Framework

In general relativity, causality can be violated due to the presence of spacetime wormholes that facilitate closed timelike curves (See Figure 1). This can lead to physical processes where a system A, starting out in a state ρ(in), interacts with a second system A ′ in state ρ CTC via a unitary U. System A then enters the wormhole and becomes system A ′ . The Deutschian model, resolves potential paradoxes by enforcing temporal self-consistency conditions.5,14

(1) ρ CTC = Tr ≠ A [ U ( ρ ( in ) ⊗ ρ CTC ) U † ] ,

where Tr ≠A represents tracing over everything apart from system A. Given a solution for ρ CTC , the final output of the process is given by

(2) ρ ( out ) = Tr A [ U ( ρ ( in ) ⊗ ρ CTC ) U † ] .

The many radical effects of CTCs rely on using specific self-interactions U to break causality in different ways.6,7,9–11

Note that since the effects of CTCs are non-linear, many conventional assumptions of linear quantum mechanics break. One consequence is that different unravellings of a density operator give different predictions, and therefore must be treated very carefully. While the above analysis does not assume ρ(in) is pure, these conditions only apply to mixed inputs if ρ(in) represents one partition of a larger composite system that is pure. In cases where the input state is deterministically prepared in a pure state, shot-by-shot, but the particular state may change shot-by-shot, there remains some ambiguity. The conventional view is taken by Bacon,10 Brun et al.6 and Pienaar et al.13—that sub-ensembles of identical input states should be treated as pure. Others argue that if this sequence is truly random, then the randomness must always be treated as if it arose from the purification of an entangled quantum state.15 The former view, which we adopt here, is supported by studies of CTCs in the context of information flow.14 Also note that if the sequence of input states were chosen via a pseudo random number generator, then both perspectives agree.

In OTCs, causality is preserved. The unitary U is the identity—such that the time-travelling system does not interact with its causal past. Any observer in the frame of reference of the time travelling system can assign a valid chronology to all the events they witness. Meanwhile, to any outside observer, all events involving interactions with the time travelling system will occur in causal sequence. From an operational standpoint, there is no breaking of causality. If all information were classical, this entire procedure would only have the effect of desynchronizing a time traveller’s clock with that of an observer.

To observe non-trivial effects we must introduce an ancilla. Suppose we have access to a bipartite system AB in state ρ AB , where only one bipartition is sent through the OTC (see Figure 2). The self-consistency relations requires ρ C T C = Tr ≠ A [ ρ A B ( in ) ⊗ ρ C T C ] = ρ A , and as a consequence

(3) ρ ( out ) = Tr ≠ A [ ρ A B ( in ) ] ⊗ Tr A [ ρ A B ( in ) ] = ρ A ⊗ ρ B

Thus, the OTC acts as a universal decorrelator on A—in sending a system A though an OTC, we erase all quantum correlations between A and the rest of the universe (and in particular, B). The resulting state, ρ A ⊗ρ B fields identical local statistics with respect to the input ρ AB , but none of its bipartite correlations. While this operation appears similar to trivial decoherence, it is non-linear, and shown to be impossible to synthesize with standard quantum dynamics.16 This decoherence ensures that the Deutschian formalism remains consistent with special relativity. If quantum correlations between A and B were preserved, the capacity for OTCs to clone quantum states (as we show in following sections) would allow for superluminal signalling via Herbert’s protocol.17

Figure 2 CTCs and OTCs in presence of ancilla. A represents the system to be sent through the spacetime wormhole, A ′ represents the same system after it passes through the spacetime wormhole, and B some chronology-respecting system initially correlated with A. (a) In general CTCs, temporal self-consistency demands that ρ CTC satisfies ρ CTC = Tr ≠ A [ U ( ρ A B ( in ) ⊗ ρ CTC ) U † ] . (b) In the case of OTCs, this implies that system A has state ρ OTC = Tr ≠ A [ ρ A B ( in ) ⊗ ρ OTC ] = ρ A after application of the protocol. Full size image

One way to understand this effect is through the monogamy of entanglement14—a particle and its past self cannot be simultaneously entangled with the same external ancilla. While OTCs produce nontrivial dynamics when the input appears completely classical (e.g., when ρ A B ( in ) = ( | 00 〉 〈 00 | + | 11 〉 〈 11 | )/2 ), this only occurs if this mixedness as being intrinsic—i.e., it arises from entanglement with some other system. If the input is deterministically prepared in either state | 00 〉 or | 11 〉 according to some classical sequence, then the OTC will have no effect (for other viewpoints, see Bennett et al.15).

OTC enhanced measurement

We first introduce OTC enhanced measurement, a procedure that harnesses OTCs to measure an arbitrary observable O ˆ to any fixed precision. Specifically, given an unknown qudit (d dimensional quantum system) in state ρ, we can determine 〈 O ˆ 〉 = Tr [ O ˆ ρ ] to any desired accuracy δ>0 with negligible failure probability. This protocol functions as a building block for more sophisticated applications of OTCs, such as the efficient solution of NP-complete problems and cloning of unknown quantum states.

The protocol is illustrated in Figure 3. Let | j 〉 : j =0,1, … , d − 1 denote a basis that diagonalizes O ˆ . On this basis, we introduce the two qudit controlled addition operator, C + | i 〉 | j 〉 = | i 〉 | j + i 〉 , where addition is done modulo d. We then

1 Prepare N identical ancillary states in an eigenstates of O ˆ , say | 0 〉 . 2 2. Apply the C + operations N times, each controlled on ρ and targeting a fresh ancilla state. This correlates ρ with each of the N ancillaries. 3 Pass each of the ancillaries through an OTC to destroy all correlations in this N+1-partite system.

Figure 3 Quantum circuit of OTC enhanced measurement. The protocol first introduces N ancilla qudits, all of which are initialized in the state ρ E = | 0 〉 〈 0 | , where | 0 〉 is an eigenstate of O ˆ . A sequence of C + gates then perfectly correlates each ancilla with ρ with respect to O ˆ basis. The erasure of these correlations via OTCs, followed by O ˆ measurements on each individual qudit, allows determination of Tr [ O ˆ ρ ] to a standard error that scales inversely with N2. Full size image

This results in N+1 uncorrelated qudits, each in state ρ diag = ∑ i =1 d ρ i i | i 〉 〈 i | , where ρ ii are the diagonal elements of ρ in the O ˆ basis. Thus, each qudit exhibits identical statistics to ρ when measured in the O ˆ basis. In taking the mean of these measurements, we obtain an estimate for 〈 O ˆ 〉 . By the central limit theorem, the error of our estimate scales linearly with 1/ N . In particular, provided the eigenvalues of O ˆ are bounded, Hoeffding’s bound implies we can estimate O ˆ to any desired accuracy δ and error rate ϵ using O[1/δ2log(1/ϵ)] OTCs (see methods for details).

We note this technique shares similarities with the proposal of Brun et al. to use closed timelike curves to perform the same task.7 Both protocols operate by harnessing spacetime wormholes to create many ‘clones’ of ρ with respect to the eigenbasis of O ˆ —each of which is measured to give a statistically independent estimate of 〈 O ˆ 〉 . The key difference is that in Brun et al., each of these clones were trapped within a closed timelike curve, and reading out information from them required direct interaction with each clone. Our proposal shows that the use of CTCs is not compulsory. A comparison between the two methods indicate the use of OTCs incurs no overhead in the number of times a spacetime wormhole is used (see methods). Hence OTCs—at least for this purpose—are as powerful as CTCs.

Solving NP-complete problems

We take inspiration from Bacon,10 who devised an efficient algorithm to solve the boolean satisfaction problem—a known NP-complete problem—using CTCs. We modify this algorithm to preserve causality—without losing efficiency. In the causality breaking algorithm, the key role of CTCs is to implement the non-linear map S that maps an input qubit in state ρ(n z ) to an output state ρ ( n z 2 ) , where ρ ( n z ) = 1 2 ( I + n z σ z ) and σ z = | 0 〉 〈 0 | − | 1 〉 〈 1 | denotes the Pauli Z matrix (see methods for details).

This non-linear map can be replicated without breaking causality (see Figure 4). Consider a special case of OTC enhanced measurement, with σ z as the observable of interest and a single ancilla. For the input qubit ρ with matrix elements ρ ij , application of the enhanced measurement protocol outputs two uncorrelated qubits, each in state ρ diag = ρ 00 | 0 〉 〈 0 | + ρ 11 | 1 〉 〈 1 | . Instead of measuring each in σ z directly, we apply a further C + gate controlled on the ancilla. After discarding the ancilla, the input qubit is now transformed to S(ρ) as required.

Figure 4 Solving NP-complete problems with OTCs. The key non-linear gate S, that takes ρ(n z ) to ρ ( n z 2 ) , can be implemented via open timelike curves. This is achieved by the use of a single OTC, applied between two successive C + gates. Full size image

In generating S(ρ) using only OTCs, we can translate Bacon’s algorithm into one that does not break causality. We note that as each call of S(ρ) only takes one OTC, the translation from CTCs to OTCs incurs no overhead on the number of times a particle needs to be sent through a spacetime wormhole. Thus, for the purpose of solving NP-complete problems, an OTC, together with one bit of entanglement, is at least as powerful as a CTC.

Cloning with OTCs

Given an unknown input ρ, OTCs allow us to generate an unlimited number of clones to arbitrary fidelity. Our approach harnesses OTC enhanced measurements as a subroutine, which allows us to accurately determine Tr[M i ρ], for any observable M i . First, observe that this remains possible even if we are supplied with

(4) ρ ′ = s ρ + 1 − s d I ,

a very noisy version of ρ. Here I is the d-dimensional identity matrix, and s is some fixed parameter such that 0<s<1.

This observation, together with an imperfect quantum cloner, forms the basis of our OTC enhanced cloning protocol (Figure 5). In conventional quantum theory, a unknown quantum state ρ can be cloned if we are given sufficiently many copies to perform accurate tomography.18 One way to do this, is to use a set of O(d2) informationally-complete measurements {M i }, whose expectation values Tr[M i ρ] have a one-to-one correspondence with the classical matrix description of ρ. If there is only a single copy of ρ, this option is no longer available. Recently, Brun et al. demonstrated that closed timelike curves circumvent this restriction, and allow the estimation of each 〈 M i 〉 to any desired accuracy.7

Figure 5 OTC Assisted Cloning An arbitrary qudit ρ can be cloned to any desired fidelity. The process involves (i) application of a standard quantum cloner C to generate O(d2) imperfect copies, and (ii) use of OTC enhanced measurements to measure different observables M i on each imperfect copy. We can choose M i to be informationally-complete, and OTCs ensure that we can determine Tr[M i ρ] to any desired precision. Thus this protocol can yield (to any fixed precision) the classical description of ρ. Full size image

OTC enhancement measurements can replicate this effect while preserving causality. We use standard methods to construct O(d2) imperfect clones in the form of equation (4), where s scales as 1/d for an optimal cloner.19 Each clone is passed through an OTC to remove all entanglement between clones. An OTC enhanced measurement is then performed on each clone with respect to a different M i . The outcomes of these measurements determine the density matrix of ρ. In methods, we show that by using O ( d 4 / δ c 2 log 1/ ε c ) OTCs, we can ensure that each 〈 M i 〉 is obtained to an accuracy of δ c with failure probability ϵ c .

A simple example

We illustrate these ideas by cloning a qubit. Here, the Pauli operators σ k , k=x, y, z form an informationally-complete set—any ρ is uniquely defined by the expectation values n k =Tr[σ k ρ]. To determine n x , n y and n z , we first apply a universal 1-to-3 quantum cloner20 to obtain three imperfect clones of ρ, each in state ρ ′ = ( I + s n → ⋅ σ → )/2 with s=5/9. All quantum correlations between these three imperfect clones may be erased by applying OTCs.