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arXiv:1909.01343 · Progress of Theoretical and Experimental Physics

Zero-correlation entanglement

Toru Ohira

Published Nov 13, 2019• 1 citation
arXivHTMLPDF

Abstract

Abstract We consider a quantum entangled state for two particles, each particle having two basis states, which includes an entangled pair of spin 1/2 particles. We show that, for any quantum entangled state vectors of such systems, one can always find a pair of observable operators $\mathcal{X}, \mathcal{Y}$ with zero correlations ($\langle{\psi}|\mathcal{X}\mathcal{Y}|{\psi}\rangle - \langle{\psi}|\mathcal{X}|{\psi}\rangle\langle{\psi}|\mathcal{Y}|{\psi}\rangle= 0$). At the same time, if we consider the analogous classical system of a “classically entangled” (statistically non-independent) pair of random variables taking two values, one can never have zero correlations (zero covariance, $E[XY] - E[X]E[Y] = 0$). We provide a general proof to illustrate the different nature of entanglements in classical and quantum theories.

DOI
10.1093/ptep/ptz149

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Abstract We consider a quantum entangled state for two particles, each particle having two basis states, which includes an entangled pair of spin 1/2 particles. We show that, for any quantum entangled state vectors of such systems, one can always find a pair of observable operators X,Y\mathcal{X}, \mathcal{Y}X,Y with zero correlations (⟨ψ∣XY∣ψ⟩−⟨ψ∣X∣ψ⟩⟨ψ∣Y∣ψ⟩=0\langle{\psi}|\mathcal{X}\mathcal{Y}|{\psi}\rangle - \langle{\psi}|\mathcal{X}|{\psi}\rangle\langle{\psi}|\mathcal{Y}|{\psi}\rangle= 0⟨ψ∣XY∣ψ⟩−⟨ψ∣X∣ψ⟩⟨ψ∣Y∣ψ⟩=0). At the same time, if we consider the analogous classical system of a “classically entangled” (statistically non-independent) pair of random variables taking two values, one can never have zero correlations (zero covariance, E[XY]−E[X]E[Y]=0E[XY] - E[X]E[Y] = 0E[XY]−E[X]E[Y]=0). We provide a general proof to illustrate the different nature of entanglements in classical and quantum theories.
It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, |E[XY]−E[X]E[Y]=0E[XY] - E[X]E[Y] = 0E[XY]−E[X]E[Y]=0|⁠), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories.
Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.
Abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum sub-systems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states—sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.
I give a simple argument that demonstrates that the state $\ensuremath{\mid}0⟩\ensuremath{\mid}1⟩+\ensuremath{\mid}1⟩\ensuremath{\mid}0⟩$, with $\ensuremath{\mid}0⟩$ denoting a state with 0 particles (or photons) and $\ensuremath{\mid}1⟩$ a one-particle state, is entangled in spite of recent claims to the contrary. I also discuss viewpoints on the old controversy about whether the above state can be said to display single-particle or single-photon nonlocality.
For any experiment with two entangled photons, some joint measurement outcomes can have zero probability for a precise choice of basis. These perfect anti-correlations would seem to be a purely quantum phenomenon. It is therefore surprising that these very anti-correlations are also evident when the input to the same experiment is analyzed via classical electromagnetic theory. Demonstrating this quantum-classical connection for arbitrary two-photon states (and analyzing why it is successful) motivates alternative perspectives concerning entanglement, the path integral, and other topics in quantum foundations.