People often talk about “creating entanglement” or “creating a superposition” in the laboratory, and quite rightly think about superpositions and entanglement as resources for things like quantum-enhanced measurements and quantum computing.
However, it’s often not made explicit that a superposition is only defined relative to a particular preferred basis for a Hilbert space. A superposition is implicitly a superposition relative to the preferred basis . Schrödinger’s cat is a superposition relative to the preferred basis . Without a there being something special about these bases, the state is no more or less a superposition than and individually. Indeed, for a spin-1/2 system there is a mapping between bases for the Hilbert space and vector directions in real space (as well illustrated by the Bloch sphere); unless one specifies a preferred direction in real space to break rotational symmetry, there is no useful sense of putting that spin in a superposition.
Likewise, entanglement is only defined relative to a particular tensor decomposition of the Hilbert space into subsystems, . For any given (possibly mixed) state of , it’s always possible to write down an alternate decomposition relative to which the state has no entanglement.
So where do these preferred bases and subsystem structure come from? Why is it so useful to talk about these things as resources when their very existence seems to be dependent on our mathematical formalism? Generally it is because these preferred structures are determined by certain aspects of the dynamics out in the real world (as encoded in the Hamiltonian) that make certain physical operations possible and others completely infeasible.
The most common preferred bases arise from the ubiquitous phenomena of decoherence, when certain orthogonal states of a system are approximately preserved under an interaction with the environment, while superpositions relative to those preferred states are quickly destroyed. For example, the overcomplete basis of wavepacket states is selected by interactions with environments that decohere through local interactions (like the scattering of photons and air molecules); wavepackets are easy to prepare, but superposition widely separated in phase space are a scarce resource. Likewise, the most common grounding of preferred subsystem tensor structure is through spatial separation. Most interactions are local in space, so making the sorts of operations and measurements that would create and reveal entanglement — i.e., non-local ones — is practically very difficult.
To summarize: Superpositions are relative to bases. Entanglements are relative to subsystems.
All of this becomes especially important when we move back and forth between the first- and second-quantized pictures. Indeed, the superposition of an electron in two locations and in a first-quantized picture,
looks like an entangled state of the electron field in a second-quantized picture:
More than one article has been published in certain flashy magazines by capitalizing on this confusion.