Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is ... more Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is inherently private. Remarkably, coherent quantum communication is not a prerequisite for privacy: there are quantum channels that are too noisy to transmit any quantum information reliably that can nevertheless send private classical information. Here, we ask how much private classical information a channel can transmit if it has little quantum capacity. We present a class of channels N(d) with input dimension d(2), quantum capacity Q(N(d)) ≤ 1, and private capacity P(N(d)) = log d. These channels asymptotically saturate an interesting inequality P(N) ≤ (1/2)[log d(A) + Q(N)] for any channel N with input dimension d(A) and capture the essence of privacy stripped of the confounding influence of coherence.
We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mec... more We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTCassisted computer is presented with a labeled mixture of states to be distinguished-the most natural formulation-we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.
We show that there exist bipartite quantum states which contain large hidden classical correlatio... more We show that there exist bipartite quantum states which contain large hidden classical correlation that can be unlocked by a disproportionately small amount of classical communication. In particular, there are (2n + 1)-qubit states for which a one bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. States exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.
The ability to perform a universal set of quantum operations based solely on static resources and... more The ability to perform a universal set of quantum operations based solely on static resources and measurements presents us with a strikingly novel viewpoint for thinking about quantum computation and its powers. We consider the two major models for doing quantum computation by measurements that have hitherto appeared in the literature and show that they are conceptually closely related by demonstrating a systematic local mapping between them. This way we effectively unify the two models, showing that they make use of interchangeable primitives. With the tools developed for this mapping, we then construct more resource-effective methods for performing computation within both models and propose schemes for the construction of arbitrary graph states employing two-qubit measurements alone. Z (j) , ∀i ∈ L,
We consider the problem of fault tolerance in the graph-state model of quantum computation. Using... more We consider the problem of fault tolerance in the graph-state model of quantum computation. Using the notion of composable simulations, we provide a simple proof for the existence of an accuracy threshold for graph-state computation by invoking the threshold theorem derived for quantum circuit computation. Lower bounds for the threshold in the graph-state model are then obtained from known bounds in the circuit model under the same noise process.
We provide a simple example that illustrates the advantage of adaptive over non-adaptive strategi... more We provide a simple example that illustrates the advantage of adaptive over non-adaptive strategies for quantum channel discrimination. In particular, we give a pair of entanglementbreaking channels that can be perfectly discriminated by means of an adaptive strategy that requires just two channel evaluations, but for which no non-adaptive strategy can give a perfect discrimination using any finite number of channel evaluations.
We consider interactions as bidirectional channels. We investigate the capacities for interaction... more We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
It is known that the number of different classical messages which can be communicated with a sing... more It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel. Interestingly, our examples are based on the exceptional simple root systems E 7 and E 8 .
We consider a communication method, where the sender encodes n classical bits into 1 qubit and se... more We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called n p Supplementary materials are available on-line at
Traditional quantum error correction involves the redundant encoding of k quantum bits using n qu... more Traditional quantum error correction involves the redundant encoding of k quantum bits using n quantum bits to allow the detection and correction of any t bit error. The smallest general t = 1 code requires n = 5 for k = 1. However, the dominant error process in a physical system is often well known, thus inviting the question: given a specific error model, can more efficient codes be devised? We demonstrate new codes which correct just amplitude damping errors which allow, for example, a t = 1, k = 1 code using effectively n = 4.6. Our scheme is based on using bosonic states of photons in a finite number of optical modes. We present necessary and sufficient conditions for the codes, and describe construction algorithms, physical implementation, and performance bounds.
The existing unconditional security definitions of quantum key distribution (QKD) do not apply to... more The existing unconditional security definitions of quantum key distribution (QKD) do not apply to joint attacks over QKD and the subsequent use of the resulting key. In this paper, we close this potential security gap by using a universal composability theorem for the quantum setting. We first derive a composable security definition for QKD. We then prove that the usual security definition of QKD still implies the composable security definition. Thus, a key produced in any QKD protocol that is unconditionally secure in the usual definition can indeed be safely used, a property of QKD that is hitherto unproven. We propose two other useful sufficient conditions for composability. As a simple application of our result, we show that keys generated by repeated runs of QKD degrade slowly.
We characterize the class of remote state preparation (RSP) protocols that use only forward class... more We characterize the class of remote state preparation (RSP) protocols that use only forward classical communication and entanglement, deterministically prepare an exact copy of a general state, and do so obliviously-without leaking further information about the state to the receiver. We prove that any such protocol can be modified to require from the sender only a single specimen of the state, without increasing the classical communication cost. This implies Lo's conjectured lower bound on the cost for these protocols. We relate our RSP protocols to the private quantum channels and establish a one-to-one correspondence between them.
Abstract: Message authentication is a cryptographic task that allows the receiver to reject a mes... more Abstract: Message authentication is a cryptographic task that allows the receiver to reject a message that is forged or has been tampered with. A protocol to authenticate quantum messages was proposed by Barnum, Crépeau, Gottesman, Tapp, and Smith [1] and stand-alone security was proved. Here, we are concerned with universal composable security (which requires a protocol to be essentially indistinguishable from the ideal functionality and ensures overall security when the protocol is part of any larger scheme). We show that the protocol in [1] is universal ...
In quantum information theory, a capacity Q(χ) of a channel χ is a theoretical maximum of the rat... more In quantum information theory, a capacity Q(χ) of a channel χ is a theoretical maximum of the rate m/n that is achievable by some communication protocol that sends m-qubit information with n uses of the channel, where n tends to infinity. The above definition of Q is defined without auxiliary resources, and additional free classi- cal communication may increase the capacity. We use Q, Q1, QB, and Q2 to denote the quantum capacities of a quantum channel when unassisted, assisted by unlim- ited forward, backward, and two-way classical communi- cation, respectively. It was proved that classical forward communication alone does not increase the quantum ca- pacity of any channel; in other words Q(χ) = Q1(χ) for all channels χ (1). In contrast, Q2 is greater than Q for some channels (1). QB is also known to be greater than Q for some channels (2), but it has been an open question whether QB(χ) = Q2(χ) for all χ. We study the capacities of the quantum erasure chan- nel, which was first int...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1998
... In §2, we show how our techniques to create effective pure states can be understood using the... more ... In §2, we show how our techniques to create effective pure states can be understood using the notion of state labelling: attaching logical, spatial or temporal labels to the quantum state in such a way that a subspace within the mixed state can be identified and used for quantum ...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013
It is known that the maximum classical mutual information, which can be achieved between measurem... more It is known that the maximum classical mutual information, which can be achieved between measurements on pairs of quantum systems, can drastically underestimate the quantum mutual information between them. In this article, we quantify this distinction between classical and quantum information by demonstrating that after removing a logarithmic-sized quantum system from one half of a pair of perfectly correlated bitstrings, even the most sensitive pair of measurements might yield only outcomes essentially independent of each other. This effect is a form of information locking but the definition we use is strictly stronger than those used previously. Moreover, we find that this property is generic, in the sense that it occurs when removing a random subsystem. As such, the effect might be relevant to statistical mechanics or black hole physics. While previous works had always assumed a uniform message, we assume only a min-entropy bound and also explore the effect of entanglement. We find that classical information is strongly locked almost until it can be completely decoded. Finally, we exhibit a quantum key distribution protocol that is 'secure' in the sense of accessible information but in which leakage of even a logarithmic number of bits compromises the secrecy of all others.
We describe a method for non-obliviously communicating a 2l-qubit quantum state by physically tra... more We describe a method for non-obliviously communicating a 2l-qubit quantum state by physically transmitting l+o(l) qubits, and by consuming l ebits of entanglement plus some shared random bits. In the non-oblivious scenario, the sender has a classical description of the state to be communicated. Our method can be used to communicate states that are pure or entangled with the sender's system; l + o(l) and 3l + o(l) shared random bits are sufficient respectively.
We show that there exist bipartite quantum states which contain large hidden classical correlatio... more We show that there exist bipartite quantum states which contain large hidden classical correlation that can be unlocked by a disproportionately small amount of classical communication. In particular, there are (2n + 1)-qubit states for which a one bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. States exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.
We present a scheme for hiding bits in Bell states that is secure even when the sharers Alice and... more We present a scheme for hiding bits in Bell states that is secure even when the sharers Alice and Bob are allowed to carry out local quantum operations and classical communication. We prove that the information that Alice and Bob can gain about a hidden bit is exponentially small in n, the number of qubits in each share, and can be made arbitrarily small for hiding multiple bits. We indicate an alternative efficient low-entanglement method for preparing the shared quantum states. We discuss how our scheme can be implemented using present-day quantum optics.
Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is ... more Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is inherently private. Remarkably, coherent quantum communication is not a prerequisite for privacy: there are quantum channels that are too noisy to transmit any quantum information reliably that can nevertheless send private classical information. Here, we ask how much private classical information a channel can transmit if it has little quantum capacity. We present a class of channels N(d) with input dimension d(2), quantum capacity Q(N(d)) ≤ 1, and private capacity P(N(d)) = log d. These channels asymptotically saturate an interesting inequality P(N) ≤ (1/2)[log d(A) + Q(N)] for any channel N with input dimension d(A) and capture the essence of privacy stripped of the confounding influence of coherence.
We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mec... more We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTCassisted computer is presented with a labeled mixture of states to be distinguished-the most natural formulation-we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.
We show that there exist bipartite quantum states which contain large hidden classical correlatio... more We show that there exist bipartite quantum states which contain large hidden classical correlation that can be unlocked by a disproportionately small amount of classical communication. In particular, there are (2n + 1)-qubit states for which a one bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. States exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.
The ability to perform a universal set of quantum operations based solely on static resources and... more The ability to perform a universal set of quantum operations based solely on static resources and measurements presents us with a strikingly novel viewpoint for thinking about quantum computation and its powers. We consider the two major models for doing quantum computation by measurements that have hitherto appeared in the literature and show that they are conceptually closely related by demonstrating a systematic local mapping between them. This way we effectively unify the two models, showing that they make use of interchangeable primitives. With the tools developed for this mapping, we then construct more resource-effective methods for performing computation within both models and propose schemes for the construction of arbitrary graph states employing two-qubit measurements alone. Z (j) , ∀i ∈ L,
We consider the problem of fault tolerance in the graph-state model of quantum computation. Using... more We consider the problem of fault tolerance in the graph-state model of quantum computation. Using the notion of composable simulations, we provide a simple proof for the existence of an accuracy threshold for graph-state computation by invoking the threshold theorem derived for quantum circuit computation. Lower bounds for the threshold in the graph-state model are then obtained from known bounds in the circuit model under the same noise process.
We provide a simple example that illustrates the advantage of adaptive over non-adaptive strategi... more We provide a simple example that illustrates the advantage of adaptive over non-adaptive strategies for quantum channel discrimination. In particular, we give a pair of entanglementbreaking channels that can be perfectly discriminated by means of an adaptive strategy that requires just two channel evaluations, but for which no non-adaptive strategy can give a perfect discrimination using any finite number of channel evaluations.
We consider interactions as bidirectional channels. We investigate the capacities for interaction... more We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
It is known that the number of different classical messages which can be communicated with a sing... more It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel. Interestingly, our examples are based on the exceptional simple root systems E 7 and E 8 .
We consider a communication method, where the sender encodes n classical bits into 1 qubit and se... more We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called n p Supplementary materials are available on-line at
Traditional quantum error correction involves the redundant encoding of k quantum bits using n qu... more Traditional quantum error correction involves the redundant encoding of k quantum bits using n quantum bits to allow the detection and correction of any t bit error. The smallest general t = 1 code requires n = 5 for k = 1. However, the dominant error process in a physical system is often well known, thus inviting the question: given a specific error model, can more efficient codes be devised? We demonstrate new codes which correct just amplitude damping errors which allow, for example, a t = 1, k = 1 code using effectively n = 4.6. Our scheme is based on using bosonic states of photons in a finite number of optical modes. We present necessary and sufficient conditions for the codes, and describe construction algorithms, physical implementation, and performance bounds.
The existing unconditional security definitions of quantum key distribution (QKD) do not apply to... more The existing unconditional security definitions of quantum key distribution (QKD) do not apply to joint attacks over QKD and the subsequent use of the resulting key. In this paper, we close this potential security gap by using a universal composability theorem for the quantum setting. We first derive a composable security definition for QKD. We then prove that the usual security definition of QKD still implies the composable security definition. Thus, a key produced in any QKD protocol that is unconditionally secure in the usual definition can indeed be safely used, a property of QKD that is hitherto unproven. We propose two other useful sufficient conditions for composability. As a simple application of our result, we show that keys generated by repeated runs of QKD degrade slowly.
We characterize the class of remote state preparation (RSP) protocols that use only forward class... more We characterize the class of remote state preparation (RSP) protocols that use only forward classical communication and entanglement, deterministically prepare an exact copy of a general state, and do so obliviously-without leaking further information about the state to the receiver. We prove that any such protocol can be modified to require from the sender only a single specimen of the state, without increasing the classical communication cost. This implies Lo's conjectured lower bound on the cost for these protocols. We relate our RSP protocols to the private quantum channels and establish a one-to-one correspondence between them.
Abstract: Message authentication is a cryptographic task that allows the receiver to reject a mes... more Abstract: Message authentication is a cryptographic task that allows the receiver to reject a message that is forged or has been tampered with. A protocol to authenticate quantum messages was proposed by Barnum, Crépeau, Gottesman, Tapp, and Smith [1] and stand-alone security was proved. Here, we are concerned with universal composable security (which requires a protocol to be essentially indistinguishable from the ideal functionality and ensures overall security when the protocol is part of any larger scheme). We show that the protocol in [1] is universal ...
In quantum information theory, a capacity Q(χ) of a channel χ is a theoretical maximum of the rat... more In quantum information theory, a capacity Q(χ) of a channel χ is a theoretical maximum of the rate m/n that is achievable by some communication protocol that sends m-qubit information with n uses of the channel, where n tends to infinity. The above definition of Q is defined without auxiliary resources, and additional free classi- cal communication may increase the capacity. We use Q, Q1, QB, and Q2 to denote the quantum capacities of a quantum channel when unassisted, assisted by unlim- ited forward, backward, and two-way classical communi- cation, respectively. It was proved that classical forward communication alone does not increase the quantum ca- pacity of any channel; in other words Q(χ) = Q1(χ) for all channels χ (1). In contrast, Q2 is greater than Q for some channels (1). QB is also known to be greater than Q for some channels (2), but it has been an open question whether QB(χ) = Q2(χ) for all χ. We study the capacities of the quantum erasure chan- nel, which was first int...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1998
... In §2, we show how our techniques to create effective pure states can be understood using the... more ... In §2, we show how our techniques to create effective pure states can be understood using the notion of state labelling: attaching logical, spatial or temporal labels to the quantum state in such a way that a subspace within the mixed state can be identified and used for quantum ...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013
It is known that the maximum classical mutual information, which can be achieved between measurem... more It is known that the maximum classical mutual information, which can be achieved between measurements on pairs of quantum systems, can drastically underestimate the quantum mutual information between them. In this article, we quantify this distinction between classical and quantum information by demonstrating that after removing a logarithmic-sized quantum system from one half of a pair of perfectly correlated bitstrings, even the most sensitive pair of measurements might yield only outcomes essentially independent of each other. This effect is a form of information locking but the definition we use is strictly stronger than those used previously. Moreover, we find that this property is generic, in the sense that it occurs when removing a random subsystem. As such, the effect might be relevant to statistical mechanics or black hole physics. While previous works had always assumed a uniform message, we assume only a min-entropy bound and also explore the effect of entanglement. We find that classical information is strongly locked almost until it can be completely decoded. Finally, we exhibit a quantum key distribution protocol that is 'secure' in the sense of accessible information but in which leakage of even a logarithmic number of bits compromises the secrecy of all others.
We describe a method for non-obliviously communicating a 2l-qubit quantum state by physically tra... more We describe a method for non-obliviously communicating a 2l-qubit quantum state by physically transmitting l+o(l) qubits, and by consuming l ebits of entanglement plus some shared random bits. In the non-oblivious scenario, the sender has a classical description of the state to be communicated. Our method can be used to communicate states that are pure or entangled with the sender's system; l + o(l) and 3l + o(l) shared random bits are sufficient respectively.
We show that there exist bipartite quantum states which contain large hidden classical correlatio... more We show that there exist bipartite quantum states which contain large hidden classical correlation that can be unlocked by a disproportionately small amount of classical communication. In particular, there are (2n + 1)-qubit states for which a one bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. States exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.
We present a scheme for hiding bits in Bell states that is secure even when the sharers Alice and... more We present a scheme for hiding bits in Bell states that is secure even when the sharers Alice and Bob are allowed to carry out local quantum operations and classical communication. We prove that the information that Alice and Bob can gain about a hidden bit is exponentially small in n, the number of qubits in each share, and can be made arbitrarily small for hiding multiple bits. We indicate an alternative efficient low-entanglement method for preparing the shared quantum states. We discuss how our scheme can be implemented using present-day quantum optics.
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Papers by Debbie Leung