We introduce a reversible theory of exact entanglement manipulation by establishing a necessary and sufficient condition for state transfer under trace-preserving transformations that completely preserve the positivity of partial transpose (PPT). Under these free transformations, we show that logarithmic negativity emerges as the pivotal entanglement measure for determining entangled states' transformations, analogous to the role of entropy in the second law of thermodynamics. Previous results have proven that entanglement is irreversible under quantum operations that completely preserve PPT and leave open the question of reversibility for quantum operations that do not generate entanglement asymptotically. However, we find that going beyond the complete positivity constraint imposed by standard quantum mechanics enables a reversible theory of exact entanglement manipulation, which may suggest a potential incompatibility between the reversibility of entanglement and the fundamental principles of quantum mechanics.
We show that Einstein--Podolsky--Rosen--Bohm (EPR) and Greenberger--Horne--Zeilinger--Mermin (GHZ) states can not generate, through local manipulation and in the asymptotic limit, all forms of three--partite pure--state entanglement in a reversible way. The techniques that we use suggest that there may be a connection between this result and the irreversibility that occurs in the asymptotic preparation and distillation of bipartite mixed states.
Reversible entanglement transfer between light and matter is a crucial requisite for the ongoing developments of quantum information technologies. Quantum networks and their envisioned applications, e.g., secure communications beyond direct transmission, distributed quantum computing or enhanced sensing, rely on entanglement distribution between nodes. Although entanglement transfer has been demonstrated, a current roadblock is the limited efficiency of this process that can compromise the scalability of multi-step architectures. Here we demonstrate the efficient transfer of heralded single-ph…
We present a family of tri-partite entangled states that, in an asymptotical sense, can be reversibly converted into EPR states shared by only two of parties (say B and C), and tripartite GHZ states. Thus we show that bipartite and genuine tripartite entanglement can be reversibly combined in several copies of a single tripartite state. For such states the corresponding fractions of GHZ and of EPR states represent a complete quantification of their (asymptotical) entanglement resources. More generally, we show that the three different kinds of bipartite entanglement (AB, AC and BC EPR states) …
We propose a scheme of continuous-variable reversible telecloning, which broadcast the information of an unknown state without loss from a sender to several spatially separated receivers exploiting multipartite entanglement as quantum channels. In this scheme, quantum information of an unknown state is distributed into $M$ optimal clones and $M-1$ anticlones using $2M$% -partite entanglement. For the perfect quantum information distribution that is optimal cloning, $2M$-partite entanglement is required to be a maximum two-party entanglement. Comparing with the quantum telecloning proposed by L…
In this letter we introduce the problem of secrecy reversibility. This asks when two honest parties can distill secret bits from some tripartite distribution $p_{XYZ}$ and transform secret bits back into $p_{XYZ}$ at equal rates using local operation and public communication (LOPC). This is the classical analog to the well-studied problem of reversibly concentrating and diluting entanglement in a quantum state. We identify the structure of distributions possessing reversible secrecy when one of the honest parties holds a binary distribution, and it is possible that all reversible distributions…
New feasible cavity QED experiment is proposed to analyse reversible quantum decoherence in consequence of quantum complementarity and entanglement. Utilizing the phase selective manipulations with enviroment, it is demonstrated how the complementarity particularly induces a preservation of visibility, whereas quantum decoherence is more progressive due to pronounced entanglement between system and enviroment. This effect can be directly observed using the proposed cavity QED measurements.
We consider the manipulation of multipartite entangled states in the limit of many copies under quantum operations that asymptotically cannot generate entanglement. As announced in [Brandao and Plenio, Nature Physics 4, 8 (2008)], and in stark contrast to the manipulation of entanglement under local operations and classical communication, the entanglement shared by two or more parties can be reversibly interconverted in this setting. The unique entanglement measure is identified as the regularized relative entropy of entanglement, which is shown to be equal to a regularized and smoothed versio…
For a multipartite system, we sort out all possible entanglements, each of which is among a set of subsystems. Each entanglement can be measured by a generalized relative entropy of entanglement, which is conserved on average under reversible local operations and classical communication (LOCC) defined for all the parties. Then we derive a series of inequalities of different entanglements that have to be satisfied by any pure state which can be generated by reversible LOCC from the set of all GHZ-like states.
An exactly soluble non-linear interaction Hamiltonian is proposed to study fundamental properties of the entanglement dynamics for a coupled non-linear oscillators. The time-evolved state is obtained analytically for initial products of two coherent and two number states and relevant informations are extracted from the dynamics of various quantities like subsystem linear and Von Neumann entropies, quadrature mean values, variances and Q-functions. We determined the re-coherence time scales and found among the interaction terms present in the Hamiltonian the one responsible for the entanglement…
We propose a dynamically-dark-mode (DDM) scheme to realize the reversible quantum conversion between microwave and optical photons in an electro-optomechanical (EOM) model. It is shown that two DDMs appear at certain times during the dynamical evolution of the EOM model. It is demonstrated that the DDMs can induce two kinds of reversible and highly efficient quantum conversion between the microwave and optical fields, the conditional quantum conversion (CQC) and the entanglement-assisted quantum conversion (EAQC). The CQC happens at the condition of vanishing of the initial-state mean value of…
Quantum operations provide a general description of the state changes allowed by quantum mechanics. Simple necessary and sufficient conditions for an ideal quantum operation to be reversible by a unitary operation are derived in this paper. These results generalize recent work on reversible measurements by Mabuchi and Zoller [Phys. Rev. Lett. {\bf 76}, 3108 (1996)]. Quantum teleportation can be understood as a special case of the problem of reversing quantum operations. We characterize completely teleportation schemes of the type proposed by Bennett {\it et al.} [Phys. Rev. Lett. {\bf 70}, 189…
Dissipative processes in physics are usually associated with non-unitary actions. However, the important resource of entanglement is not invariant under general unitary transformations, and is thus susceptible to unitary "dissipation". In this note we discuss both unitary and non-unitary dissipative processes, showing that the former is ultimately of value, since reversible, and enables the production of entanglement; while even in the presence of the latter, more conventional non-unitary and non-reversible, process there exist nonetheless invariant entangled states.
Transformations from pure to mixed states are usually associated with information loss and irreversibility. Here, a protocol is demonstrated allowing one to make these transformations reversible. The pure states are diluted with a random noise source. Using this protocol one can study optimal transformations between states, and from this derive the unique measure of information. This is compared to irreversible transformations where one does not have access to noise. The ideas presented here shed some light on attempts to understand entanglement manipulations and the inevitable irreversibility…
We derive quantitative relations among several naturally defined measures of classical and nonclassical correlations in a bipartite quantum state. We also obtain an upper bound of entanglement irreversibility and a sufficient condition for reversible entanglement. The additivity of entanglement of formation is directly related to the additivity of quantum discord as well as a certain measure of classical correlation.
Coherent and reversible storage of multi-photon entanglement with a multimode quantum memory is essential for scalable all-optical quantum information processing. Although single photon has been successfully stored in different quantum systems, storage of multi-photon entanglement remains challenging because of the critical requirement for coherent control of photonic entanglement source, multimode quantum memory, and quantum interface between them. Here we demonstrate a coherent and reversible storage of biphoton Bell-type entanglement with a holographic multimode atomic-ensemble-based quantu…
Landauer's principle asserts that any computation has an unavoidable energy cost that grows proportionally to its degree of logical irreversibility. But even a logically reversible operation, when run on a physical processor that operates on different energy levels, requires energy. Here we quantify this energy requirement, providing upper and lower bounds that coincide up to a constant factor. We derive these bounds from a general quantum resource-theoretic argument, which implies that the initial resource requirement for implementing a unitary operation within an error~$ε$ grows like $1/\sqr…
Reversible algorithms play a crucial role both in classical and quantum computation. While for a classical bit the only nontrivial reversible operation is the bit-flip, nature is far more versatile in what it allows to do to a quantum bit. The reversible operations that a quantum computer can perform on a qubit are group of linear unitary transformations. However, laws of quantum mechanics prohibit implementation of anti-linear anti-unitary gates, even though they are perfectly reversible. Here we show that such a restriction on possible set of reversible operations is, remarkably, a fundament…
We analyze the dynamics of a system qubit interacting by means a sequence of pairwise collisions with an environment consisting of just two qubits. We show that the density operator of the qubits approaches a common time averaged equilibrium state, characterized by large fluctuations, only for a random sequence of collisions. For a regular sequence of collisions the qubitstates of the system and of the reservoir undergo instantaneous periodic oscillations and do not relax to a common state. Furthermore we show that pure bipartite entanglement is developed only when at least two qubits are init…
We propose a protocol, based on entanglement procedures recently suggested by [D. Jaksch et al., Phys. Rev. Lett. 82, 1975 (1999)], which allows the teleportation of an unknown state of a neutral atom in an optical lattice to another atom in another site of the lattice, without any irreversible detection.
We investigate the irreversibility of entanglement distillation for a symmetric d-1 parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension d x d. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, such that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure state entanglement and every operation is allowed, which preserves the positivity of the partial…
I give a pedagogical account of Shor's nine-bit code for correcting arbitrary errors on single qubits, and I review work that determines when it is possible to maintain quantum coherence by reversing the deleterious effects of open-system quantum dynamics. The review provides an opportunity to introduce an efficient formalism for handling superoperators. I present and prove some bounds on entanglement fidelity, which might prove useful in analyses of approximate error correction.
Consider two quantum systems A and B interacting according to a product Hamiltonian H = H_A x H_B. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all non-local features of a product Hamiltonian -- including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians -- depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for…
In the statistical description of dynamical systems, an indication of the irreversibility of a given state change is given geometrically by means of a (pre-)ordering of state pairs. Reversible state changes of classical and quantum systems are shown to be represented by isometric state transformations. An operational distinction between reversible and irreversible dynamics is given and related to the geometric characterisation of the associated state transformations.
It has long been known that to minimise the heat emitted by a deterministic computer during it's operation it is necessary to make the computation act in a logically reversible manner\cite{Lan61}. Such logically reversible operations require a number of auxiliary bits to be stored, maintaining a history of the computation, and which allows the initial state to be reconstructed by running the computation in reverse. These auxiliary bits are wasteful of resources and may require a dissipation of energy for them to be reused. A simple procedure due to Bennett\cite{Ben73} allows these auxiliary bi…
We experimentally demonstrate phase-insensitive linear optical amplification which preserves the idler at the output. Since our amplification operation is unitary up to small excess noise, it is reversible beyond the classical limit. The entanglement between the two output modes is the resource for the reversibility. The amplification gain of 2.0 is demonstrated. In addition, combining this amplifier with a beamsplitter, we also demonstrate approximate cloning of coherent states where an anticlone is present. We investigate the reversibility by reconstructing the initial state from the output …
We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical logic from the 1940s. It is a step toward the ambitious goal of classifying all possible quantum gate sets acting on qubits. Our theorem implies a linear-time algorithm (which we have implemented), that takes as input the truth tables of reversible gates G and…
Quantum operations provide a general description of the state changes allowed by quantum mechanics. The reversal of quantum operations is important for quantum error-correcting codes, teleportation, and reversing quantum measurements. We derive information-theoretic conditions and equivalent algebraic conditions that are necessary and sufficient for a general quantum operation to be reversible. We analyze the thermodynamic cost of error correction and show that error correction can be regarded as a kind of ``Maxwell demon,'' for which there is an entropy cost associated with information obtain…
A reversible logic has application in quantum computing. A reversible logic design needs resources such as ancilla and garbage qubits to reconfigure circuit functions or gate functions. The removal of garbage qubits and ancilla qubits are essential in designing an efficient quantum circuit. In the literature, there are multiple designs that have been proposed for a reversible multiplication operation. A multiplication hardware is essential for the circuit design of quantum algorithms, quantum cryptanalysis, and digital signal processing (DSP) applications. The existing designs of reversible qu…
General relativity allows for the existence of closed time-like curves, along which a material object could travel back in time and interact with its past self. This possibility raises the question whether certain initial conditions, or more generally local operations, lead to inconsistencies and should thus be forbidden. Here we consider the most general deterministic dynamics connecting classical degrees of freedom defined on a set of bounded space-time regions, requiring that it is compatible with arbitrary operations performed in the local regions. We find that any such dynamics can be rea…
We introduce and study the concept of a reversible transfer of the quantum state of two internally-translationally entangled fragments, formed by molecular dissociation, to a photon pair. The transfer is based on intracavity stimulated Raman adiabatic passage and it requires a combination of processes whose principles are well established.
We consider an open bipartite quantum system with dissipative Lindblad type dynamics. In order to study the entanglement of the stationary states, we develop a perturbative approach and apply it to the physically significant case when a purely dissipative perturbation is added to the unperturbed generator which by itself would produce reversible unitary dynamics.
Pure state entanglement transformations have been thought of as irreversible, with reversible transformations generally only possible in the limit of many copies. Here, we show that reversible entanglement transformations do not require processing on the many copy level, but can instead be undertaken on individual systems, provided the amount of entanglement which is produced or consumed is allowed to fluctuate. We derive necessary and sufficient conditions for entanglement manipulations in this case. As a corollary, we derive an equation which quantifies the fluctuations of entanglement, whic…
We show that the entanglement of cost and entanglement of distillation can be vague when we consider a more general form of entanglement manipulation in which we collectively deal with not only states of our concern but also other states. We introduce the most general entanglement manipulation in which the formation and distillation can be simultaneously performed. We show that in a certain case entanglement manipulations are reversible with respect to the most general entanglement manipulation. This broadens our scope of vision on the irreversibility of entanglement manipulations.
Our main models of computation (the Turing Machine and the RAM) make fundamental assumptions about which primitive operations are realizable. The consensus is that these include logical operations like conjunction, disjunction and negation, as well as reading and writing to memory locations. This perspective conforms to a macro-level view of physics and indeed these operations are realizable using macro-level devices involving thousands of electrons. This point of view is however incompatible with quantum mechanics, or even elementary thermodynamics, as both imply that information is a conserv…
The reversible transfer of quantum states of light in and out of matter constitutes an important building block for future applications of quantum communication: it allows synchronizing quantum information, and enables one to build quantum repeaters and quantum networks. Much effort has been devoted worldwide over the past years to develop memories suitable for the storage of quantum states. Of central importance to this task is the preservation of entanglement, a quantum mechanical phenomenon whose counter intuitive properties have occupied philosophers, physicists and computer scientists sin…
We show how to perform reversible universal quantum computation on a translationally invariant pure state, using only global operations based on next-neighbor interactions. We do not need not to break the translational symmetry of the state at any time during the computation. Since the proposed scheme fulfills the locality condition of a quantum cellular automata, we present a reversible quantum cellular automaton capable of universal quantum computation.
Entanglement is one of the most striking features of quantum mechanics, and yet it is not specifically quantum. More specific to quantum mechanics is the connection between entanglement and thermodynamics, which leads to an identification between entropies and measures of pure state entanglement. Here we search for the roots of this connection, investigating the relation between entanglement and thermodynamics in the framework of general probabilistic theories. We first address the question whether an entangled state can be transformed into another by means of local operations and classical co…
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT-operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this easily computable entanglement measure. As examples we discuss general Werner states and arbitrary bi-partite Gaussian states. Equipped with this result we then prove that for the anti-symmetric Werner state PPT-cost and PPT-entanglement of distillation coincide giving the first example of a truly mixed state for which entanglement ma…
We consider the transformation of multisystem entangled states by local quantum operations and classical communication. We show that, for any reversible transformation, the relative entropy of entanglement for two parties must remain constant. This shows, for example, that it is not possible to convert 2N three party GHZ states into 3N singlets, even in an asymptotic sense. Thus there is true three-party non-locality (i.e., not all three-party entanglement is equivalent to two-party entanglement). Our results also allow us to make {\em quantitative} statements about concentrating multi-particl…
We give the logical description of a new kind of quantum measurement that is a reversible operation performed by an hypothetical insider observer, or, which is the same, a quantum measurement made in a quantum space background, like the fuzzy sphere. The result is that the non-contradiction and the excluded middle principles are both invalidated, leading to a paraconsistent, symmetric logic. Our conjecture is that, in this setting, one can develop the adequate logic of quantum computing. The role of standard quantum logic is then confined to describe the projective measurement scheme.
We study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distilla…
We consider composability of quantum channels from a limited amount of entanglement via local operations and classical communication (LOCC). We show that any $k$-partially entanglement breaking channel can be composed from an entangled state with Schmidt number of $k$ via one-way LOCC. From the entanglement assisted construction we can reach an alternative definition of partially entanglement breaking channels.
Based on the monogamy of entanglement, we develop the technique of quantum conditioning to build an {\it additive} entanglement measure: the conditional entanglement of mutual information. Its {\it operational} meaning is elaborated to be the minimal net "flow of qubits" in the process of partial state merging. The result and conclusion can also be generalized to multipartite entanglement cases.
Proposed below is a reversible digital computer modeled after the natural behavior of a quantum system. Using approaches usually reserved for idealized quantum computers, the Reversible CAM, or State Vector Parallel (RSVP) processor can easily find keywords in an unstructured database (that is, it can solve a needle in a haystack problem). The RSVP processor efficiently solves a SAT (Satisfiability of Boolean Formulae) problem; also it can aid in the solution of a GP (Global Properties of Truth Table) problem. The power delay product of the RSVP processor is exponentially lower than that of a …
Bipartite operations underpin both classical communication and entanglement generation. Using a superposition of classical messages, we show that the capacity of a two-qubit operation for error-free entanglement-assisted bidirectional classical communication can not exceed twice the entanglement capability. In addition we show that any bipartite two-qubit operation can increase the communication that may be performed using an ensemble by twice the entanglement capability.
We show that the classification of bi-partite pure entangled states when local quantum operations are restricted yields a structure that is analogous in many respects to that of mixed-state entanglement. Specifically, we develop this analogy by restricting operations through local superselection rules, and show that such exotic phenomena as bound entanglement and activation arise using pure states in this setting. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. In particular, we demonstrate that several types of quantum optica…
Two-qubit operations may be characterized by their capacities for communication, both with and without free entanglement, and their capacity for creating entanglement. We establish a set of inequalities that give an ordering to the capacities of two-qubit unitary operations. Specifically, we show that the capacities for entanglement creation and bidirectional communication without entanglement assistance are at least as great as half the bidirectional communication capacity with entanglement assistance. In addition, we show that the bidirectional communication that can be performed using an en…
We consider reversible work extraction from identical quantum batteries. From an ensemble of individually passive states, work can be produced only via global unitary (and thus entangling) operations. However, we show here that there always exists a method to extract all possible work without creating any entanglement, at the price of generically requiring more operations (i.e. additional time). We then study faster methods to extract work and provide a quantitative relation between the amount of generated multipartite entanglement and extractable work. Our results suggest a general relation b…
We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the $3\otimes3$ antisymmetric subspace, the amount of dis…
Classical engines turn thermal resources into work, which is maximized for reversible operations. The quantum realm has expanded the range of useful operations beyond energy conversion, and incoherent resources beyond thermal reservoirs. This is the case of entanglement generation in a driven-dissipative protocol, which we hereby analyze as a continuous quantum machine. We show that for such machines the more irreversible the process the larger the concurrence. Maximal concurrence and entropy production are reached for the hot reservoir being at negative effective temperature, beating the limi…
We show that the entanglement of a $2\times 2$ bipartite state can be improved and maximized probabilistically through single-qubit operations only. An experiment is proposed and it is numerically simulated.
Operator entanglement of two-qubit joint unitary operations is revisited. Schmidt number is an important attribute of a two-qubit unitary operation, and may have connection with the entanglement measure of the unitary operator. We found the entanglement measure of two-qubit unitary operators is classified by the Schmidt number of the unitary operators. The exact relation between the operator entanglement and the parameters of the unitary operator is clarified too.
