Quantum no-deletion theorem for entangled states

10 June 2002

Physics Letters A 298 (2002) 225–228 www.elsevier.com/locate/pla

Quantum no-deletion theorem for entangled states Jian Feng a,b,c,∗ , Yun-feng Gao c , Ji-suo Wang a,c , Ming-sheng Zhan a a State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,

Chinese Academy of Sciences, Wuhan 430071, PR China b Laser Spectroscopy Laboratory, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, PR China c Institute of Optical Communication, Liaocheng Teachers University, Liaocheng 252059, Shandong, PR China 1

Received 16 October 2001; received in revised form 12 December 2001; accepted 12 December 2001 Communicated by P.R. Holland

Abstract We demonstrate that there are no physical means for deleting an unknown entangled state against its copy in either pure or mixed state case.  2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.-a; 03.65.Bz; 89.70.+c Keywords: Deletion; Entangled state

The quantum mechanical principles have been used to realize quantum computers, quantum teleportation and quantum cryptography. These principles not only enhance the possibility of information processing but also put on some limitations on manipulations with quantum information. These limitations tell us what we can do with the information contained in unknown states and what we cannot. For instance, linearity of quantum theory forbids perfect cloning [1,2] and exact disentanglement [3,4] of arbitrary unknown state. The unitarity of quantum theory prohibits accurate cloning of non-orthogonal states with certainty [5,6]. Recently, the quantum no-deleting principle for the pure states of a single qubit was presented [7]. It tells us that lin-

* Corresponding author.

E-mail address: [email protected] (J. Feng). 1 Mailing address.

earity of quantum theory forbids the perfect deletion of an unknown pure state of a qubit against its copy. This principle has been extended to the case of a single qudit (a qudit is a d-dimensional single quantum system). It was shown that the complete deletion of an unknown pure state of a qudit against its copy is impossible [8]. It is necessary to notice that above quantum no-deleting principles are limited to the pure states of a single qubit or qudit. The next step will thus be to find out how quantum mechanics poses the restriction on deleting arbitrary entangled state against its copy since the entangled states are central to the study of quantum non-locality, quantum teleportation, quantum cryptography, quantum dense coding and so on. In this Letter, we formulate and prove an impossibility theorem that extends no-deleting principle for the pure states of a single qubit (qudit) to arbitrary pure and mixed entangled states. The theorem answers the question: Are there any physical means for deleting

0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 4 6 - 5

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J. Feng et al. / Physics Letters A 298 (2002) 225–228

an unknown entangled state against its copy in either pure or mixed state case? In the following, we define the quantum deleting operations and demonstrate the quantum no-deleting theorem of entangled states for pure and mixed states, respectively. First, we consider the case of entangled pure state. The quantum deleting operation is defined for two initially identical entangled pure states |Φ|Φ such that the linear operator T acts on the combined Hilbert space of input states and ancilla and deletes one of two copies of |Φ together with keeping another copy intact. It is defined by
 T |Φ|Φ|A = |Φ|β|AΦ , (1) where |A and |AΦ  are the initial and final states of the ancilla, |β is the blank state in Hilbert space with the same dimension as the one where |Φ lives in. |Φ, |β, |A and |AΦ  are normalized states. It is necessary to emphasize that although the final state of ancilla |AΦ  may generally depend on the original state |Φ, in actuality it should not. Otherwise, the copy would be swapped onto the final state of the ancilla as shown below. However, the swapping of an unknown copy onto ancilla is explicitly excluded as a proper quantum deletion [7,8] since the swapping is just hiding of quantum information in the final state of the ancilla which can be retrieved by a unitary transformation. It should be also pointed that if two input pure states are different then the deleting processing can yield an arbitrary state such as 
T |Φi |Φj |A = |Ψij A  (i, j = 1, 2, i = j ), (2) where |Ψij A  might be any normalized state of the combined input–ancilla system. The Schmidt decomposition [9,10] allows one to always write arbitrary entangled pure state of a bipartite system as a linear combination of two orthogonal product states built from local base states. For twoqubit (denoted as A and B) it reads [11,12]

where |AΦ1  and |AΦ2  are the normalized final states of ancilla. Therefore T also transforms copies of arbitrary entangled pure state |Φ which can always be represented by Eq. (3) with a and b being unknown non-zero real numbers in the same manner as Eqs. (4), i.e.,
 T |Φ|Φ|A = |Φ|β|AΦ  = a|Φ1 |β|AΦ  + b|Φ2 |β|AΦ . (5) On the other hand, by the linearity of transformation T we have
 T |Φ|Φ|A = a 2 |Φ1 |β|AΦ1  + b2|Φ2 |β|AΦ2  
+ ab |Ψ12A  + |Ψ21A  , (6) where |Ψ12A  and |Ψ21A  are two arbitrary states of the combined input–ancilla system according to Eq. (2). Since the state |Ψ12A  + |Ψ21A  is independent of a and b, therefore the only solution to both Eqs. (5) and (6) is
 |Ψ12A  + |Ψ21A  = |Φ1 |AΦ2  + |Φ2 |AΦ1  |β, (7) |AΦ  = a|AΦ1  + b|AΦ2 .

(3)

where |Φ1  ≡ |0A ⊗ |0B and |Φ2  ≡ |1A ⊗ |1B are two orthonormal product states, both local bases {|0A , |1A } and {|0B , |1B } are orthonormal eigenbases of the reduced density matrices ρA = TrB |ΦΦ|,

(8)

Furthermore, since the final state (5) has to be normalized for all possible a and b, it follows that AΦ1 |AΦ2  = 0.

|Φ = a|0A ⊗ |0B + b|1A ⊗ |1B ≡ a|Φ1  + b|Φ2 ,

ρB = TrA |ΦΦ|, respectively, and therefore depends on the state |Φ. The Schmidt coefficients a and b are non-zero positive real numbers with a 2 +b2 = 1 which guarantees the arbitrary state |Φ to be entangled and normalized. If a or b is equal to zero, the state |Φ reduces to a non-entangled (separable) state. Suppose that a transformation T for two copies of above two orthogonal product states |Φ1  and |Φ2  exists such that
 T |Φ1 |Φ1 |A = |Φ1 |β|AΦ1 ,
 T |Φ2 |Φ2 |A = |Φ2 |β|AΦ2 , (4)

(9)

Eqs. (8) and (9) indicate that the transformation T merely swaps the copy |Φ onto a four-dimensional subspace of the ancilla. As demonstrated above, the swapping has been explicitly excluded as a proper quantum deletion. Therefore, the final state of ancilla |AΦ  cannot in fact depend on the original state |Φ.

J. Feng et al. / Physics Letters A 298 (2002) 225–228

Except for above unreasonable unique solution (7)– (9), Eqs. (5) and (6) generally contradict each other, which implies that the transformation T cannot exist. Hence, the linearity of quantum mechanics forbids deleting an unknown entangled pure state against its copy. This is called quantum no-deletion theorem for entangled pure states. In the above proof, the entangled nature of the state |Φ plays an essential role. Since any entangled pure state of two-qubit system can always be represented by Eq. (3) with both a and b are non-zero positive real numbers, the entangled property of the state |Φ is reflected in a = 0 and b = 0. If a or b is equal to zero, the state |Φ reduces to a non-entangled state and Eqs. (5) and (6) no longer contradict each other but always hold simultaneously. So the entangled nature of the state |Φ play a crucial function in the above proof. Next, we discuss the situation of entangled mixed states. First, we review the concepts of mixed state and entangled mixed state. In practice, we often deal with mixed states due to decoherence effects. When the state of any quantum system is |Ψi  with probability pi (i = 1, 2, . . . , N), it is said that the system is in a mixed state which must be described by a density matrix defined as follows [13]: ρ=

N 

pi |Ψi Ψi |.

(10)

i=1

This definition is suitable for both a single quantum system and a composite quantum system. A mixed state of a composite quantum system consisting of two subsystems is called separable (or non-entangled) if it can be written in the form ρ=

k  j =1

Pj ρjA ⊗ ρjB ,

Pj
0,

k 

Pj = 1,

(11)

j =1

where ρjA and ρjB denote the density matrices of subsystems, respectively. Conversely, ρ is entangled (nonseparable) if it cannot be written in this form [14,15]. Physically, a state described by an entangled (nonentangled) density operator ρ can never (always) be prepared locally. Secondly, we propose the definition of deletion of mixed states. The quantum deleting operation is defined for two initially identical mixed states ρ ⊗ ρ

227

such that the linear operator U acts on the combined Hilbert space of input states and ancilla and deletes one of two copies of ρ together with keeping another copy intact, namely, f

U (ρ ⊗ ρ ⊗ ρA ) = ρ ⊗ ρ0 ⊗ ρA ,

(12)

f

where ρA and ρA are the initial and final states of the f ancilla, ρA may depend on original state ρ generally; ρ0 is the blank state in Hilbert space with the same dimension as the one where ρ lives in. If two input mixed states are different, then the deleting processing can yield an arbitrary state such as U (ρi ⊗ ρj ⊗ ρA ) = ρij A

(i, j = 1, 2, i = j ),

(13)

where ρij A might be any mixed state of the combined input–ancilla system. Now we prove that the linearity of quantum theory prohibits deleting an unknown mixed state against its copy. Suppose that a transformation U for two copies of two arbitrary mixed states ρ1 and ρ2 exists such that (1)

U (ρ1 ⊗ ρ1 ⊗ ρA ) = ρ1 ⊗ ρ0 ⊗ ρA , (2)

U (ρ2 ⊗ ρ2 ⊗ ρA ) = ρ2 ⊗ ρ0 ⊗ ρA ,

(14)

ρA(1) , ρA(2)

where the final states of ancilla may generally depend on ρ1 , ρ2 , respectively. Therefore U also transforms the convex combination of ρ1 and ρ2 , ρη = ηρ1 + (1 − η)ρ2

(0 < η < 1),

(15)

in the same manner as Eqs. (14) [16], i.e., U (ρη ⊗ ρη ⊗ ρA ) (η)

= ρη ⊗ ρ0 ⊗ ρA

(η)

(η)

= ηρ1 ⊗ ρ0 ⊗ ρA + (1 − η)ρ2 ⊗ ρ0 ⊗ ρA ,

(16)

where ρη is arbitrary density matrix for any real η sat(η) isfying (0 < η < 1), ρA may depend on ρη generally. But the linearity of quantum mechanics requires that U (ρη ⊗ ρη ⊗ ρA ) (1)

(2)

= η2 ρ1 ⊗ ρ0 ⊗ ρA + (1 − η)2 ρ2 ⊗ ρ0 ⊗ ρA + η(1 − η)(ρ12A + ρ21A ),

(17)

where ρ12A and ρ21A are two arbitrary mixed states of the combined input–ancilla system according to Eq. (13).

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Since ρA , ρA(1) , ρA(2) generally depends on ρη , ρ1 , ρ2 , respectively, and ρ12A , ρ21A are two arbitrary mixed states, obviously, Eqs. (16) and (17) in general contradict each other, which indicates that the transformation U cannot exist. Therefore, the linearity of quantum theory prohibits deleting an unknown mixed state against its copy. This is known as quantum nodeletion theorem for mixed states. Above proof manifests that this conclusion holds for both entangled mixed state and non-entangled mixed state since Eq. (15) is valid for any kind of mixed state (regardless of their being entangled or separable). Therefore, we can immediately conclude that the linearity of quantum theory prohibits deleting an unknown entangled mixed state against its copy. This is known as quantum no-deletion theorem for entangled mixed states. In conclusion, we demonstrate that the linearity of quantum theory does not allow us to delete an unknown entangled state against its copy in either pure or mixed state case. We call it as quantum no-deletion theorem for entangled states which is the expansion of the no-deleting principle for the pure state of a single qubit [7]. As a fundamental limitation on quantum information stored in entangled states, the theorem may offer security to copies of files in a quantum computer and possibility in both quantum dense coding and quantum cryptographic protocols based on entangled states. Moreover, we expect our results to play a fundamental role in future understanding of quantum information theory. Finally, we should note that we have concentrate on the situation where the two subsystems are provides to the deleting machine together. Recently, Mor has proved that if the subsystems are only available one after the other then there are various cases where

orthogonal states of combined systems cannot be cloned [17]. It is natural to ask whether two (or more) orthogonal states of composite systems can be deleted if the subsystems are only available one after the other. We hope to solve this mystery in the future.

Acknowledgements This work has been financially supported by the National Natural Science Foundation of China under the Grants No. 19734006, 10074072, and the Natural Science Foundation of Shandong Province of China.

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