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On the minimum quantum dimension for a given quantum correlation Zhaohui Wei (NTU and CQT, Singapore) Joint work with Jamie Sikora and Antonios Varvitsiotis arXiv:1507.00213 We will show a new application of PSD-rank in quantum mechanics We will show a new application of PSD-rank in quantum mechanics PSD-rank has been shown to be related to quantum information. We will show a new application of PSD-rank in quantum mechanics PSD-rank has been shown to be related to quantum information. ◦ Communication to compute a function in expectation *1 *1. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015. We will show a new application of PSD-rank in quantum mechanics PSD-rank has been shown to be related to quantum information. ◦ Communication to compute a function in expectation *1 ◦ Correlation complexity of distribution *2 *1. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015. *2. Jain, Shi, Wei, Zhang, IEEE T INFORM THEORY, 2013. An introduction to quantum information An introduction to device-independent style The target problem and known results Our main result and its proof Applications Further work and open problems Quantum state: A positive semidefinite (psd) matrix with the trace Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., we call pure Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: we call pure . Usually they are huge. Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: we call pure . Usually they are huge. Quantum measurement: A measurement is formed by psd matrices with Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: we call pure . Usually they are huge. Quantum measurement: A measurement is formed by psd matrices with ◦ The outcome is not deterministic: Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: we call pure . Usually they are huge. Quantum measurement: A measurement is formed by psd matrices with ◦ The outcome is not deterministic: ◦ Measurement will disturb the state Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: Quantum measurement: A measurement is formed by psd matrices with ◦ The outcome is not deterministic: ◦ Measurement will disturb the state we call pure . Usually they are huge. Quantum advantages: Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: Quantum measurement: A measurement is formed by psd matrices with ◦ The outcome is not deterministic: ◦ Measurement will disturb the state we call pure . Usually they are huge. Quantum advantages: ◦ More secure: QKD Quantum state: A positive semidefinite (psd) matrix with the trace ◦ Pure state: when is rank 1, i.e., ◦ Composite systems: Quantum measurement: A measurement is formed by psd matrices with ◦ The outcome is not deterministic: ◦ Measurement will disturb the state we call pure . Usually they are huge. Quantum advantages: ◦ More secure: QKD ◦ More efficient: Shor’s algorithm The Bell setting: A source S distribute two physical systems to two separated players Alice and Bob: The Bell setting: A source S distribute two physical systems to two separated players Alice and Bob: ◦ After departure, no communication any more The Bell setting: A source S distribute two physical systems to two separated players Alice and Bob: ◦ After departure, no communication any more ◦ Alice chooses a measurement , and gets outcome The Bell setting: A source S distribute two physical systems to two separated players Alice and Bob: ◦ After departure, no communication any more ◦ Alice chooses a measurement , and gets outcome ◦ Bob has similar and ; the outputs are immediate The Bell setting: A source S distribute two physical systems to two separated players Alice and Bob: ◦ ◦ ◦ ◦ After departure, no communication any more Alice chooses a measurement , and gets outcome Bob has similar and ; the outputs are immediate After repeating many times, they record the statistics , and we call this a correlation – the correlation between the (classical) inputs and the (classical) outputs Classical correlation: Classical correlation: Quantum correlation: Classical correlation: Quantum correlation: A major physical discovery: Classical correlation: Quantum correlation: A major physical discovery: The main idea to achieve this: Bell inequality: Classical correlation: Quantum correlation: A major physical discovery: The main idea to achieve this: Bell inequality: is a convex polytope, thus described by linear inequalities, and each of them is called a Bell inequality Classical correlation: Quantum correlation: A major physical discovery: The main idea to achieve this: Bell inequality: is a convex polytope, thus described by linear inequalities, and each of them is called a Bell inequality ◦ For a same scenario, points in can violate some inequality Classical correlation: Quantum correlation: A major physical discovery: The main idea to achieve this: Bell inequality: is a convex polytope, thus described by linear inequalities, and each of them is called a Bell inequality ◦ For a same scenario, points in can violate some inequality ◦ An example - CHSH inequality: vs. Classical correlation: Quantum correlation: A major physical discovery: The main idea to achieve this: Bell inequality: is a convex polytope, thus described by linear inequalities, and each of them is called a Bell inequality ◦ For a same scenario, points in can violate some inequality ◦ An example - CHSH inequality: vs. They can be verified by physical experiments! Suppose , and . A correlation in this scenario can be expressed as the following block matrix: Suppose , and . A correlation in this scenario can be expressed as the following block matrix: Suppose , and . A correlation in this scenario can be expressed as the following block matrix: with each block being The difficulties in physical realizations: The difficulties in physical realizations: ◦ Measurement disturbs states, cannot clone unknown information: error-correcting is hard, though possible The difficulties in physical realizations: ◦ Measurement disturbs states, cannot clone unknown information: error-correcting is hard, though possible ◦ Quantum states are fragile; memory is short The difficulties in physical realizations: ◦ Measurement disturbs states, cannot clone unknown information: error-correcting is hard, though possible ◦ Quantum states are fragile; memory is short ◦ The accuracy of quantum operations is limited The difficulties in physical realizations: ◦ Measurement disturbs states, cannot clone unknown information: error-correcting is hard, though possible ◦ Quantum states are fragile; memory is short ◦ The accuracy of quantum operations is limited Quantum control is hard. Especially, The difficulties in physical realizations: ◦ Measurement disturbs states, cannot clone unknown information: error-correcting is hard, though possible ◦ Quantum states are fragile; memory is short ◦ The accuracy of quantum operations is limited Quantum control is hard. Especially, ◦ The internal working of a quantum device is hard to monitor Question: Can we know nontrivial internal properties of a quantum system using very limited classical in-out? Question: Can we know nontrivial internal properties of a quantum system using very limited classical in-out? Challenge1: The cost to describe a quantum system classically is huge ◦ exponential Question: Can we know nontrivial internal properties of a quantum system using very limited classical in-out? Challenge1: The cost to describe a quantum system classically is huge ◦ exponential Challenge2: One single measurement only reveals very limited information of the system ◦ Typically quantum state collapses Question: Can we know nontrivial internal properties of a quantum system using very limited classical in-out? Challenge1: The cost to describe a quantum system classically is huge ◦ exponential Challenge2: One single measurement only reveals very limited information of the system ◦ Typically quantum state collapses The answer is YES: the idea of Device-independent (DI) How to understand? How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically The values of DI: How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically The values of DI: ◦ Theoretical values on its own How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically The values of DI: ◦ Theoretical values on its own ◦ A huge convenience for quantum tasks How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically The values of DI: ◦ Theoretical values on its own ◦ A huge convenience for quantum tasks Known applications of DI: How to understand? ◦ It is helpful to think the quantum system as a quantum box. The correctness of DI comes from quantum mechanics, and has nothing to do with how to realize the box specifically The values of DI: ◦ Theoretical values on its own ◦ A huge convenience for quantum tasks Known applications of DI: ◦ quantum key distribution(QKD) ◦ Entropy ◦ Entanglement Task: Two separated quantum players want to prepare and share a secret classical key Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: *1. Artur Ekert, Phys. Rev. Lett, 1991. Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: ◦ They share a lot of EPR pairs *1. Artur Ekert, Phys. Rev. Lett, 1991. Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: ◦ They share a lot of EPR pairs ◦ The choose the following binary POVMs randomly to measure the qubits they have: *1. Artur Ekert, Phys. Rev. Lett, 1991. Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: ◦ They share a lot of EPR pairs ◦ The choose the following binary POVMs randomly to measure the qubits they have: Alice *1. Artur Ekert, Phys. Rev. Lett, 1991. Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: ◦ They share a lot of EPR pairs ◦ The choose the following binary POVMs randomly to measure the qubits they have: Alice Bob *1. Artur Ekert, Phys. Rev. Lett, 1991. Task: Two separated quantum players want to prepare and share a secret classical key ◦ They have quantum and classical channels, but unsafe A DI flavor scheme*1: ◦ They share a lot of EPR pairs ◦ The choose the following binary POVMs randomly to measure the qubits they have: Alice Bob ◦ After all measurements, they announce the choices and outcomes *1. Artur Ekert, Phys. Rev. Lett, 1991. The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The secrete key: If the shared states are EPR, the first group of POVMs always give the same random outcomes The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The secrete key: If the shared states are EPR, the first group of POVMs always give the same random outcomes Advantage: The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The secrete key: If the shared states are EPR, the first group of POVMs always give the same random outcomes Advantage: ◦ It works even if the channel and the EPRs are prepared by enemy The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The secrete key: If the shared states are EPR, the first group of POVMs always give the same random outcomes Advantage: ◦ It works even if the channel and the EPRs are prepared by enemy ◦ Do not have to check the internal working of quantum devices The main idea: the second group of POVMs are used to calculate the value of the CHSH inequality: ◦ If the value is , the shared states must be EPR ◦ Eve cannot be entangled to any qubit of EPR ◦ If the value is smaller than , start it over The secrete key: If the shared states are EPR, the first group of POVMs always give the same random outcomes Advantage: ◦ It works even if the channel and the EPRs are prepared by enemy ◦ Do not have to check the internal working of quantum devices ◦ The base of realistic DI QKD Question: What is the optimum size of quantum state needed to generate a given quantum correlation? Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem For a given , if there exists a quantum state on , POVMs and s.t. then admits a d-dimensional representation. We denote by the minimum such d. Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem For a given , if there exists a quantum state on , POVMs and s.t. then admits a d-dimensional representation. We denote by the minimum such d. For an arbitrarily given , , how to estimate ? Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem For a given , if there exists a quantum state on , POVMs and s.t. then admits a d-dimensional representation. We denote by the minimum such d. For an arbitrarily given , , how to estimate ◦ Dimension is a most fundamental quantum property ? Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem For a given , if there exists a quantum state on , POVMs and s.t. then admits a d-dimensional representation. We denote by the minimum such d. For an arbitrarily given , , how to estimate ◦ Dimension is a most fundamental quantum property ◦ Dimension is a kind of computational resource ? Question: What is the optimum size of quantum state needed to generate a given quantum correlation? ◦ Size means dimension - a DI style problem For a given , if there exists a quantum state on , POVMs and s.t. then admits a d-dimensional representation. We denote by the minimum such d. For an arbitrarily given , , how to estimate ◦ Dimension is a most fundamental quantum property ◦ Dimension is a kind of computational resource ? For a fixed Bell scenario, the set of all quantum correlations is convex. However, if restricting dimension, it is usually not. For a fixed Bell scenario, the set of all quantum correlations is convex. However, if restricting dimension, it is usually not. By allowing free classical correlations, the set of quantum correlations for a given dimension becomes convex: For a fixed Bell scenario, the set of all quantum correlations is convex. However, if restricting dimension, it is usually not. By allowing free classical correlations, the set of quantum correlations for a given dimension becomes convex: ◦ The setting is changed a little bit For a fixed Bell scenario, the set of all quantum correlations is convex. However, if restricting dimension, it is usually not. By allowing free classical correlations, the set of quantum correlations for a given dimension becomes convex: ◦ The setting is changed a little bit ◦ The standard convex analysis method applies For a fixed Bell scenario, the set of all quantum correlations is convex. However, if restricting dimension, it is usually not. By allowing free classical correlations, the set of quantum correlations for a given dimension becomes convex: ◦ The setting is changed a little bit ◦ The standard convex analysis method applies The main known method: dimension witness ◦ Others: entropy method A d-dimensional witness is a linear function of the correlation described by a vector s.t. is valid for all correlations admitting d-dimensional representation, and can be violated by some others*1 *1. Brunner et al., Phys. Rev. Lett, 2008. A d-dimensional witness is a linear function of the correlation described by a vector s.t. is valid for all correlations admitting d-dimensional representation, and can be violated by some others*1 ◦ If violation happens, then the dimension must be larger than d *1. Brunner et al., Phys. Rev. Lett, 2008. A d-dimensional witness is a linear function of the correlation described by a vector s.t. is valid for all correlations admitting d-dimensional representation, and can be violated by some others*1 ◦ If violation happens, then the dimension must be larger than d ◦ For different Bell scenarios, the values of ‘s are different *1. Brunner et al., Phys. Rev. Lett, 2008. Cannot handle the case without public randomness Cannot handle the case without public randomness Even for the case with classical correlation, it is not a direct function or bound of quantum correlations. Cannot handle the case without public randomness Even for the case with classical correlation, it is not a direct function or bound of quantum correlations. The quantum region for a fixed quantum dimension is very hard to characterize: Cannot handle the case without public randomness Even for the case with classical correlation, it is not a direct function or bound of quantum correlations. The quantum region for a fixed quantum dimension is very hard to characterize: ◦ The values ‘s are hard to compute Cannot handle the case without public randomness Even for the case with classical correlation, it is not a direct function or bound of quantum correlations. The quantum region for a fixed quantum dimension is very hard to characterize: ◦ The values ‘s are hard to compute ◦ Dimension witnesses were found only on some very small quantum systems We go back to the initial setting, i.e., no free shared randomness is allowed, thus not convex any more. We go back to the initial setting, i.e., no free shared randomness is allowed, thus not convex any more. ◦ The convex analysis approach fails We go back to the initial setting, i.e., no free shared randomness is allowed, thus not convex any more. ◦ The convex analysis approach fails ◦ The idea of PSD-factorization plays the key role We go back to the initial setting, i.e., no free shared randomness is allowed, thus not convex any more. ◦ The convex analysis approach fails ◦ The idea of PSD-factorization plays the key role We provide an easy-to-compute lower bound for , which is composed by simple functions of the entries of : Recall that a correlation is a block matrix: Let be a nonnegative matrix, then a PSD factorization of size is given by two sets of PSD matrices and satisfying that *1*2 *1. Gouveia, Parrilo, Thomas, Math. Oper. Res., 2013 *2. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015. Let be a nonnegative matrix, then a PSD factorization of size is given by two sets of PSD matrices and satisfying that *1*2 The PSD-rank of smallest integer of size . , denoted by , is the such that has a PSD factorization *1. Gouveia, Parrilo, Thomas, Math. Oper. Res., 2013 *2. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015. Proof: Proof: ◦ Suppose it is generated by , and Proof: ◦ Suppose it is generated by ◦ Purify the state onto , and with Proof: ◦ Suppose it is generated by ◦ Purify the state onto ◦ Set and , and with Proof: ◦ ◦ ◦ ◦ Suppose it is generated by Purify the state onto Set and Choose , and with Proof: ◦ ◦ ◦ ◦ Suppose it is generated by Purify the state onto Set and Choose , and with Proof: ◦ ◦ ◦ ◦ Suppose it is generated by Purify the state onto Set and Choose ◦ Then , and with A new characterization for quantum correlations*1 *1. Sikora, Varvitsiotis, 2015. A new characterization for quantum correlations*1 *1. Sikora, Varvitsiotis, 2015. A new characterization for quantum correlations*1 Thus, is the minimum d to make the above conditions satisfied *1. Sikora, Varvitsiotis, 2015. Normalizing the columns of a nonnegative matrix does not change its PSD-rank Normalizing the columns of a nonnegative matrix does not change its PSD-rank Let be a nonnegative column summing to 1, if exists a PSD factorization each has trace 1 and all *1. Lee, Wei, de Wolf, 2014. matrix with each , then there such that sum to identity*1 Normalizing the columns of a nonnegative matrix does not change its PSD-rank Let be a nonnegative column summing to 1, if exists a PSD factorization each has trace 1 and all matrix with each , then there such that sum to identity*1 ◦ Each column corresponds the outcome probability distribution of one quantum state under the same POVM *1. Lee, Wei, de Wolf, 2014. Measurement increase the fidelity: Measurement increase the fidelity: Measurement increase the fidelity: Measurement increase the fidelity: This is valuable to DI Suppose and , and matrices with size satisfy Suppose and , and matrices with size satisfy Wlog we let the summation above be full rank, then there exists an invertible matrix such that Suppose and , and matrices with size satisfy Wlog we let the summation above be full rank, then there exists an invertible matrix such that Then for any choice of valid POVM. , is a Let factor such that , where is a proper is a valid quantum state. Let factor such that , where is a proper is a valid quantum state. Then we have that for any fixed , distribution of the outcome the POVM of . , which means is the probability when is measured by Let factor such that , where is a proper is a valid quantum state. Then we have that for any fixed , distribution of the outcome the POVM of . , which means is the probability when is measured by Note that the measurement does not decrease the fidelity, thus the following is valid for any Recall that that for all quantum state , then we have the fact . Thus for all , we have a valid . Recall that that for all quantum state , then we have the fact . Thus for all , we have a valid . Note that . Since we have that for any , , then it holds that is independent in , . Recall that that for all quantum state , then we have the fact . Thus for all , we have a valid . Note that . Since we have that for any , Lastly, note that that , then it holds that is independent in , . is a quantum state on , we have This means that This means that This means that Thus This means that Thus Notice the fact that , then This means that Thus Notice the fact that implying that , then This means that Thus Notice the fact that implying that Recall that for any , then Combining the last two together, we have that Combining the last two together, we have that Since this is valid for any , it holds that If a correlation matrix satisfies the normalization condition, can it always be generated physically? If a correlation matrix satisfies the normalization condition, can it always be generated physically? Answer: no If a correlation matrix satisfies the normalization condition, can it always be generated physically? Answer: no The correlation cannot violate relativity: If a correlation matrix satisfies the normalization condition, can it always be generated physically? Answer: no The correlation cannot violate relativity: and are well-defined, i.e., non-signaling Non-signaling polytope: the set of all correlations that respect the non-signaling rule. A PR-box is a nonlocal vertex of this polytope Non-signaling polytope: the set of all correlations that respect the non-signaling rule. A PR-box is a nonlocal vertex of this polytope ◦ Question: can it be quantum? No Non-signaling polytope: the set of all correlations that respect the non-signaling rule. A PR-box is a nonlocal vertex of this polytope ◦ Question: can it be quantum? No Set and PR box can be given by Then a Non-signaling polytope: the set of all correlations that respect the non-signaling rule. A PR-box is a nonlocal vertex of this polytope ◦ Question: can it be quantum? No Set and PR box can be given by Then a The lower bound is infinite, meaning that any finitedimensional quantum system cannot generate it A magic square is a odd column sums Boolean matrix with even row sums and A magic square is a odd column sums ◦ No such a square exists Boolean matrix with even row sums and A magic square is a odd column sums Boolean matrix with even row sums and ◦ No such a square exists Consider the following game: Alice and Bob each gets an input corresponding to the index of a row and a column, they are required to give the row and the column themselves, satisfying the above condition (no communication is allowed) ◦ A Bell setting: A magic square is a odd column sums Boolean matrix with even row sums and ◦ No such a square exists Consider the following game: Alice and Bob each gets an input corresponding to the index of a row and a column, they are required to give the row and the column themselves, satisfying the above condition (no communication is allowed) ◦ A Bell setting: No classical scheme can win for sure A magic square is a odd column sums Boolean matrix with even row sums and ◦ No such a square exists Consider the following game: Alice and Bob each gets an input corresponding to the index of a row and a column, they are required to give the row and the column themselves, satisfying the above condition (no communication is allowed) ◦ A Bell setting: No classical scheme can win for sure ◦ The best winning probability is 8/9 A magic square is a odd column sums Boolean matrix with even row sums and ◦ No such a square exists Consider the following game: Alice and Bob each gets an input corresponding to the index of a row and a column, they are required to give the row and the column themselves, satisfying the above condition (no communication is allowed) ◦ A Bell setting: No classical scheme can win for sure ◦ The best winning probability is 8/9 Surprisingly, this game can be wined for sure quantumly ◦ Because of stronger correlations quantum provides A choice of the state is A choice of the state is and by choosing proper POVMs the correlation can be given as A choice of the state is and by choosing proper POVMs the correlation can be given as The lower bound is which is tight If , i.e., each player only has one POVM, the new lower bound goes back to a lower bound for the PSD-rank If , i.e., each player only has one POVM, the new lower bound goes back to a lower bound for the PSD-rank ◦ Has been given: Lee, Wei, de Wolf, 2014. This algebraic approach still works: This algebraic approach still works: ◦ Though classical correlation is free, the combination needs more than one quantum state This algebraic approach still works: ◦ Though classical correlation is free, the combination needs more than one quantum state Suppose the target quantum correlation is generated by mixing settings with quantum dimensions , then the sum of them is lower bounded by the result We have given a lower bound for We have given a lower bound for ◦ The bound is easy to compute We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations How to improve this? We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations How to improve this? ◦ It can be loose for some cases We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations How to improve this? ◦ It can be loose for some cases Similar lower bound for classical correlations? We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations How to improve this? ◦ It can be loose for some cases Similar lower bound for classical correlations? ◦ The gap will be interesting: one POVM case is known We have given a lower bound for ◦ The bound is easy to compute ◦ The bound is tight on a lot of famous examples ◦ Composed by simple functions, thus robust to perturbations How to improve this? ◦ It can be loose for some cases Similar lower bound for classical correlations? ◦ The gap will be interesting: one POVM case is known Other physical applications of the PSD-factorization idea? ◦ Works for the prepare-and-measure setting Thank you