What do the terms "hyperparallel algorithm" and "hyperentangled states" mean? I found it mentioned here[1]. In the abstract they say: "Hyperentangled states, entangled states with more than one degree of freedom, are considered as a promising resource in quantum computation", but I'm not sure they mean by "degree of freedom" in this context.

[1]: Quantum hyperparallel algorithm for matrix multiplication (Zhang et al., 2016)

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Hyperentanglement refers to entanglement between multiple degrees of freedom of a given system. It is a concept commonly encountered in some fields of quantum information processing, typically in the context of photonics. As an example, this can mean that you have entanglement between the polarisation and the position degrees of freedom of a single photon. A review paper on the subject can be found in 1610.09896.

Note first of all that there isn't anything fundamental about hyperentanglement. At the end of the day, a hyperentangled system is a "regularly entangled" system in which you just happen to have nonseparability between degrees of freedom that somehow "look different". The reason for having a specific name for it is more practical than fundamental.

The simplest example of a hyperentangled system that comes to mind is a quantum walk. More specifically, one can implement a photonic quantum walk in which the walker degree of freedom is implemented as the position (or orbital angular momentum, or frequency, or time, or something else) of a single photon, and the coin degree of freedom is the photon's polarisation. Already after a single step (assuming a bunch of things about the kind of quantum walk under consideration) one has a state of the form $$\lvert-1,\uparrow\rangle+\lvert1,\downarrow\rangle,$$ where the first label characterises the walker (path, OAM, or anything else) dof, while the second one the polarisation dof. This is probably the simplest example of a "hyperentangled" photonic state. Again, note that this differs from a standard Bell state only in the physical substrate associted with the degrees of freedom, so there is nothing fundamental about it (which does not mean it is not interesting of course). An example of such a OAM+polarisation quantum walk is studied in 1407.5424 (even though they don't use the term "hyperentanglement" there). More generally, any vector beam (class of beams characterised by a space-variant polarisation in the transverse plane, see e.g. ncomms8706) qualifies.

More interesting states can be obtained by "hyperentangling" different degrees of freedom of different photons. An example of this is found in 1602.03769, in which entanglement of path and polarisation of two photons is exploited to get a four-qubit entangled state. Another example is 1507.08887, in which the authors implement a state in which there it entanglement between different hyperentangled states.

"Hyperparellel algorithm" is not a popular term at all and all instances of this phrase seem to be coming from the same group of co-authors.

Hyper-entanglement is at least used more in the "mainstream", although it is still not a very common word to hear. In this presentation by NASA they define it as a system being entangled in more than one degree-of-freedom (DOF) at the same time. Here we have two photons that are entangled with respect to three different DOFs at the same time:

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In this paper there's two photons that are entangled with respect to polarization, and simultaneously also entangled in terms of their spatial DOF:

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Using this definition, the authors of the paper you mention defined a hyper-CNOT and talked about "hyper-parallel quantum computation" as operating on two DOFs at the same time:

"In this paper, we investigate the possibility of achieving scalable hyper-parallel quantum computation based on two DOFs of photon systems without using the auxiliary spatial modes or polarization modes"

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    So states in $(H_1 \otimes H_2 \otimes \cdots H_n)^{\otimes 2}$ such that if you can write them in $H_1^{\otimes 2} \otimes \cdots H_n^{\otimes 2}$ as $| \psi_1 \rangle \otimes \cdots | \psi_n \rangle$ with at least 2 of the $\psi_i$ having rank $>1$ when viewed as matrices. Or do they have to be fully entangled instead of merely the open rank condition? Or might they require more summands as $\sum a_i | \psi_{i,1} \rangle \otimes \cdots | \psi_{i,n} \rangle$ like the most general state would? What exactly is the definition? – AHusain Oct 9 at 13:44

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