Entanglement between Two Uses of a Noisy Multipartite Quantum Channel Enables Perfect Transmission of Classical Information
Suppose that senders want to transmit classical information to receivers with zero probability of error using a noisy multipartite communication channel. The senders are allowed to exchange classical, but not quantum, messages among themselves, and the same holds for the receivers. If the channel is classical, a single use can transmit information if and only if multiple uses can. In sharp contrast, we exhibit, for each and with or , a quantum channel of which a single use is not able to transmit information yet two uses can. This latter property requires and is enabled by quantum entanglement.
The maximum rate at which a communication channel carries information is characterized by a quantity called its channel capacity. Different constraints on the channel give rise to variants of channel capacity. For example, the most commonly studied capacity, referred to simply as the Shannon capacity, can be attained using encoding and decoding on a large number of uses of the channel, with the error probability approaching .
In 1956 Shannon introduced the notion of zero-error capacity to characterize the ability of noisy channels to transmit classical information with zero probability of error SHA56 (). The study of this notion and the related topics has since then grown into a vast field called zero-error information theory KO98 (), partly motivated by the fact that no error may be allowed in some applications, or the channel is not available for the large number of uses required for attaining a small error probability. Unlike Shannon capacity, the calculation of zero-error capacity is in essence a combinatorial optimization problem on graphs and may be extremely difficult even for very simple graphs. This difficulty has in turn stimulated a great deal of research in graph theory (e.g. LOV79 ()), a recent highlight of which is the resolution of the Strong Perfect Graph Conjecture CRST06 ().
Since the physical process underlying all communication channels, such as optical fibers, is quantum mechanical, a quantum information theory is necessary to capture the full potential of communication channels. While much progress has been made on understanding the quantum analogy of Shannon capacity, little is known about quantum zero-error capacity for transmitting classical information. Some basic facts on the latter subject were observed in Ref. MA05 (), while it was shown that the capacity is in general also extremely difficult to compute BS07 ().
The purpose of this Letter is to demonstrate that quantum zero-error capacity behaves dramatically different from the corresponding classical capacity, and the difference is due to the effect of quantum entanglement. We consider the following “multi-user” scenario: a set of senders want to send classical information with zero probability of error to a set of receivers through a noisy channel . We impose the following LOCC (Local Operations and Classical Communication) requirement: The senders are allowed to exchange classical, but not quantum, messages, and the same holds for the receivers. Note that if is a classical channel, the LOCC restriction does not reduce the capacity. However, when is a quantum channel, the LOCC requirement may reduce the capacity. Multi-user channels in which no communication is allowed within senders and receivers have been studied widely in classical information theory (see, e.g., Chapter 14 of Ref. CT91 ()). Our definition of multi-user channels is a natural extension of such channels, and captures realistic settings where quantum communication is expensive. Multipartite quantum communication was studied in Ref. DCH04 (). The scenario considered there differs from ours in several important aspects: in their model, (a) the quantum channel may be assisted with one- or two-way classical channels; (b) their purpose was to study the additivity of the channel capacity for transmitting quantum information with vanishing error probability, while our focus is on the capacity for transmitting classical information with zero-error probability.
We now describe our main result. When is classical, it is straightforward to see that its capacity is if and only if one use of cannot transmit information. In sharp contrast, we show that this is not true for quantum channels in general. In particular, we construct a quantum channel , for each pair of and , or , that one use of is not able to transmit information yet two uses can. The later property can be achieved in two different ways: (1) The senders apply to create a maximally entangled state between the receivers. The receivers then distinguish the output states of the second use of by teleportation. (2) The senders locally prepare maximally entangled states between the two uses of . The effect of the second case cannot be observed under the assumptions of Refs. MA05 (); BS07 () where only product input states between two uses are allowed. Fig. 1 demonstrates our construction for and .
Our construction in Case (2) uses the notion of completely entangled subspace, which has been studied by several authors recently PAR04 (); HLW06 (); DFJY07 (); CMW07 (); WS07 (); GW07 (). More precisely, for any , we construct a partition of an -partite state space into two orthogonal subspaces, each of which contains no nonzero product state. Such partitions were found in Refs. DY07 (); CHL+08 (), and can be used to construct counterexamples to the additivity of minimum output -Rnyi entropy for close to CHL+08 (). Unfortunately, previous partitions are not sufficient for our purpose.
We define some notions necessary for describing our constructions. The set of -bit binary strings is denoted by . Denote the complement of by . We associate each Hilbert space a fixed orthonormal basis, referred to as the computational basis and is usually denoted by or when . The operator space on is denoted by . If , denote by its transpose with respect to the computational basis. Denote the application of an operator on the ’th component of a multipartite system by .
Let be positive integers. Let be a set of senders and be a set of receivers. Their state spaces are , , and , , respectively. Denote by and . An multi-user quantum channel is a completely positive trace-preserving map from to , and is used as follows. The senders start with , and encode a message into a state through an LOCC protocol. The receivers receive , and decode the message by LOCC.
Define to be the maximum integer with which there exist a set of states such that: (a) Each can be locally prepared by the senders, and (b) can be perfectly distinguished by the receivers using LOCC. It follows from the linearity of superoperators that a set achieving can be assumed without loss of generality to be orthogonal product pure states. Intuitively, one use of can be used to transmit bits of classical information perfectly. When it is clear that by a single use of the senders cannot transmit any classical information to the receivers perfectly.
The local zero-error classical capacity of , , is defined as follows:
Suppose that is classical (a so-called memoryless stationary channel), that is, for some states diagonalized under the computational basis . Then if and only if for all pairs of and , . Thus if and only if for any . Therefore, if and only if . Our main theorem is:
For any and with or , there exists a multi-user channel from senders to receivers such that and .
Proof. For any positive integers , an channel can be extended to an channel , which ignores the inputs from the additional senders and provides to all the additional receivers. Then for any , . Thus we need only to prove the theorem for and .
Let and , where . Consider the following channel , which is from qubit to qubits:
The only output pairs that are orthogonal are and . However, they cannot be distinguished by an LOCC protocol. Thus . On the other hand, when is used twice, the two receivers, Alice and Bob, can distinguish the output states by the following LOCC protocol: Alice first teleports her second qubit to Bob using the first qubit (which is maximally entangled with the first qubit of Bob), then Bob applies a local measurement to distinguish and . Thus it follows from Eq. (1) that .
Now we turn to the construction of a channel. The input space , and the output space is . Each of and is a dimensional space and is a qubit. Let be the subspace spanned by the following vectors:
and . Let be the projector onto , . The channel is defined as:
Let . The projections and have the following useful properties:
and , for any and . As a consequence, . (Note that ).
Neither nor contains a product state, i.e. both and are completely entangled.
Property (i) can be verified by inspection. We now prove Property (ii). Let be a product vector orthogonal to , . Then
Suppose that . Assume without loss of generality that . Then
By and we have . Substituting and into we have . If then we have , which together with implies . However, this is impossible as we have . Thus . This together with implies . However, we have already shown that . Thus , again a contradiction. Therefore . Note that if and for a nonzero constant , , then . Applying this inference rule many times, one concludes that all , in both and cases. Thus , and contains no product state. By Property (i), this implies that does not contain a product state, either.
It follows from Property (ii) that is a mixed state for any product state . Thus . We now consider using twice. Let . Define and as follows:
where . Note that for any operators and , . Thus for any , applying Property (i), we have
Thus and are orthogonal. This can also be verified by the following facts:
Therefore , and .
The quantum channel constructed above has the following desirable property: when it is used twice, each sender is able to transmit one classical bit without leaking any information about this bit to the other sender. This is based on the fact that on and , the channel outputs the same state orthogonal to the output on . While tensoring with trivial channels does not preserve this property, we are able to construct a family of channels , , that have this property. Each is a channel from qubits to qubit defined in analogy to with the following set of base vectors for ():
The proof for and is similar to that for , thus we leave it to the interested reader.
We note that if the “privacy” property is not required, the input dimension of can be reduced to with the following set of base vectors for :
Now only Bob (the party with the dimensional state space) can transmit a private bit with two uses of the channel. This is simply due to the fact that both Properties (i) and (ii) still hold with the restriction that in Property (i). We omit the details of the proof as it is similar to that for .
Our construction of and the above variant has another application on the additivity of the minimum output -Rnyi entropy . For and a quantum channel , is defined as
where is a density operator, and at , the right hand side takes the limit. The additivity problem on asks if
The case of is a central open problem in quantum information theory Shor04 (). This motivates the study of the same question for other values of by many authors. It has been shown that is not additive for any HW02 (). Very recently Ref. CHL+08 () showed that is not additive for in a neighborhood of , by constructing a counterexample for . This is done by first constructing two completely entangled subspaces and of a bipartite space so that is not completely entangled. Two channels and are then defined based on and , respectively, through the Choi-Jamiołkowski isomorphism. It remains open if is additive when . We answer this question negatively: setting both and to be the in (and its variant) gives a channel so that . We omit the details of the argument as it is similar to that in Ref. CHL+08 ().
In conclusion, we have shown that for a broad class of multi-user quantum channels, of which the communication within the senders and receivers is restricted to be LOCC, a single use of the channel cannot be used to transmit classical information with zero probability of error, while multiple uses can. The latter property requires, and is a consequence of, quantum entanglement between different uses, thus cannot be achieved by classical channels.
For all the channels we know, the LOCC restriction on encoding and decoding is necessary. It remains an intriguing open problem if such channels exist for a single sender and a single receiver. We do not know the exact values of for the constructed channels, despite the simplicity of their definitions. The difficulty in computing them is not unexpected given that generalizes the classical concept of zero-error capacity, which can be notoriously difficult to compute even for simple channels. Developing methods for estimating is thus of great theoretical interest, besides its obvious practical usefulness.
We are grateful to Andreas Winter for very useful discussions at AQIS2007 and for sending us a copy of Ref. CHL+08 () prior to publication. This work was partially supported by the National Science Foundation of the United States under Awards 0347078 and 0622033. R. Duan was also partially supported by the National Natural Science Foundation of China (Grant Nos. 60702080, 60736011, 60503001, and 60621062) and the Hi-Tech Research and Development Program of China (863 project) (Grant No. 2006AA01Z102).
- (1) C. E. Shannon, IRE Trans. Inf. Theory 2, 8 (1956).
- (2) J. Krner and A. Orlitsky, IEEE Trans. Inf. Theory 44, 2207 (1998).
- (3) L. Lovsz, IEEE Trans. Inf. Theory 29, 1 (1979); W. Haemers, IEEE Trans. Inf. Theory 25, 231 (1979); N. Alon, Combinatorica 18, 301 (1998); N. Alon and E. Lubetzky, IEEE Trans. Inf. Theory 52, 2172 (2006).
- (4) M. Chudnovsky, N. Robertson, P. D. Seymour, and R. Thomas, Ann. Math. 164, 51 (2006).
- (5) R. A. C. Medeiros and F. M. de Assis, Int. J. Quant. Inf. 3, 135 (2005); R. A. C. Medeiros, R. Alleaume, G. Cohen, and F. M. de Assis, quant-ph/0611042.
- (6) S. Beigi and P. W. Shor, arXiv:0709.2090 [quant-ph] (2007).
- (7) T. Cover and J. Thomas, Elements of Information Theory, John Wiley&Sons (1991).
- (8) W. Dr, J. I. Cirac, and P. Horodecki, Phys. Rev. Lett. 93, 020503 (2004).
- (9) K. R. Parthasarathy, Proc. Indian Acad. Sci. 114, 365 (2004).
- (10) P. Hayden, D. Leung, and A. Winter, Comm. Math. Phys. 265, 95 (2006).
- (11) R. Y. Duan, Y. Feng, Z. F. Ji, and M. S. Ying, Phys. Rev. Lett. 98, 230502 (2007).
- (12) T. S. Cubitt, A. Montanaro, and A. Winter, arXiv: 0706.0705 [quant-ph] (2007).
- (13) J. Walgate and A. J. Scott, arXiv: 0709.4238 [quant-ph] (2007).
- (14) G. Gour and N. R. Wallach, Phys. Rev. A 76, 042309 (2007).
- (15) R. Y. Duan and M. S. Ying, arXiv: 0708.3559 [quant-ph] (2007).
- (16) T. Cubitt, A. Harrow, D. Leung, A. Montanaro, and A. Winter, Counterexamples to additivity of minimum output -Rnyi entropy for close to , QIP 2008, arXiv: 0712.3628 [quant-ph] (2007).
- (17) P. W. Shor, Comm. Math. Phys. 246, 453 (2004).
- (18) A. S. Holevo and R. F. Werner, J. Math. Phys. 43, 4353 (2002); A. Winter, arXiv: 0707.0402 [quant-ph] (2007); P. Hayden, arXiv: 0707.3291 [quant-ph] (2007).