The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.
The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.
We present an inequality on the purity of a bipartite state depending solely on the length difference of the local Bloch vectors. For two qubits this inequality is tight for all marginal states and so extends the previously known solution for the two-qubit marginal problem. With this inequality we construct a three-dimensional Bloch model of the two-qubit quantum state space in terms of Bloch lengths, providing a pleasing visualization of this high-dimensional state space. This allows to characterize quantum states by a strongly reduced set of parameters and to investigate the interplay between local properties of the marginal systems and global properties encoded in the correlations.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
Imaginary numbers play an important role in physics, especially in quantum physics. They are not only a part of describing the dynamic behavior of quantum systems, but also play a key role in the phase and interference phenomena of quantum states. Similar to coherence and entanglement, the imaginarity has been regarded as a quantum resource, further revealing the nonclassical properties of quantum systems and providing new research directions and applications for fields such as quantum computing and quantum communication. In our work, we mainly explore the quantitative relationship between imaginarity and entanglement. For an arbitrary quantum state, there exists a general quantitative relationship between imaginarity and entanglement. For a given class of quantum states, where the closest real states to them are diagonal states, there exists an imaginarity-entanglement inequality. We demonstrate that for these quantum states, any degree of imaginarity with respect to some reference basis can be transformed into entanglement through incoherent real operations. This discovery allows us to define an entanglement-based imaginarity quantifier for these states in quantum systems of arbitrary dimensions and to study their properties. It is shown that for these quantum states, imaginarity and entanglement are operationally equivalent. Our work provides a clear quantitative and operational link between imaginarity and entanglement.
The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space. General remarks General system . At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N 2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N 2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3. Pure state . If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2 N –1 real numbers are needed. Out of these 2 N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).