknown state via a universally unitary evolution. We also indicate that the partial conjugating operations are not anti-unitary and it is impossible to universally partially conjugate an unknown state via a unitary evolution. We compute the fidelities of the optimal physical approximations to conjugating and conjugating the bit-flip and phase-flip operations.
In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states (qubits or quregisters) of a convenient (n-fold tensor product) Hilbert space ⊗nC2 (where C2 is the bidimensional complex Hilbert space). Consequently, quantum gates represent reversible time evolution of the system. According to [1] however, such a representation is unduly restrictive insomuch it does not encompass open systems where coupling with environment and measurement processes may occur. In this case, time evolution is no longer reversible. In [1] the authors formulate a more general model of quantum computational processes, where pure states and unitary operators are replaced by density operators and quantum operations, respectively. Let H be a Hilbert space, L(H) the vector space of all linear operators of H and D(H) the set of density operators in H. A quantum operation [2] is a linear and completely positive operator O: L(H1) → L(H2). By [2], for every quantum operation O: L(H1) → L(H2), there exists a set of operators {Ai} such that iA iAi = I and ∀ρ ∈ D(H1): E(ρ) = iAiρA
The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers R, or complex numbers 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.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.