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Progress of Theoretical and Experimental Physics

An entanglement analogue in light cones

Fumiaki Morikoshi

Published Dec 1, 2015• 1 citation
Publisher

Abstract

The four-vector formalism is commonly used in Minkowski spacetime, but there exists an alternative formalism with spinors. We revisit the spinor formalism in the context of quantum information and establish an unexpected connection between directions in light cones and the notion of entanglement. This allows us to interpret that timelike directions correspond to entangled states of two imaginary qubits, whereas null directions correspond to separable states. With the correspondence, the speed of a particle can be expressed in terms of the amount of entanglement in concurrence. Furthermore, the entanglement also leads to a novel interpretation of the fact that moving clocks run slowly.

DOI
10.1093/ptep/ptv164

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The four-vector formalism is commonly used in Minkowski spacetime, but there exists an alternative formalism with spinors. We revisit the spinor formalism in the context of quantum information and establish an unexpected connection between directions in light cones and the notion of entanglement. This allows us to interpret that timelike directions correspond to entangled states of two imaginary qubits, whereas null directions correspond to separable states. With the correspondence, the speed of a particle can be expressed in terms of the amount of entanglement in concurrence. Furthermore, the entanglement also leads to a novel interpretation of the fact that moving clocks run slowly.
Formal similarity between Minkowski tetrads and Bell bases allows to think of metric tensors in terms of quantum teleportation protocols. The role of null tetrads for quantum information processing is different. They define qubits resistant to a special kind of noise that occurs if coding and decoding of quantum information is performed in different reference frames. These examples show that mutual links between quantum information and the 2-spinor calculus may be nontrivial and worthy of further studies.
Relying on the mathematical analogy of the pure states of a two-qubit system with four-component Dirac spinors, we provide an alternative consideration of quantum entanglement using the mathematical formulation of Cartanʼs pure spinors. A result of our analysis is that the Cartan equation of a two-qubit state is entanglement sensitive in the same way that the Dirac equation for fermions is mass sensitive. The Cartan equation for unentangled qubits is reduced to a pair of Cartan equations for single qubits as the Dirac equation for massless fermions separates into two Weyl equations. Finally, we establish a correspondence between the separability condition in qubit geometry and the separability condition in spinor geometry.
The Feynman propagator has nonzero values outside of the forward light cone. That does not allow messages to be transmitted faster than the speed of light, but it is shown here that it does allow entanglement and mutual information to be generated at space-like separated points. These effects can be interpreted as being due to the propagation of virtual photons outside of the light cone or as a transfer of pre-existing entanglement from the quantum vacuum. The differences between these two interpretations are discussed.
Relying on a mathematical analogy of the pure states of the two-qubit system of quantum information theory with four-component spinors we introduce the concept of the intrinsic entanglement of spinors. To explore its physical sense we study the entanglement capabilities of the spin representation of (pseudo-) conformal transformations in (3+1)-dimensional Minkowski space-time. We find that only those tensor product structures can sensibly be introduced in spinor space for which a given spinor is not entangled.
This paper focuses on the reason behind the entanglement phenomena, which is still an unknown fundamental question. It argues that while two entangled particles are separated in the 3D space the flow of information between them is instantaneous through the 4D space. The fourth dimension of space has the unit of measurement of length like the other three space dimensions. We shall call it Q, from the word quatre (four) in French. Its magnitude is equal to the invariant proper time τ times the speed of light c. Q = cτ. Each point in the 4D space corresponds to an event. Distances between events in the four-dimensional space can be computed using the Pythagoras theorem. It is argued that time t as measured by clock in our 3D space times the speed of light gives the length of the position vector of an event in the 4D space. At time t, when an observation is made at one of the entangled particles, the length of the position vectors of both entangled particles, sharing the same wave function, in 4D space are equal to t times the speed of light, i.e., ct. The time t is also the same for all the points on the line connecting the two events in the 3D space. Consequently, the length of the position vectors of all the points on the line connecting the two events in the 3D space is equal to ct in the 4D space. The corresponding points in the 4D space make an arc of equidistance to the origin. It is conjectured that the flow of information between the two entangled particles through this arc in the 4D space is not carried by energy or mass particles. It is, therefore, not bound by the time nor the speed of light constraint in 4D space and is instantaneous.
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.