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arXiv:1003.4221 · Physical Review A

Purification of mixed states with closed timelike curve is not possible

Arun Kumar Pati, Indranil Chakrabarty, Pankaj Agrawal

Published Dec 22, 2011• 11 citations
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Abstract

In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a ``proper'' mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures.

DOI
10.1103/physreva.84.062325

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In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a ``proper'' mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures.
We discuss the nature of quantum states (density operators) and entanglement in quantum theory with closed timelike curves. One of the key concepts in the standard quantum theory is that given any mixed state we can always purify it in an enlarged Hilbert space by bringing an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact and on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed time like curve (CTC) and allowed to interact in a consistent way with a causality‐respecting (CR) quantum system. Thus, in general for arbitrary interactions between the CR and the CTC systems there is no universal ‘Church of the larger Hilbert space’ for mixed states with CTC. This shows that in quantum theory with CTCs there can exist ‘proper’ and ‘improper’ mixtures. Also, we suggest what kind of entangled states are allowed in the presence of CTCs. We argue that the nature of entanglement can be different in the presence of such exotic objects. Towards the end we discuss how the no‐hiding theorem in the presence of CTCs may allow quantum information pop‐up.
Although general relativity allows the existence of closed timelike curves (CTCs), self-consistency problems arise (the ‘grandfather paradox’ among others). It is known that quantum mechanical consideration of the matter formally removes all the paradoxes, but the questions about causal structure remain. On the other hand, the idea of postselected CTCs (P-CTC) in quantum teleportation has know been put forward and experimentally implemented. We consider these problems with the aid of quantum causal analysis, where causality is defined without invoking the time relation. It implements the Cramer principle of weak causality, which admits time reversal in entangled states. We analyze Deutsch CTCs (D-CTC) with different kinds of interactions between the chronology-violating and chronology-respecting particles, with refined inferences about mixedness, quantum/classical correlations, entanglement and thermodynamics in the D-CTC. The main result is that time reversal causality can really exist, however, the final quantum state does not place retrospective constraints on the initial state, instead the final state can influence the state inside the D-CTC. This is effectively the implementation of Novikov self-consistency principle. The P-CTC has radically different properties; in particular, if the initial state was pure, the final state is always pure too. Self-consistency is controlled by the initial state-dependent traversability of the P-CTC.
The possibility of performing quantum cloning in the presence of closed time like curves is considered. We analyze a model of the quantum cloning process involving a source qubit, a target qubit, and a quantum copy machine which is itself a composite quantum system. A subsystem of the copy machine is allowed to traverse a closed time like curve. Under these circumstances the evolution of the chronology respecting parts of the system is nonlinear. This suggests (Deutsch, 1991) that quantum cloning might be possible. We prove that, if only one subsystem of the copy machine is allowed to traverse a closed time like curve and the copy machine’s Hilbert space is of finite dimension, universal quantum cloning is impossible.
The possible existence of closed timelike curves (CTCs) draws attention to fundamental questions about what is physically possible and what is not. An example is the "no cloning theorem" in quantum mechanics, which states that no physical means exists by which an unknown arbitrary quantum state can be reproduced or copied perfectly. Using the Deutsch approach, we show here that this theorem can be circumvented in the presence of closed timelike curves, allowing the cloning of an unknown arbitrary quantum state chosen from a finite alphabet of states. Since the "no cloning theorem" has played a central role in the development of quantum information science, it is clear that the existence of CTCs would radically change the rules for quantum information technology. Nevertheless we show that this type of cloning does not violate no-signalling criteria.
Closed Timelike Curves (CTCs) are intriguing relativistic objects that allow for time travel to the past and can be used as computational resources. In Deutschian Closed Timelike Curves (D-CTCs), due to the monogamy of entanglement, non-local correlations between entangled states are destroyed. In contrast, for Postselected Closed Timelike Curves (P-CTCs), a second variant of CTCs, the non-local correlations are preserved. P-CTCs can be harnessed for the signaling of non-orthogonal states to the past without a disruption of causality. In this paper, we take up signaling to the past and show a method of sending four non-orthogonal states to the past using P-CTCs. After constructing our signaling protocol, we study the causality violations that our protocol results in and put forward two consistency relations to prevent them.
We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two noncommuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.