Distinguishing quantum states using time travelling qubits in a presence of thermal environments

TL;DR

This paper investigates how thermal environments impact quantum circuits with time travel designed to distinguish nonorthogonal quantum states, revealing that dephasing environments preserve some quantum distinguishing power.

Contribution

It introduces an analysis of thermal noise effects on time-travel quantum circuits and discusses the validity of Deutsch's maximum entropy rule under these conditions.

Findings

Dephasing environments preserve the distinguishing power of time-travel circuits.

Thermal noise generally degrades quantum state discrimination capabilities.

The paper provides conditions under which Deutsch's maximum entropy rule remains valid.

Abstract

We consider quantum circuits with time travel designed for distinguishing specific nonorthogonal quantum states in two most popular models: Deutschs and postselected. We modify them by a presence of weakly coupled thermal environment. Using the Davies approximation we study how the thermal noise affects an ability of the circuits to distinguish nonorthogonal quantum states. We show that for purely dephasing environment a paradoxial power of such circuits remains preserved. We also present a physics based argument for conditions of validity of the maximum entropy rule introduced by David Deutsch for resolving the uniqueness ambiguity in a circuit with time travel.

Equations

$$H$$

$$H_Q$$

$$\begin{array}[]{ll}D\bigg{[}|1\rangle\langle 1|\bigg{]}=[1-(1-p)(1-e^{-At})]|1\rangle\langle 1|+\\ +(1-p)(1-e^{-At})|0\rangle\langle 0|,\\ D\bigg{[}|1\rangle\langle 0|\bigg{]}=e^{i\omega t-Gt}|1\rangle\langle 0|,\\ D\bigg{[}|0\rangle\langle 1|\bigg{]}=e^{-i\omega t-Gt}|0\rangle\langle 1|,\\ D\bigg{[}|0\rangle\langle 0|\bigg{]}=p(1-e^{-At})|1\rangle\langle 1|+[1-(1-e^{-At})p]|0\rangle\langle 0|\end{array},$$

$$p$$

$$\underset{t\to \infty }{\lim }D(p,A,G,\omega ,t)\rho =p|1⟩⟨1|+(1-p)|0⟩⟨0|.$$

$$G$$

$$\begin{array}[]{l@{}l}\tau&{}=\Lambda(\tau),\\ \Lambda(\cdot)&{}=\mbox{Tr}_{CR}\{U(\rho_{i}\otimes\cdot)U^{\dagger}\},\end{array}$$

$${\rho }_f$$

$$\tau$$

$$U$$

$$Q_{-}$$

$$Q_0$$

$$R$$

$${\rho }_f$$

$${\chi }_{CB}$$

$$2b_1$$

$$2a_1$$

$$2c$$

$$2b_0$$

$$2a_0$$

$${\rho }_f$$

$$L_I$$

$$L_{II}$$

$$L_{III}$$

$$L_{IV}$$

$$L_V$$

$$L_{VI}$$

$$|1⟩⟨1|$$

$$|+⟩⟨+|$$

$$|1⟩⟨1|$$

$$|+⟩⟨+|$$

$$a_1$$

$$a_0$$

$$U$$

$${\tau }_{\alpha }$$

Full text

I Introduction

Impossibility of distinguishing non–orthogonal quantum states is a bedrock granting safety of quantum communication protocols Nielsen and I. Chuang (2000); Scarani et al. (2009); Gisin et al. (2002). This bedrock, however, can be eroded by closed time–like curves (CTC) which existence (under certain assumptions) has already been predicted long time ago Gödel (1949). Potential time travelers could utilize the ’paradoxial power’ of such circuits to solve problems which are hard to solve or even impossible to perform, cf. Ref.  Brun and Wilde (2015) for recent a review. In particular they may be able to distinguish non–orthogonal quantum states Brun et al. (2009); Brun and Wilde (2012).

There are at least three non–relativistic models, utilizing quantum circuit formalism, of how the quantum computation is affected by the presence of CTCs. In other words, there are at least three models of quantum time travel useful for quantum information. (i) David Deutsch Deutsch (1991) was the first who began to investigate properties of quantum systems in a presence of CTCs. He proposed an effective (nonrelativistic) description utilizing the quantum circuit formalism to describe quantum systems built of interacting the chronology respecting (CR) and chronology violating (CV) constituents. This proposal allowed to resolve at least some of the paradoxes caused by CTCs. Despite of experimental attempts of mimicking the Deutsch model Ringbauer et al. (2014); Brun and Wilde (2015) this proposal remains controversial Wallman and Bartlett (2012); Allen (2014). The second (ii), utilizes a nowadays experimentally accessible teleportation protocol equipped with a post–selection Svetlichny (2011); Lloyd et al. (2011a, b) and the third (iii), most recent Allen (2014), uses transition probabilities. For examples of other approaches one can consult e.g. Ref.  Elze, Hans-Thomas (2013) or  Vaidman (2013).

One can expect that the ’paradoxial’ computational power of time travelers, originating from non–linearity of quantum models in the presence of CTCs, becomes weakened by omnipresent decoherence. In this paper we consider how the ability of distinguishing states of qubits is affected by thermal environment of time traveling qubit. We apply the Davies weak coupling approach Alicki and Lendi (2007) for a model of decoherence which is reviewed in Sec 2. of our paper. We limit our attention quantum circuits distinguishing non–orthogonal qubit’s states in the Deutsch model in Sec 3. of the paper and the post–selected teleportation in Sec. 4. In Sec. 5 we analyze the circuit for the unproven theorem Allen (2014), designed to exemplify a celebrated paradox of information originating out of nowhere, and we present, utilizing general consideration of Ref.Allen (2014), how the effect of thermal decoherence can serve as a physical justification of the Deutsch’s maximum entropy rule introduced ad hoc in Ref.Deutsch (1991) in order to resolve the uniqueness ambiguity Allen (2014) present circuits with CTCs.

II Davies decoherence

Quantum decoherence is caused by the environment. Its influence on the qubit $Q$ is modeled by the Hamiltonian in the form:

[TABLE]

where $H_{Q}$ is the Hamiltonian of the qubit, $H_{env}$ models the environment and $H_{int}$ describes the qubit–environment interaction.

For the qubit:

[TABLE]

where $\omega$ is the energy splitting of the qubit and $|0\rangle,|1\rangle$ span a Hilbert space of $Q$.

We assume that the interaction between the qubit and its environment satisfies the Davies weak coupling conditions Alicki and Lendi (2007) dedicated for rigorous construction of the qubit reduced dynamics calculated with respect to the environment. It is formulated in terms of a completely positive (strictly Markovian) semigroup using parameters of the microscopic Hamiltonian of the full system Alicki and Lendi (2007). As the Davies semigroups can be rigorously and consistently derived from microscopic models of open systems they satisfy thermodynamic and statistical–mechanical properties of open quantum systems such as the detailed balance condition and the Gibbs canonical distribution in the stationary regime Alicki and Lendi (2007). The Davies method has been successfully used in recent studies of various problems in quantum information and physics of open quantum systems including teleportation Kłoda and Dajka (2014), entanglement dynamics Lendi and Wonderen (2007), quantum discord Dajka et al. (2012); Dajka and Łuczka (2013), properties of geometric phases of qubits Dajka et al. (2011), thermodynamic properties of nano–systems Szela̧g et al. (2008) and quantum games Dajka et al. (2015). In this paper we consider only certain elements of Davies semi–groups: the Davies maps $D=D(p,A,G,\omega,t)$ which acts as follows Roga et al. (2010):

[TABLE]

where $p\in[0,1/2]$ is related to the temperature $T$ of the environment via:

[TABLE]

We set $k_{B}=1$. Let us notice that in long time limit the Davies map transforms any qubit state $\rho$ into the equilibrium Gibbs state:

[TABLE]

The case $T=0$ corresponds to the value $p=0$ and for $T\to\infty$ the parameter $p\to 1/2$.

The parameters $A=1/\tau_{R}$ and $G=1/\tau_{D}$ interpreted in terms of spin relaxation Levitt (2008) are related to the energy relaxation time $\tau_{R}$ and the dephasing time $\tau_{D}$ respectively Roga et al. (2010). There is a relation between $A$ and $G$ which guarantee that the Davies map is a trace-preserving completely positive map. It is given by the inequalities Levitt (2008)

[TABLE]

The limiting case $A=0$ and $G\neq 0$ corresponds to (Markovian) pure dephasing without dissipation of energy. Let us notice that the pure dephasing despite its apparent simplicity can be effective applied to modeling of realistic systems c.f. Ref.Schuster et al. (2007), in which no energy dissipation occurs for a time scale significantly larger than other time scales in the system.

III Deutschian model of CTC

The simplest Deutsch’s circuit Deutsch (1991) designed to mimic quantum dynamics in a presence of closed time–like curves (CTC) consists of a pair of qubits: the one is chronology respecting (CR) whereas the second violating the chronology (CV). This two qubits are coupled by the unitary $U$. The CV qubit enters the circuit and interacts with the CR qubit. Then it violates the chronology and is identified with its past. Formally the CV time evolution reads as follows:

[TABLE]

with the partial trace $\mbox{Tr}_{CR}$ calculated with respect to the CR qubit.

At the same time the state of the CR qubit which enters the circuit in a state $\rho_{i}$ changes into its final form $\rho_{f}$ given by:

[TABLE]

with the partial trace calculated with respect to the CV qubit.

Fundamentals of the Deutsch’s consistence condition Eq.(7) is a subject of an important debate Wallman and Bartlett (2012). In this paper we do not intend to enter such philosophical topics and simply assume the ontic interpretation of quantum states both pure and, what is probably more unconventional, mixed.

Instead, our aim is to investigate an effect of thermal noise affecting the CV qubit. We consider the Deutsch’s consistence condition Eq.(7) modified by the presence of thermal environment in the Davies approximation discussed in the previous section. It is given by a composition $\circ$ of maps:

[TABLE]

where $D=D(p,A,G,\omega,t)$ is the Davies map Eq.(3), with $t$ denoting a time period when the CV qubit interacts with thermal bath. Equation (9) has a natural interpretation: the CV qubit, before it returns to its past, interacts with thermal environment in the Markovian Davies approximation given by the map $D=D(p,A,G,\omega,t)$. Let us notice that the position of the $D$ in Eq.(9) rather than formal has a physical meaning reflecting our intention of making time travel ’noisy’.

Quantum circuits with CTCs can do tasks which are essentially inaccessible for the ’ordinary’ (linear) quantum mechanics. One of the most spectacular examples of such a task is an ability of distinguishing non–orthogonal quantum states. This ability influences security of most quantum key distribution protocols Scarani et al. (2009) with the celebrated archetype - the B92 Bennett (1992). There is a quantum circuit with the CTC Brun et al. (2009) which can be utilized to distinguish non–orthogonal qubit states. It is presented in Fig.1.

Formally its action is given by Eq.(8) and, in the presence of thermal environment, by Eq.(9) with the unitary $U$ given by

[TABLE]

where $|\pm\rangle=[|0\rangle\pm|1\rangle]/\sqrt{2}$. It the noise–less case, when Eq.(7) instead of Eq.(9) is used, the circuit transforms the indistinguishable states $|-\rangle$,$|0\rangle$ into $|1\rangle$,$|0\rangle$ which are orthogonal and hence can be distinguished Brun et al. (2009). It is not surprising that the effect of thermal noise is to weaken this ability.

In order to qualify an effect of noise we compare an output $\rho_{f}$ Eq.(8) of the noise–disturbed circuit with the noise–less output (which is $\xi_{-}=|1\rangle\langle 1|$ for $\rho_{i}=|-\rangle\langle-|$ and $\xi_{0}=|0\rangle\langle 0|$ for $\rho_{i}=|0\rangle\langle 0|$ respectively). We quantify an effect of noise by the trace distance $Q(\rho_{f},\xi)=\mbox{Tr}[\sqrt{(\rho_{f}-\xi)^{2}}]/2$ Nielsen and I. Chuang (2000) which is known Nielsen and I. Chuang (2000) to indicate distinguishability the states $\rho_{f},\xi$. For both inputs $\rho_{i}=|-\rangle\langle-|$ and $\rho_{i}=|0\rangle\langle 0|$ the corresponding states of the CV ($\tau$) and CR ($\rho_{f}$) qubits with the details of their calculation are given in the Appendix. For $\rho_{i}=|-\rangle\langle-|$ the trace distance $Q_{-}=Q(\rho_{f},\xi_{-})$ reads as follows

[TABLE]

For $\rho_{i}=|0\rangle\langle 0|$ the corresponding trace distance $Q_{0}=Q(\rho_{f},\xi_{0})$ is given by:

[TABLE]

There are three parameters $p$, $A$ and $G$ describing thermal environment affecting the CV qubit via the Davies map. The first two affect qualitatively the value of $Q=Q_{-},Q_{0}$. Increasing the last, $G$, has only a quantitative impact and results in faster growth of both $Q_{-}$ and $Q_{0}$. It is not the case if one considers $A$. The most important feature is that for purely dephasing environments the trace distance between the noisy and the noise–less output of the circuit in Fig.(1) vanishes i.e. for $A=0$ both $Q_{-}=0$ and $Q_{0}=0$. In other words, in the case of pure dephasing the CTC–assisted distinguishing of non–orthogonal quantum states works as as good as in the noise–less case. Moreover, with decreasing $A$ the corresponding trace distance $Q_{0},Q_{-}$ decreases as presented in Fig.(2).

Let us also notice that in the low temperature limit $p=0$ the trace distance $Q_{0}=0$ and that for larger values of $p$ the trace distance $Q_{-}$ grows slower than $Q_{0}$. As one infers from Fig.(3) for fixed time instant $t$ and ordered values $p_{1}<p_{2}$ the corresponding time derivatives $\partial Q_{0}/\partial t|_{t,p_{1}}<\partial Q_{0}/\partial t|_{t,p_{2}}$ whereas $\partial Q_{-}/\partial t|_{t,p_{1}}<\partial Q_{-}/\partial t|_{t,p_{2}}$.

This seemingly counter–intuitive property results from the particular and distinguished role played by the pure dephasing limit and related symmetry Alicki (2004).

A natural quantifier of an effect of thermal environment on the ’paradoxial’ power of distinguishing non–orthogonal states is a difference between the trace distance of two inputs $R(\rho_{i1},\rho_{i2})$ and the corresponding outputs $R(\rho_{f1},\rho_{f2})$ for $\rho_{f1},\rho_{f2}$ calculated via Eq.(8). As the circuit in Fig.(1) is dedicated to distinguish two very particular states $|0\rangle,|-\rangle$, cf. Ref. Brun et al. (2009), with $R(\rho_{i1},\rho_{i2})=\sqrt{2}/2$, the figure of merit is the quantity $R(\rho_{f1},\rho_{f2})$ which reads as follows:

[TABLE]

For $R=1$ the output states $\rho_{f1},\rho_{f2}$ are distinguishable. The smaller value of $R$ is the more ineffective the circuit in Fig.(1) is. Let us notice that for $R<\sqrt{2}/2$ distinguishability of the output states becomes, due to Davies decoherence, even worse than initially. The threshold condition $R(\rho_{f1},\rho_{f2})=\sqrt{2}/2$, indicated by the horizontal line in Fig.(4), depends not only on time instant $t$ but also on parameters of the system. Decreasing $A$ (for given $G$ and $p$) allows to keep the circuit useful despite longer exposition on decoherence. Again, for $A=0$ the $R=1$ i.e. the output states in a presence of a purely dephasing environment are as good distinguishable as they where in a absence of decoherence as presented in Fig.(4). It is natural to attempt to generalize this result beyond a limited class of input states which is the circuit in Fig.(1) designed for. Although we cannot present a formal proof, we conjecture, upon numerical experiments performed on randomly chosen pairs of non–orthogonal initial states, that a thermal environment never enhance state distinguishability which is ’best’ in the pure dephasing limit $A=0$. It is known that non–completely positive maps describing e.g. time–evolution of quantum systems initially entangled with their environment are not contractive Dajka and Łuczka (2010); Laine et al. (2010). As the Davies map Eq.(3) is, under the condition Eq.(6), contractive, one expects that any enhancement of distinguishability is solely due to peculiar character of the Deutsch map Eq.(8) and Eq.(7) originating from its non–linearity.

IV Post–selected CTC and thermal noise

The Deutsch’s model Deutsch (1991) of time travel operates essentially beyond standard quantum mechanics. However, there is the second most popular circuit–based model of quantum dynamics in a presence of CTCs in which one mimics the CV motion by a post–selected teleportation Svetlichny (2011); Lloyd et al. (2011a, b); Brun and Wilde (2012). Contrary to various difficulties arising in attempts of implementing Deutsch model Ringbauer et al. (2014); Brun and Wilde (2015) there are no fundamental experimental obstructions to post–select a desired outcome of teleportation procedure. However, let us notice that this apparent simplification occurs at cost of deterministic post–selection introduced ad hoc leading the well defined quantum teleportation protocol out of quantum mechanics per se. In Fig. (5) we present a well known circuit designed to transform (in an absence of noise) non–orthogonal states $|1\rangle$ and $|-\rangle$ into a pair of orthogonal, and hence distinguishable, states $|1\rangle$ and $|0\rangle$, respectively Brun and Wilde (2012).

The only but crucial difference between circuits in Fig(5) and Fig.(1) is in a way how an evolution of the CV qubit is modeled. Mimicking CTC with a post–selected teleportation utilizes a maximally entangled state as a resource Brun and Wilde (2012) which, however, can be imperfect due to a presence of thermal noise. Here we consider a state of two qubits and we assume that only one of parties in this resource is affected by thermal environment. Let us notice that such a setting is physically different to that which we adopt in previous studies of the Deutsch model where the time travel itself was assumed to be ’noisy’. Postselected CTC with thermal noise affecting the maximally entangled Bell state $|\Phi\rangle=[|00\rangle+|11\rangle]/\sqrt{2}$ of the CV qubits, cf. Fig.(5), is given by

[TABLE]

where

[TABLE]

is the noisy Bell state obtained tensor product $D\otimes I$ of the Davies and an identity map. It is assumed that only the CV qubit in $\chi_{CB}$ labeled by $C$ is coupled to the thermal Davies environment. Let us notice formal analogy of this scenario with a recently studied thermally modified teleportation protocol Kłoda and Dajka (2014) or entanglement swapping Dajka and Łuczka (2013). In particular

[TABLE]

where

[TABLE]

For $\rho_{i}=|\psi\rangle\langle\psi|$ pure, the action of the circuit in Fig.(5) is, in the presence of Davies environment Eq.(22) is given by the following transformation:

[TABLE]

where (notice that $U=U_{SYS,C}$),

[TABLE]

In a general case the transformation in Eq.(22) transforms non–orthogonal states into the states which remain non–orthogonal i.e. thermal Markovian noise divests the circuit in Fig.(5) of its ’paradoxial’ power (below we skip normalization constants):

[TABLE]

However, if an energy exchange between the CV qubit and the environment is negligible, i.e. the circuit operates in the pure dephasing regime $A=0$, the situation changes. Non–orthogonal input states are transformed into an output states which are orthogonal and hence can be distinguished:

[TABLE]

The reason of that becomes clear if one notices that both in the original noise–less case Brun and Wilde (2012) (i.e. when $D=I$) and in the pure decoherence limit $A=0$

[TABLE]

the transformation of non–orthogonal into the orthogonal states occurs since it follows that either $L_{i}|1\rangle\rightarrow|1\rangle$ and $L_{i}|+\rangle\rightarrow|0\rangle$ or $L_{i}|1\rangle=L_{i}|+\rangle=0$ for $i=I,\ldots,VI$.

In the above equations we used Eqs (22) and (15) but skip (non-vanishing) normalization constants which does not affect orthogonality of states. From Eq.(26) one infers that also in the case of postselected teleportation model pure dephasing plays a distinguished role exactly as it was in the Deutsch’s model.

V Uniqueness ambiguity

According to the Schauder’s fixed point theorem, there is a solution $\tau$ of the Deutsch’s condition Eq.(7). However, such a solution may not be unique resulting in the uniqueness ambiguity Deutsch (1991); Allen (2014). Using the Deutsch model of quantum time travel one faces with a problem which state $\tau$ (among many possibilities) is the ’proper’ one. The original proposal of David Deutsch Deutsch (1991) is the maximum entropy rule which states that the physical $\tau$ is the one which contains minimum information. This condition introduced ad hoc Allen (2014) is not universal and can be replaced by other proposals Politzer (1994); DeJonghe et al. (2010). As an example of a Deutsch’s circuit with uniqueness ambiguity can serve a circuit designed for the unproven theorem paradox. It is an example of a knowledge–generating circuit: a mathematician $M$, equipped with a knowledge about her/his modern mathematics read from a book $B$, becomes a time traveler $T$ and travels back in time in order to write the book $B$. A simplest example of a circuit playing such a role is presented in Fig.(6), cf. Ref. Allen (2014).

Such a circuit describes interaction of three qubits: $B$, $M$ which are CR and the last one $T$ which violates chronology. The interaction is given by a unitary

[TABLE]

and an input of the circuit is $|0\rangle_{B}|0\rangle_{M}$.

The Deutsch’s consistency condition Eq.(7) for this circuit is solved by a family of states

[TABLE]

where $\alpha\in[0,1]$ and hence is ambiguous. In Ref.Allen (2014) it is shown that an effect of depolarization can resolve this ambiguity.

Here we consider probably the most natural and omnipresent source of noise. We assume that the time travel is disturbed by a thermal environment. In such a case the state of the time traveler is a solution of Eq.(9) i.e. the time travel of $T$ is affected by thermal Davies noise. This solution is unique and is given by the Gibbs state:

[TABLE]

Let us notice that in the zero–temperature limit $p=0$ one obtains $\tau\rightarrow\tau_{0}$. In the $p=1/2$ limit one arrives at the state which maximizes entropy. Let us also notice that, for the model of Davies decoherence considered here, the Deutsch’s rule holds only approximately (in the regime of high temperature) and that in general the unique solution is not always maximally mixed.

The solution to the uniqueness ambiguity discussed here is essentially the same as in Ref. Allen (2014), but the source of noise that resolves the ambiguity is physically rather than formally motivated. In other words, a very natural condition that the CV qubit is weakly disturbed by its thermal environment can serve as physics–based justification for the choice of the solution $\tau$ of the Deutsch consistency condition Eq.(7), instead of otherwise ad hoc, maximum entropy rule introduced by David Deutsch in Ref.Deutsch (1991).

VI Summary

If time travels were possible, the world would be essentially different. Quantum cryptography Gisin et al. (2002) and in particular quantum key distribution Scarani et al. (2009) essentially changed basic objectives of communication which, comparing to a pre–quantum age, became much safer. However, most of the quantum no–go theorems – sine qua non conditions for security of quantum protocolsScarani et al. (2005, 2009) – originate from linearity of quantum mechanicsNielsen and I. Chuang (2000). An existence of closed time–like curves can change (almost) everything. There are quantum circuits which in a presence of CTCs can break security of quantum protocols. In this work we analyzed only one of them: the circuit designed to distinguish non–orthogonal qubit’s states and to break e.g. the B92 quantum crypto–protocol. Our aim was to check if and how such a ’paradoxial power’ becomes reduced by the omnipresent decoherence caused by thermal environment affecting time–traveling qubits. We consider only two among many approaches to CTCs: the one proposed by David Deutsch Deutsch (1991) and the second based on the post–selected teleportation protocol Svetlichny (2011); Lloyd et al. (2011a, b). Our intention was to investigate possibly wide class of open systems modeled in way which is both tractable and rigorous. That is why we assumed the general type of coupling to the environment: the Davies weak coupling approach Alicki and Lendi (2007). Using Davies approach one can describe the broadest class of open quantum systems with finite–dimensional space of states with the only restriction imposed: the coupling to environment must be weak. We showed for both Deutsch’s and post–selected model a distinctive role played by pure decoherence when, despite of a presence of environment and resulting information loss, circuits with CTCs do not lose their ’paradoxial power’ of distinguishing non–orthogonal quantum states. This result can serve as a potentially useful guideline for experimentalists who attempt to mimic circuits with CTCs in order to implement ’linearity–free quantum computations’. Physically pure decoherence describe open quantum systems operating at time scales which are short comparing with a time scale of a system–environment energy exchange Schuster et al. (2007).

In addition to practical there is also a fundamental aspect of decoherence which needs to be taken into account in all the applications of quantum phenomena Schlosshauer (2007). In the last section of our paper, inspired by Ref. Allen (2014), we investigated the circuit for an unproven theorem to show that thermal decoherence, present in any real system, can help to resolve the uniqueness ambiguity originating from non–uniqueness of a solution of the Deutch’s consistence condition Eq.(7). We showed that in a particular case considered in our work thermal noise not only allows to select the ’proper’ state of chronology violating qubit, which is not necessarily maximally mixed, but also justifies the Deutsch’s maximum entropy rule in the regime of high temperature.

There are many physical concepts affecting human imagination ranging from confining light black holes, dilatation of time, butterfly effect up to teleportation and the celebrated but piteous Schrödinger’s cat. All of them are strange but the closed time–like curves are stranger than the other. We hope that our work will modestly contribute to both better understanding hypothetical behavior of quantum systems in a presence of CTCs and, as a guideline, to experimental attempts of mimicking such systems.

Acknowledgments

The work has been supported by the NCN project UMO-2013/09/B/ST2/03382 (B.D. and M.R.) and the NCN grant 2015/19/B/ST2/02856 (J.D)

Appendix

In this Appendix we provide detail of calculations leading to the results presented in Sec III and V for the Deutchian model of CTC. Further in the Appendix we adopt the following notation $p(i,j)=|i\rangle\langle j|,p(ij,kl)=|ij\rangle\langle kl|$ and $p(ijm,kln)=|ijm\rangle\langle kln|$ where $i,j,k,l,m,n=0,1$ labels our computational basis. In this notation partial traces of a two–qubit matrix $X$ with respect to CR and CV qubits and for $x(ij,kl)=\mbox{Tr}(Xp(kl,ij))$ read as follows:

[TABLE]

and the partial traces of a three–qubit matrix $X$ (in Sec. V) with respect to CR qubits and for $x(ijm,kln)=\mbox{Tr}(Xp(klm,ijn))$ read as follows:

[TABLE]

The unitary coupling between the CV and CR qubits is a product of controlled Hadamard $H_{C}$ and the $SWAP$ i.e. $U=H_{C}\,SWAP$ where

[TABLE]

For an input $\rho_{i}=|-\rangle\langle-|=[p(0,0)+p(1,1)-p(1,0)-p(0,1)]/2$ the corresponding CV qubit

[TABLE]

satisfying Eq.(9) with real $a,b_{r},b_{i}$ can be calculated in the following steps: (i) An output of the circuit $X=U\rho_{i}\otimes\tau U^{\dagger}$ is traced with respect to the CR qubit and then (ii) subjected to thermal noise via Eq.(3) and finally (iii) selfconsistently compared to the input i.e.:

[TABLE]

resulting in a set of linear equations which allows to calculate the parameters $a,b_{r},b_{i}$. The CV qubit $\tau=\tau[1,1]p(0,0)+\tau[1,2]p(0,1)+\tau[2,1]p(1,0)+\tau[2,2]p(1,1)$ is then given by

[TABLE]

and the output $\rho_{f}$ of the circuit calculated via Eq.(8) reads as follows

[TABLE]

For an input $\rho_{i}=|0\rangle\langle 0|=p(0,0)$ the CV qubit, calculated via the same steps, is given by:

[TABLE]

and the corresponding output of the circuit calculated via Eq.(8) reads as follows

[TABLE]

In the case of the unproven theorem paradox considered in Sec. V the circuit acts as a unitary

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

An input $\rho_{i}=p(00,00)$ and $\tau$ is given in Eq.(39). Further one follows the same steps as in the previous case but with $X=U\rho_{i}\otimes\tau U^{\dagger}$ then traced with respect to the CR qubits according to Eq.(35) and subjected to thermal noise via Eq.(3) obtaining $\tau[1,1]=1-\tau[2,2]=p$ and $\tau[1,2]=\tau[2,1]=0$ corresponding to the Gibbs state of the time travelling qubit.