The Lorentz metric represented by the diagonal matrix G = diag(1,-1,-1,-1) acts on Minkowski space-time quadrivectors. In the language of Quantum Information the operator G can be viewed as an entangling gate this because it acts in a similar way as the Controlled-Z gate on the computational basis of a 2-qubit separable quantum vector. The entangling power corresponds to the fact that the resulting vector, considered as a 2-qubit vector, cannot be put into a Kronecker product of two 1-qubit vectors.One can represent the generators of the Lorentz group by 4x4 matrices. An example is given by the Lorentz rotation considered as a controlled 2-qubit gate structure similar to the Control-NOT logical quantum gate.The Minkowski metric, according to the positive energy theorem in General Relativity, represents the ground state of the metric field, its maximally entangled character analogous to the singlet Bell state suggests that entanglement and hyperbolic space are intimately interconnected.Quantum Information associated with the logic linear algebraic structures, as proposed in Eigenlogic, could provide a new tool to revisit the geometric structures (rotational and hyperbolic) of the Lorentz group linking Quantum Mechanics and Relativity Theory.
The Lorentz metric represented by the diagonal matrix G = diag(1,-1,-1,-1) acts on Minkowski space-time quadrivectors. In the language of Quantum Information the operator G can be viewed as an entangling gate this because it acts in a similar way as the Controlled-Z gate on the computational basis of a 2-qubit separable quantum vector. The entangling power corresponds to the fact that the resulting vector, considered as a 2-qubit vector, cannot be put into a Kronecker product of two 1-qubit vectors.One can represent the generators of the Lorentz group by 4x4 matrices. An example is given by the Lorentz rotation considered as a controlled 2-qubit gate structure similar to the Control-NOT logical quantum gate.The Minkowski metric, according to the positive energy theorem in General Relativity, represents the ground state of the metric field, its maximally entangled character analogous to the singlet Bell state suggests that entanglement and hyperbolic space are intimately interconnected.Quantum Information associated with the logic linear algebraic structures, as proposed in Eigenlogic, could provide a new tool to revisit the geometric structures (rotational and hyperbolic) of the Lorentz group linking Quantum Mechanics and Relativity Theory.
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.
A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group SO+(1,3) on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the 'W-matrix' is an SL(2,C)⊗n invariant scalar. Consequently, the linear n-partite quantum mutual information is shown to be an SL(2,C)⊗n invariant for all n-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group SO(1,3). In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.
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.
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.
This article discusses the intimate relationship between quantum mechanics, information theory, and relativity theory. Taken together these are the foundations of present-day theoretical physics, and their interrelationship is an essential part of the theory. The acquisition of information from a quantum system by an observer occurs at the interface of classical and quantum physics. The authors review the essential tools needed to describe this interface, i.e., Kraus matrices and positive-operator-valued measures. They then discuss how special relativity imposes severe restrictions on the transfer of information between distant systems and the implications of the fact that quantum entropy is not a Lorentz-covariant concept. This leads to a discussion of how it comes about that Lorentz transformations of reduced density matrices for entangled systems may not be completely positive maps. Quantum field theory is, of course, necessary for a consistent description of interactions. Its structure implies a fundamental tradeoff between detector reliability and localizability. Moreover, general relativity produces new and counterintuitive effects, particularly when black holes (or, more generally, event horizons) are involved. In this more general context the authors discuss how most of the current concepts in quantum information theory may require a reassessment.
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of S7 over the quaternionic projective space HP1≃S4 with an SU(2)≃S3 fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on HP1 between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.