Comparison of D-CTCs and P-CTCs#
In ordinary quantum mechanics, any mixed state can be purified in an enlarged Hilbert space through the use of an ancillary system. In the Deutsch model however, purification of the CV state is not possible since it has to be in a proper mixture (that is, it can not be a subsystem of a pure entangled state) [8]. The very power of the prescription lies in the fact that it allows for the information trapped within the CTC to exist in a mixed state, and it is this distinguishing feature which enables it to solve classically paradoxical scenarios.
P-CTCs on the other hand provide solutions by ignoring the precise mechanism behind antichronological time travel. Instead, the theory posits that the effect is simply mathematically equivalent to the operational protocol of preselecting and postselecting against the same maximally entangled state.
Evidently, these two prescriptions are distinct, yet they each resolve time-travel paradoxes, albeit in usually very different ways. While P-CTCs provide unique resolutions, D-CTCs alternatively require an additional condition (see [2, 1, 4, 5, 6, 7]) to arrive at a unique solution. However, since these prescriptions are treatments of interacting quantum systems near CTCs, we observe non-unitarity in the evolutions of CR systems in both models, which in turn means that the output states are non-linear functions of the input state(s). In the case of P-CTCs, the output states are unnormalized (thereby requiring renormalization), and such an effect jeopardizes the usual probabilistic interpretation of quantum states. The non-linearity and non-unitarity of D-CTCs on the other hand is manifestly visible in the production of entropy through mixing of the input states. This is clear from its equations of motion (223) and (224), which essentially describe an open interaction between two quantum systems.
Another important difference between the two prescriptions is that for D-CTCs there is no restriction on the initial data, whilst with P-CTCs, the renormalization leads to limits on the initial data as a function of the future. This aspect of the model means that, for certain interactions, there are forbidden input states, which mathematically is due to destructive interference in the P-CTC path-integral picture. This in turns leads to antichronological and superluminal influence [2, 3, 9, 10], which is contrary to many of the expectations we have of any physical theory that is to be considered “good”.
Non-linearity is a highly non-trivial issue, as its presence fundamentally changes the structure of the theory. This is because the standard proofs of many key theorems and concepts in quantum mechanics, including the no-signalling theorem, the no-cloning theorem, and the inability to distinguish between non-orthogonal states, depend on linearity. Therefore, given the non-linearity and non-unitarity of both D-CTCs and P-CTCs, it unsurprising that these prescriptions possess some powerful features with very interesting consequences.
When used as computational resources, time machines provide significant increases in computational speed and efficiency. According to D-CTCs, both classical and quantum computers with access to a CTC can solve PSPACE-complete [11, 12] problems while a quantum computer can solve NP-complete [11, 13] problems efficiently. Alternatively, P-CTCs tell us that CTC-assisted quantum computers, while being able to solve decision problems efficiently in NP \(\cap\) co-NP and probabilistically in NP [14], can only generally solve problems in PP [15]. This means that P-CTCs are (putatively) computationally inferior to D-CTCs, though both are still vastly superior to the regular BQP power of quantum computers and BPP power of classical computers.
In addition to their use as computational resources, both D-CTCs [16] and P-CTCs [14] permit non-orthogonal quantum states to be distinguished to some degree. D-CTCs [17, 18] can also clone arbitrary states, and P-CTCs can both signal to the past [9] and delete arbitrary states (as a consequence of being able to solve problems in PP).
References
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