Treating Time Travel Quantum Mechanically John-Mark A. Allen∗ Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this question by utilising the quantum circuit formalism has given rise to two theories: Deutschian- CTCs (D-CTCs) and “postselected” CTCs (P-CTCs). In this paper the quantum circuit approach is thoroughly reviewed, and the strengths and shortcomings of D-CTCs and P-CTCs are presented in view of their non-linearity and time travel paradoxes. In particular, the “equivalent circuit model”— which aims to make equivalent predictions to D-CTCs, while avoiding some of the difficulties of the original theory—is shown to contain errors. The discussion of D-CTCs and P-CTCs is used to motivate an analysis of the features one might require of a theory of quantum time travel, following which two overlapping classes of new theories are identified. One such theory, the theory of “transition probability” CTCs (T-CTCs), is fully developed. The theory of T-CTCs is shown not to have certain undesirable features—such as time travel paradoxes, the ability to distinguish non- orthogonal states with certainty, and the ability to clone or delete arbitrary pure states—that are present with D-CTCs and P-CTCs. The problems with non-linear extensions to quantum mechanics are discussed in relation to the interpretation of these theories, and the physical motivations of all three theories are discussed and compared. PACS numbers: 03.65.-w, 03.65.Ta, 03.67.-a 04.20.Gz I. INTRODUCTION possibility of precluding such solutions. Spacetimes con- taining CTCs are globally non-hyperbolic and therefore are incompatible with standard relativistic quantum field There are few areas of physics in which one confronts theory [11]. Such spacetimes typically contain no Cauchy the idea of time travel to the past. When it is discussed, surfaces [12] and so the initial value problem is not well time travel is often associated with particular ways of defined, making the analysis of matter in such spacetimes thinking about quantum mechanics and quantum field tricky. theory. One often hears the view that in quantum tele- portation, for example, the teleported information travels Deutsch [13] began a programme for investigating back in time to the point at which entanglement was cre- quantum mechanics along CTCs that uses the quantum ated, before proceeding forward in time to the recipient circuit formalism of quantum computation [14]. In this [1–4]. Another example might be remarks about anti- approach a circuit is used to ferry qubit states along clas- particles in quantum field theory being akin to particles sical paths between localised regions of spacetime where “travelling backwards in time”. The time travel con- they evolve and interact according to unitary quantum sidered in this work is rather different. Here it is assumed theory; the relativistic effects of CTCs are present only in that time travel into the past is a possible physical pro- the ability to send qubits back in time. In this way, the cess. The question is then: How must quantum mech- resulting theories abstract away from the specific mech- anics be modified to account for time travel? The mo- anism of time travel, and may, therefore, be applicable to tivation for this comes not from within quantum theory any such mechanism, be it TARDIS, DeLorean, or CTC. Deutsch’s theory of quantum mechanics with time arXiv:1401.4933v3 [quant-ph] 10 Oct 2014 itself, but rather from general relativity and the existence of spacetime solutions containing closed timelike curves travel has come to be known as the theory of D-CTCs. (CTCs). A second theory of quantum mechanics with time travel based on the same quantum circuit approach has come CTCs are paths through exotic spacetimes along to be known as the theory of P-CTCs, since it uses the which massive particles may travel, apparently forwards mathematics of postselection [15–17]. P-CTCs are motiv- through time, only to return to their own past. As such, ated by “teleporting” quantum states into the past and they represent a mechanism for time travel to the past. are equivalent to being able to postselect on quantum The discovery that the Einstein field equations of general measurement outcomes. relativity permit spacetime solutions that contain CTCs [5–7] came as something of a surprise [8]. Many believe Science fiction is rife with naïve classical examples of that such spacetimes containing CTCs should not occur time travel to the past, almost inevitably leading to vari- in nature [9, 10]; however, there is no consensus on the ous paradoxes. To some extent, both D-CTCs and P- CTCs mitigate such paradoxes. However, both theories still contain undesirable features that may also be patho- logical or paradoxical. ∗Electronic address: [email protected] The evolution of a state in a region involving time 2 travel is expected to be both non-unitary [18–20] and ternative model of D-CTCs [25, 26]—is shown to contain non-linear [21]. Non-linearity, if present, fundament- errors. Readers familiar with these theories may want to ally changes the structure of quantum theory. The skim through this section rather rapidly. The meat of standard proofs of most central theorems in quantum the paper is in Sec. IV where certain desirable features mechanics (including no-signalling, no-cloning, general- of any theory of quantum time travel are identified, and ised uncertainty principles, and indistinguishably of non- used to find two classes of new theories; from these new orthogonal states) depend on the linearity of the the- theories that of T-CTCs is selected and fully developed. ory. It is therefore of little surprise that: both D-CTCs Finally, in Sec. V, the theory of T-CTCs is compared and P-CTCs can distinguish non-orthogonal states with to the previous theories, focussing on their motivation, a single measurement and (in certain ontologies) signal shortcomings, and successes. faster than light; D-CTCs can clone arbitrary pure states; and P-CTCs can delete arbitrary states. Linearity is also necessary for the equivalence of proper and improper II. MODELLING TIME TRAVEL WITH mixed states in ordinary quantum theory [22, 23] and the QUANTUM CIRCUITS ability of quantum theory to be compatible with many different interpretations and ontologies [24]. Non-linear A. The Standard Form of Quantum Circuits with extensions of quantum theory, such as D-CTCs and P- Time Travel CTCs, therefore, require careful examination of any on- tological assumptions as the predictions of the theories The quantum circuit approach to time travel is based generally depend on them. on a particular form of qubit circuit introduced by Such issues surrounding theories of time travel in Deutsch [13]. This convenient building block, from which quantum circuits are subtle, and different theories give all other circuits involving time travel can be built, is different perspectives. As well as containing undesirable here called the standard form circuit. It allows different features, both D-CTCs and P-CTCs are more difficult to circuits and theories to be concisely specified. interpret than standard quantum mechanics. Since they The advantage of the quantum circuit model in this are constructed from the abstract quantum circuit form- context is that quantum evolution of qubit states is sep- alism, any interpretational assumptions behind them are arated from spatial motion. Quantum interactions de- unclear. One may therefore ask if there could be a the- scribed by unitary gates are assumed to only occur in ory with fewer undesirable features. Could such a theory small, freely falling, non-rotating regions of spacetime so have a more robust physical motivation? that they obey non-relativistic quantum mechanics. To In this paper, the first of these questions is answered include time travel to the past in such a quantum circuit squarely in the affirmative. Several alternative theories of model is therefore equivalent to saying that the classical time travel in quantum mechanics, falling into two over- paths qubits take between gates are allowed to go back lapping classes, are presented. One such theory, called in time. The restriction to circuits of qubits may seem the theory of T-CTCs (since it uses transition probabil- unphysical but it is convenient and expected to be com- ities), is thoroughly developed and seen to have far fewer putationally universal [13]; circuits with systems other problematic features than either D-CTCs or P-CTCs; in than qubits may also be constructed by analogy. Within particular, cloning and deleting are impossible and non- this approach, different theories are then defined by their orthogonal states cannot be distinguished with a single behaviour when sending qubits back in time. In this pa- measurement. The second question is more subjective in per only finite-dimensional Hilbert spaces are considered its nature, but the theories presented here all have phys- by considering only finite numbers of qubits. ical motivations that may be, depending one’s prejudice, The standard form circuit is a circuit of n + m < at least as compelling as those of D-CTCs and P-CTCs. qubits, a single time travel event that occurs in a local-∞ This strongly suggests that the question as to how ised spacetime region [70], and a single unitary quantum quantum mechanics behaves in the presence of time interaction U. This paper does not consider situations in travel, and specifically CTCs, is far from decided. With which the time travel region is not localised in spacetime.