Local cloning of entangled states

Local cloning of entangled states Vlad Gheorghiu Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. March 16, 2010 Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 1 / 18

Outline 1 Introduction 2 LOCC vs Separable Operations 3 Local cloning entangled states The problem Previously known results Necessary conditions Local cloning of entangled qubits Local cloning of locally-shifted entangled qudits 4 Conclusions 5 Bibliography A summary of this talk is available online at http://quantum.phys.cmu.edu/qip Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 2 / 18

Introduction Introduction Hypothesis: Let ψ 0 and ψ 1 be two non-identical known quantum states, living in a Hilbert space H. Goal: build a quantum xerox that can make perfect copies of ψ 0 and ψ 1, without knowing which state will be fed in at a given time. Answer: Impossible, unless ψ 0 ψ 1 = 0. This is the famous no-cloning theorem (W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802803). Proof: Ingredients: Kraus operators F k, quantum resource (or blank ) φ. Idea: F k ψ j φ = σ kj ψ j ψ j, with k σ kj 2 = 1. Details: exercise. Can you think of a simpler proof? Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 3 / 18

LOCC vs Separable Operations LOCC vs Separable Operations Assume H = H A H B. LOCC paradigm: 1 Alice performs a generalized measurement with Kraus operators {A (0) k }, and obtains the result k 1. 2 Bob learns the result k 1 through a classical channel (his iphone). Depending on the result k 1 Alice gets, Bob performs {B (k1) k }. 3 Now Bob communicates the result k 2 to Alice, who performs {A (k1,k2) k }, and so on... 4 Tree-like structure. It eventually stops after a finite number of steps. LOCC is a composition of local quantum operations. The overall quantum operation is described by a (often quite large) set of product Kraus operators, {A m B m }, satisfying the closure condition A ma m B mb m = I A I B. m Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 4 / 18

LOCC vs Separable Operations Separable Operation (SEP): a quantum operation Λ whose Kraus operators are product, Λ(ρ) = m=0(a m B m )ρ(a m B m ). Every LOCC is (trivially) a SEP. The converse is not true. There are SEPs that are not LOCC, so LOCC is a strictly smaller subset of SEPs. We recently proved [1] in [V. Gheorghiu and R. B. Griffiths, Phys. Rev. A 78 (R) 020304, 2008] that SEPs and LOCC coincide when the input is a pure state. SEPs are sometimes more convenient to use because of their mathematical simplicity. Every result proven for SEPs automatically holds for LOCC. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 5 / 18

Local cloning entangled states The problem Local cloning entangled states: The problem Consider a set S = { ψ i AB } of partially entangled states, in H A H B. Let φ ab be the resource (blank state) in H a H b. Goal: implement ψ i AB φ ab ψ i AB ψ i ab, i deterministically using a SEP. Basic questions: Which states can be cloned? What are the necessary resources? (how entangled the blank has to be?) If one allows sufficient entanglement, i.e. 2 maximally entangled states (MES), Alice and Bob can perform 2 rounds of teleportation, and the problem becomes trivial. In the following, we will assume that all states, including the resource, have Schmidt rank D, the same as the dimension of the local Hilbert space H A, H a, H B and H b. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 6 / 18

Local cloning entangled states Previously known results Previously known results D = 2 [2]. Any 2 Bell states (and not more) can be locally cloned using 1 MES blank. Extension to (prime) D > 2 [3, 4]. D MES in a D D system can be locally cloned using 1 MES blank if and only if the states are locally-shifted, ψ i = 1 D 1 r i A r B. D r=0 No more than D MES can be locally cloned with 1 MES blank. Extension to (prime) D > 2, partially entangled states (PES) [5]. D locally-shifted PES ψ i = 1 D 1 λr r i A r B D r=0 in a D D system can be locally cloned using 1 MES blank. Although the authors claim that the above is also a necessary condition, their proof is incorrect. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 7 / 18

3 No information in the environment about which state was cloned Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 8 / 18 Necessary conditions Local cloning entangled states Theorem If the local cloning of S = { ψ i AB } D 1 i=0 using a blank state φ ab H a H b is possible by a separable operation, then the following must hold: 1 Ent( φ ) max Ent( ψ i ), i where Ent( ) denotes any pure-state entanglement measure 2 All states in S must be equally entangled with respect to the G-concurrence measure, C G ( ψ i AB ) = C G ( ψ j AB ), i, j, where C G ( ψ AB ) = D ( D 1 r=0 λ r ) 1/D

Local cloning entangled states Interconvertible states 2 bipartite states ψ and φ are called interconvertible under SEP if either ψ φ or φ ψ (or both) is possible by a deterministic SEP. ψ φ by a SEP if and only if the following majorization condition holds λ 0 µ 0 λ 0 + λ 1 µ 0 + µ 1. λ 0 + λ 1 +... + λ D 1 µ 0 + µ 1 +... + µ D 1 where λ 0 λ 1... λ D 1 are the Schmidt coefficients of ψ and µ 0 µ 1... µ D 1 are the Schmidt coefficients of φ. Any 2 qubit entangled states are interconvertible. Not true for D > 2. Majorization induces a partial order. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 9 / 18

Corollary Local cloning entangled states Majorization is often denoted as λ µ. Important concept, see Nielsen and Chuang (Chapter 12.5) for more details... Corollary We have the following corollary Let S = { ψ i AB } D 1 i=0 be a set of interconvertible states under SEP. If the local cloning of S using a blank state φ ab H a H b is possible by a separable operation, then all states in S must share the same set of Schmidt coefficients. In other words: any 2 states in S must either share the same set of Schmidt coefficients or otherwise be incomparable (not interconvertible). Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 10 / 18

Local cloning entangled states Local cloning of entangled qubits For D = 2 (local cloning of qubit entangled states), we can further show that ψ 0 AB = λ 00 AB + 1 λ 11 AB ψ 1 AB = λ 01 AB + e iϕ 1 λ 10 AB are the only possible states that can be locally cloned. An LOCC protocol that clones them with an MES blank already exists [5]. It is not known if one can use a PES blank. We derived a lower bound on the necessary entanglement of the blank state. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 11 / 18

Local cloning entangled states Minimum required entanglement of the blank Without loss of generality, the blank can be chosen φ ab = γ 00 ab + 1 γ 11 ab, where γ is the largest Schmidt coefficient, 1/2 γ < 1. 1.0 0.9 0.8 Γ 0.7 0.6 0.5 0.5 0.6 0.7 0.8 0.9 1.0 Λ Figure: Theorem 1 (i) forbids cloning in the light yellow region γ > λ. We can extend the forbidden region to γ > λ 2 + (1 λ) 2 by the additional light green area. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 12 / 18

Local cloning entangled states Local cloning of locally-shifted entangled qudits Consider now a set of D shifted entangled states S = { ψ i AB } D 1 i=0 on H A H B, where the dimension of both Hilbert spaces H A and H B is equal to D, where e iϑ ir ψ i AB = D 1 r=0 are complex phases. λr e iϑ ir r i A r B, As mentioned before, it was recently proved in [5] that the local cloning of the above set of states (with all phases e iϑ ir chosen to be 1) is possible using a MES blank. It is not yet known if one can use less than a maximally entangled blank state. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 13 / 18

Local cloning entangled states Minimum required entanglement of the blank Theorem (Qudit shifted entangled states) If the local cloning of S = { ψ i AB } D 1 i=0 using a blank state φ ab is possible by a separable operation, then 1 The entanglement of the blank state has to be at least E( φ ab ) H({q x }) min Ψ AB Span(S) E( Ψ AB ), where E( ) denotes the entropy of entanglement and H({q x }) is the Shannon entropy of the probability distribution {q x }, q x := r λ r λ r x }, x q x = 1. 2 As a corollary, if there exist a product state in the span of the states of S, the blank state has to have a strictly greater entanglement than that of any of the states in S, E( φ ab ) H({q x }) > E( ψ i AB ), i. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 14 / 18

Local cloning entangled states The proof is based on the fact that if ψ i AB is deterministically mapped to ψ i AB ψ i ab, then a superposition of the ψ i AB will be deterministically mapped to a corresponding superposition of the ψ i AB ψ i ab. We show that the output superposition is always more entangled than the states of S, often by a wide margin, and if the input superposition has vanishing entanglement, then all the entanglement in the output must have been present to begin with in the blank state. We still do not know any protocol that can use a PES blank state, and conjecture that a MES blank is necessary for the deterministic local cloning. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 15 / 18

Local cloning entangled states D = 3 (qutrit) example Figure: Local cloning of 3 qutrit locally-shifted entangled states. The vertical axis labels the minimum required entanglement difference between the blank and the states to be cloned, E( φ ) E( ψ i ); the horizontal axes label the Schmidt coefficients λ 0 and λ 1 of the states to be cloned. There is no gap in the required entanglement difference only when λ 0 = 1 or λ 1 = 1 (product states) or λ 0 = λ 1 = 1/3 (maximally entangled states), cases that we exclude. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 16 / 18

Conclusions Conclusions Found a set of necessary conditions for the local cloning by SEPs. No general sufficient conditions yet... Do not know any local cloning protocol that works with a PES blank. Based on numerical studies, conjecture that a MES blank is necessary! Cannot prove. Rewards for a solution or a counterexample :) Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 17 / 18

Bibliography Vlad Gheorghiu and Robert B. Griffiths. Separable operations on pure states. Phys. Rev. A, 78(2):020304, 2008. Sibasish Ghosh, Guruprasad Kar, and Anirban Roy. Local cloning of bell states and distillable entanglement. Phys. Rev. A, 69(5):052312, 2004. Fabio Anselmi, Anthony Chefles, and Martin B. Plenio. Local copying of orthogonal entangled quantum states. New J. Phys., 6:164, 2004. Masaki Owari and Masahito Hayashi. Local copying and local discrimination as a study for nonlocality of a set of states. Phys. Rev. A, 74(3):032108, 2006. Alastair Kay and Marie Ericsson. Local cloning of arbitrarily entangled multipartite states. Phys. Rev. A, 73(1):012343, 2006. Vlad Gheorghiu (CMU) Local cloning of entangled states March 16, 2010 18 / 18