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Separable States can be Used to Distribute Entanglement Toby Cubitt 1, Frank Verstraete 1, Wolfgang Dür 2, and Ignacio Cirac 1 1 Max Planck Institüt für Quantenoptik, Garching, Germany 2 Ludwig Maximilians Universität, Munich, Germany Conclusions Separable states can be used to distribute entanglement. We are forced to abandon intuitive pictures of entanglement being sent through a quantum channel. (Discrete system) Similarly, we must abandon intuitive pictures of flows of entanglement. (Continuous system) (at least for general – i.e. mixed – states...) Impossibility for Pure States For pure states, the ancilla does have to become entangled and the effect is impossible. E.g. for the continuous system: Consider evolving separable state for an infinitesimal time-step under a Hamiltonian of the form Multiplying by gives, so which is separable. The condition on the separability of the ancilla is then Therefore, if the ancilla is required to remain separable, and can not become entangled. a B A Discrete and Continuous Systems A continuous process can always be turned into discrete procedure by expanding the evolution operator using the Trotter formula: Continuous system: B A a Particles and interact continuously via a mediating particle. For certain initial states, remains separable from and at all times, yet and become entangled. AB a AB A B a Discrete system: (2) is sent to Bob, who interacts it with his particle. B a (3) At the end they share some entanglement, although has remained separable throughout. a (1) Alice interacts an ancilla with her particle. A a Continuous Example B A a Consider e.g. a system of 2 qubits, one qutrit, and Hamiltonian a BA If the interactions are weak ( small), the most significant processes in the evolution of the system are second order virtual transitions of the ancilla from an energy level back to itself accompanied by an interaction between and. If starts in an energy eigenstate, it will be left in that eigenstate, not entangled with or. It will become entangled due to higher order processes, but this entanglement will be weak. The interaction between and, however, can strongly entangle them. We then have a situation close to the desired one: and become entangled while remains almost separable. However, assume there is noise in the system. Mixing a barely entangled state with an amount of noise proportional to the entanglement destroys the entanglement. We can therefore add sufficient noise to remove any entanglement with, but not so large that it destroys the strong entanglement between and, thereby achieving the effect. B A a a B A B A B A a a B A Discrete Example (0) Alice and Bob prepare the separable initial state (1) Alice applies a CNOT operation to her particle and the ancilla (with as the control qubit), preparing A aa ( except ) (2) Alice sends a to Bob, who applies a CNOT to his particle and (with as control), preparing B aa (3) Alice and Bob share entanglement, which can be extracted e.g. by measuring ancilla in the computational basis. The ancilla has remained separable throughout. a Einstein associated entanglement with “spooky action at a distance ”. Modern quantum information theory, however, takes a different view: entanglement is a physical quantity, which can be used as a resource. We investigate more closely what is required to entangle two distant particles: Surprisingly, we show that entanglement can be created by sending an ancilla that never becomes entangled. Introduction LOCC is not sufficient by definition. One might expect that sending a separable ancilla is not sufficient. Sending an entangled ancilla trivially is sufficient – entanglement is sent through the channel. Explanation Entanglement properties of tripartite systems play the key role in understanding the counter-intuitive effect presented in the examples. For pure states, entanglement properties of bipartite partitions are inter-dependent. E.g. However, for mixed states the entanglement properties of bipartite partitions are independent. E.g. a state can simultaneously have the following properties: & A a B A a B A a B & A a B A a B A a B In fact, step (1) of the discrete example creates a state with precisely these properties. Step (2) then entangles with (i.e. ) without entangling with (i.e. throughout ). The continuous example goes through a similar sequence of entanglement properties. B A a a B A

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