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The recent paper entitled by K. C. Lee et al. (2011) establishes nonlocal macroscopic quantum correlations, which they term “entanglement”, under ambient conditions. Photon(s)-phonon entanglements are established within each interferometer arm. However, our analysis demonstrates, the phonon fields between arms become correlated as a result of single-photon wavepacket path indistinguishability, not true nonlocal entanglement. We also note that a coherence expansion (as opposed to decoherence) resulted from local entanglement which was not recognized. It occurred from nearly identical Raman scattering in each arm (importantly not meeting the Born and Markovian approximations). The ability to establish nonlocal macroscopic quantum correlations through path indistinguishability rather than entanglement offers the opportunity to greatly expand quantum macroscopic theory and application, even though it was not true nonlocal entanglement.

The ability to observe and control nonlocal macroscopic quantum coherence/correlations, under ambient conditions, would likely have a powerful influence across a wide range of fields. This was achieved recently by Lee et al., in

The work in the Lee et al. paper is essentially a two-arm extension of the DLCZ (Duan, Lukin, Cirac, and Zoller) experiments [

This diagram is a simplified version of the interferometer used in the Lee et al. experiments. Components have been removed which are needed for practical application but not for understanding the physical principles.

Our analysis is that Lee’s explanation, in the

Second, the pump photon/diamond interactions do not (and must not) meet the Born (system-environment coupling weak) or Markovian (memory effects of the environment are negligible) approximations of decoherence theory [

In the next several paragraphs, the topics addressed will be as follows. First, nonlocal correlations will be examined, which can be represented by entangled states or states generated by indistinguishable paths. Second, we review the general definition of entanglement demonstrating why the nonlocal phonon field correlations in the Lee study are not accurately described as being entangled. Third, we discuss that path indistinguishability and the quantum correlations that can be generated. This and the previous paragraphs draw heavily from the work by pioneers that include von Neumman, Mandel, and Shih, as well as insights from recent decoherence theory by Zurcek and Zeh [

In order to discuss quantum correlations, including entangled states and those from path indistinguishability, density operators and their nonseparability will be discussed. The density operator is a Hermitian operator acting on Hilbert space with nonnegative eigenvalues whose sum is 1 (it is not a classical statistical operator). It should not be confused with a classical statistical matrix and it has its greatest value in calculating expectation values of physical properties [

Described more formally below, a state describing a pair of nonlocal quantum correlated entities (photons or phonons) has an unfactorizable density operator for the pair that progresses forward in time via linear unitary operators. But in performing the trace operation to obtain the subsystems (e.g., a given diamond phonon field), these subsystems are represented by reduced density operators that move forward in time, unlike the true principal, via nonlinear unitary operators (i.e., the trace gives information on the subsystem statistical averages but is not the complete description of the subsystems) [

Entanglement, a type of quantum correlation, is a function of superposition and the linearity of Schrödinger’s equation, but not generally path indistinguishability (which will be dealt with in a subsequent section) [

This is a form which would be used to describe decoherence (or a one-arm Lee experiment) where the principal is given by the wavefunction (

Equation (

As noted, in addition to the Lee study being an extension of the DLCZ experiments, it is analogous to the pioneering experiments by Brune, entangling atoms with fields (and then a second atom) [

For a more formal description of entanglement and its subsystems, we will provide the mathematical framework for one EPR-B particle state. There are two observers of these particles,

The density operator product is nonfactorizable. If we examine a subsystem, it is an inseparable state as the trace operation of each observer (here, observer

A reduced density operator is generated by the trace operation representing an improper mixed state, losing information about coherences. It is an expectation value. To paraphrase Schrödinger, the best possible knowledge of a whole does not include the best possible knowledge of its parts (if that knowledge is even available) [

Path indistinguishability can lead to nonlocal macroscopic correlations but generally not entanglement. A more complete discussion of coherence and indistinguishability can be found in the pioneering work of Mandel [

We begin looking at path indistinguishability for a single photon entering a beam splitter with the two arms as exit ports (essentially equivalent to the pump photon in the Lee paper). All first-order interference is a single-photon wavepacket interference (as per Dirac), no matter what the intensity, along indistinguishable paths. Second-order correlations are generally the interference of biphoton wavepackets and are reviewed elsewhere [

Here the subscripts 1 and 2 are the two paths and the value in the ket represents occupation number. The alpha and beta terms take into account beam splitter ratios. Note that this is the form of (

Returning to (

The first two terms, the diagonal terms, are the DC terms that reduce fringe visibility to a maximum of 50% unless they can be removed (for true entanglement, there are no DC terms and maximum visibility is 100%). When paths are distinguishable, these are the only nonzero terms. The third and fourth terms represent indistinguishable paths and generate interference (h.c. is the Hermitian conjugate or adjoint) (see Figure 3 in the Lee paper, as off-diagonal elements are not exclusive to entanglement as suggested). These off-diagonal elements are complex. It is important to note that the density operator is inseparable only within the constraints of path indistinguishable (e.g., wavepacket width, detector time, path lengths, etc.). Coherence time is an example. For an optical pulse, delay times must be within the coherence time. In contrast, for most entangled states, coherence time is not an issue except when demonstrating interference.

Young’s interferometer is useful for illustrating the concepts of path indistinguishability. We will use diamonds similar to the Lee experiment before each slit in the Young’s interferometer. Examining the Young’s interferometer (Figure

Illustration path indistinguishability and the influence environmental entanglements (diamonds) with Young’s interferometer. The I is an interference pattern and the NI is no interference pattern.

Now we extend (

The first two terms are again DC terms and the second two represent interference terms. The wavefunction (in the bras-kets) incorporates all properties of the photons (polarization, bandwidth, photon numbers, etc.) now and not just occupation number. As can be seen from the density operator, the interference pattern is independent of whether the photons come individually or at high intensity (if one of the wavefunctions was zero, interference would still occur). In the density operator equation, 1 and 2 correspond to the two potential paths the photon can take. The density operator contains an inner product (

To illustrate the counter-intuitive interaction of the photons and phonons leading to indistinguishable paths and coherence, Young’s experiment will be examined by varying the Raman scattering. As a basic rule of quantum mechanics, which can be found in any introductory quantum mechanics textbook, until a measurement is made potentials are added then squared but once a measurement occurs, intensities (squared potentials) are added. If we initially ignore the

But another critical point is that the Born and Markovian approximations are not met hence decoherence will result. The Born approximation is that the diamond-principal interaction is sufficiently weak and environment (diamond) large such that the principal does not significantly change the diamond. Obviously the coupling is strong (Raman scattering) and the diamond changes significantly (change in phonon frequency). The Markovian approximation, having no memory effects, means that self-correlations within the diamond/environment decay for all practical purposes instantly into the environment. If these two are not met (along with the diamond interactions being identical), then the diamonds become part of the coherent system rather than a source of decoherence. Together, the indistinguishable paths of single-photon coherence, near identical nature of Raman scattering, and not meeting the Born/Markovian approximations resulted in expansion of the coherence (the two diamonds become part of the principal, resulting in quantum correlations). This describes why the two phonon fields become correlated and why it does not require (or include) an explanation of true non-local entanglement between arms.

We suggest that confusion over the distinction between quantum correlations due to entanglement versus path indistinguishability has arisen, at least in part, over a misunderstanding of the type II spontaneous parametric downconversion (SPDC) source and overextending interpretations of Dirac notation, which is presented in the Appendix. This speculative topic is addressed in the Appendix.

So to summarize, in the Lee paper the state, when using a single arm/diamond, is initially a Stokes-phonon(s) entanglement then Stokes-phonon(s)-anti-Stokes entanglement, arising from and remaining consistent with (

As pointed out, the coherence expansion that results requires very specific conditions with respect to the diamonds. First, the high phonon frequency minimizes thermal decoherence. Second, the generated Stokes photons must be essentially identical with respect to detection. Third, the Born and Markovian approximations must not be met. Together, along with the path indistinguishability, this results in quantum correlations between the diamond phonons.

Just briefly discussing the probe photons, what is being measured is second-order correlation between detectors

The four-quantitative/qualitative results for discussion from the

True entanglement between the phonon fields neither needs to be elicited as an explanation for the results nor leads to be proven in the paper. Though the phrase “entanglement of diamonds” attracts considerable attention, we believe that the establishment of quantum correlations/coherence between two macroscopic objects using path indistinguishability without nonlocal entanglements is far more important to the field. We point out that we have also achieved this with two macroscopic distant reflectors [

The recent paper in

Unfortunately, many examples exist in the literature that treat quantum correlations from path indistinguishability and entanglement as essentially identical, an obstruction to the field and in part likely due to misunderstanding of the widely used SPDC II source (spontaneous parametric downconversion) and misuse of Dirac’s notation. Two prominent examples are a 2008

SPDC sources generally use a CW pumped nonlinear crystal to produce two energy entangled photon pairs (including entanglement of uncertainty) [

Now, using a SPDC II source with an interferometer (Figure

An SPDC type II source using a beam splitter used to generate Bell states.

The example also illustrates the misuse of Dirac notation, which seems particularly common in the quantum communication and computer fields. Dirac notation is a powerful shorthand technique for describing quantum information flow. But it is frequently treated as representing the state of a system, which it generally does not do. If we represent a vacuum and photon by

This paper is sponsored by the National Institutes of Health, Contracts R01-AR44812, R01-EB000419, R01 AR46996, R01- HL55686, R21 EB015851-01, and R01-EB002638.